{ "name": "Quantum Information and Quantum Computation Open Problem Zoo", "shortName": "QIQCOP Zoo", "siteUrl": "https://qiqc-op.com/", "repositoryUrl": "https://github.com/Naixu-Guo/quantum-open-problems", "generated": "2026-09-24", "updated": "2026-09-24", "counts": { "unit": "records", "distinctQuestions": { "total": 212, "unsolved": 185, "solved": 27 }, "total": 212, "unsolved": 185, "solved": 27, "fields": 6, "topics": 62, "references": 851, "equations": 793 }, "problems": [ { "id": "op_89fb664ba06ba5de", "ulid": "01M1Q787QRHF57Y6BJ3D34S0H6", "aliases": [ "op_89fb664ba06ba5de", "01M1Q787QRHF57Y6BJ3D34S0H6", "op-89fb664ba06ba5de" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 1, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-04T22:04:59Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "source-stated", "posed": null, "areaIds": [ "quantum-communication" ], "topicIds": [ "bosonic-channels", "classical-capacity", "one-shot-and-finite-blocklength-bounds" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [] }, "title": "Candidate pure-loss second-order converse", "status": "Solved", "fields": [ "Quantum Communication" ], "topics": [ "Bosonic channels", "Classical capacity", "One-shot and finite-blocklength bounds" ], "tags": [ "Quantum Communication", "Bosonic channels", "Classical capacity", "One-shot and finite-blocklength bounds" ], "statement": "Does the pure-loss bosonic channel admit the following candidate second-order classical converse under a maximum-photon-number occupation constraint? Let $\\mathcal N_\\eta$ be the single-mode pure-loss channel with transmissivity $0<\\eta<1$, defined in the Heisenberg picture by\n\n \\begin{equation}\n \\hat b=\\sqrt{\\eta}\\,\\hat a+\\sqrt{1-\\eta}\\,\\hat e,\n\\tag{1}\n\\end{equation} \nwhere the environment mode $E$ is in the vacuum. In an $n$-use code, the channel in Eq. (1) is used to transmit one of $M$ input states $\\rho_m^{A^n}$, with a decoding POVM $\\{\\Lambda_m^{B^n}\\}_{m=1}^M$. Write $\\overline\\rho_{A^n}:=M^{-1}\\sum_m\\rho_m^{A^n}$ and let $\\Pi_{\\lceil nN_S\\rceil}$ project onto the $n$-mode subspace of total photon number at most $\\lceil nN_S\\rceil$. For fixed $N_S>0$, $\\varepsilon\\in(0,1)$, and $c>0$, impose\n\n \\begin{equation}\n \\frac1M\\sum_{m=1}^M\n \\operatorname{Tr}\\!\\left[\n \\Lambda_m\\mathcal N_\\eta^{\\otimes n}(\\rho_m)\n \\right]\n \\geq1-\\varepsilon,\n \\qquad\n \\operatorname{Tr}\\!\\left[\n \\Pi_{\\lceil nN_S\\rceil}\\overline\\rho_{A^n}\n \\right]\n \\geq1-\\delta_n,\n \\qquad\n 0\\leq\\delta_n\\leq2^{-cn}.\n\\tag{2}\n\\end{equation} \nLet $M^*_{\\rm occ}(n,\\eta,N_S,\\varepsilon,c)$ be the largest $M$ satisfying Eq. (2). Define the thermal entropy and its entropy variance by\n\n \\begin{equation}\n g(x):=(x+1)\\log_2(x+1)-x\\log_2x,\n \\qquad\n v(x):=x(x+1)\n \\left[\\log_2(x+1)-\\log_2x\\right]^2,\n\\tag{3}\n\\end{equation} \nwhere $0\\log_2 0:=0$. With the functions in Eq. (3), is the following upper bound valid as $n\\to\\infty$?\n\n \\begin{equation}\n \\log_2 M^*_{\\rm occ}(n,\\eta,N_S,\\varepsilon,c)\n \\leq ng(\\eta N_S)\n +\\sqrt{n\\,v(\\eta N_S)}\\,\\Phi^{-1}(\\varepsilon)\n +O(\\log n),\n\\tag{4}\n\\end{equation} \nHere $\\Phi^{-1}$ in Eq. (4) is the inverse standard-normal cumulative distribution function, and the implicit constant may depend on $\\eta,N_S,\\varepsilon,$ and $c$, but not on $n$.", "url": "https://qiqc-op.com/problem/op_89fb664ba06ba5de/", "json": "https://qiqc-op.com/api/problems/op_89fb664ba06ba5de.json", "tex": "https://qiqc-op.com/problem/op_89fb664ba06ba5de/op_89fb664ba06ba5de.tex", "created": "2026-09-03", "updated": "2026-09-24", "createdAt": "2026-09-03T02:22:57.000Z", "updatedAt": "2026-09-24T04:27:44.000Z", "sha256": "dd286abb24d07b0a4d11dbb543bef94c3ca90b0073673f1e207b93eaacaef33c" }, { "id": "op_3b3bfcda365a83a9", "ulid": "01M2M9FBG3PKXNFV5BMDWQYVTR", "aliases": [ "op_3b3bfcda365a83a9", "01M2M9FBG3PKXNFV5BMDWQYVTR", "op-3b3bfcda365a83a9" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 2, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-16T05:01:47.907Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "source-stated", "posed": null, "areaIds": [ "quantum-algorithm" ], "topicIds": [ "quantum-circuit-complexity" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [] }, "title": "Linear-size exact quantum Fourier transform", "status": "Unsolved", "fields": [ "Quantum algorithm" ], "topics": [ "Quantum circuit complexity" ], "tags": [ "Quantum algorithm", "Quantum circuit complexity" ], "statement": "Can the exact quantum Fourier transform on $n$ qubits be implemented with $O(n)$ one- and two-qubit gates?\n\nDefine\n\n \\begin{equation}\nF_{2^n}|x\\rangle=2^{-n/2}\\sum_{y=0}^{2^n-1}e^{2\\pi ixy/2^n}|y\\rangle.\n\\tag{1}\n\\end{equation} \nLet $C_{\\mathrm{exact}}(n)$ be the minimum size of a uniform circuit that implements the unitary in Eq. (1) exactly on arbitrary inputs. Gates may be arbitrary efficiently specified one- or two-qubit unitaries; clean ancillas may be used but must be restored, and all gates on them count. Is\n\n \\begin{equation}\nC_{\\mathrm{exact}}(n)=O(n)?\n\\tag{2}\n\\end{equation} \nThe target in Eq. (2) concerns exact coherent implementation, not approximate QFT or sampling only.", "url": "https://qiqc-op.com/problem/op_3b3bfcda365a83a9/", "json": "https://qiqc-op.com/api/problems/op_3b3bfcda365a83a9.json", "tex": "https://qiqc-op.com/problem/op_3b3bfcda365a83a9/op_3b3bfcda365a83a9.tex", "created": "2026-09-16", "updated": "2026-09-16", "createdAt": "2026-09-16T05:56:05.000Z", "updatedAt": "2026-09-16T10:04:19.000Z", "sha256": "af122c2bb34d2b5a126bf0753e0b9662876e9e7ecf4cce626bddf2dca02a5ce1" }, { "id": "op_0ff53a57552ddd7f", "ulid": "01M2M9FBMJ72NN6ZVW727YNWRX", "aliases": [ "op_0ff53a57552ddd7f", "01M2M9FBMJ72NN6ZVW727YNWRX", "op-0ff53a57552ddd7f" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 2, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-16T05:01:48.050Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "source-stated", "posed": null, "areaIds": [ "quantum-algorithm" ], "topicIds": [ "computational-complexity-and-computability" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [] }, "title": "Unrestricted quantum time–space tradeoff for collision finding", "status": "Unsolved", "fields": [ "Quantum algorithm" ], "topics": [ "Computational complexity and computability" ], "tags": [ "Quantum algorithm", "Computational complexity and computability" ], "statement": "Does every unrestricted quantum collision-finding algorithm satisfy $T^2S=\\Omega(N\\log N)$?\n\nLet $f:[N]\\to[N]$ be uniformly random and available through a coherent value oracle. An algorithm must output distinct $x,x'$ with $f(x)=f(x')$ with probability at least $2/3$. Let $T$ be its number of oracle queries and $S$ its maximum retained workspace in qubits, including quantum memory and classical data retained for later use.\n\nIs the tradeoff\n\n \\begin{equation}\nT^2S=\\Omega(N\\log N)\n\\tag{1}\n\\end{equation} \nvalid without any symmetry restriction? In Eq. (1), the external oracle storage is excluded from $S$, but query registers and accumulated tables are included.", "url": "https://qiqc-op.com/problem/op_0ff53a57552ddd7f/", "json": "https://qiqc-op.com/api/problems/op_0ff53a57552ddd7f.json", "tex": "https://qiqc-op.com/problem/op_0ff53a57552ddd7f/op_0ff53a57552ddd7f.tex", "created": "2026-09-16", "updated": "2026-09-16", "createdAt": "2026-09-16T05:56:05.000Z", "updatedAt": "2026-09-16T06:51:51.000Z", "sha256": "5195ca437db542d7401da25314b65d743eb18b131b71680e9c90608df2663eb5" }, { "id": "op_15f8ca2e9d5d272b", "ulid": "01M2M9FBBE2TAHMFT2MAQA57WB", "aliases": [ "op_15f8ca2e9d5d272b", "01M2M9FBBE2TAHMFT2MAQA57WB", "op-15f8ca2e9d5d272b" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 2, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-16T05:01:47.758Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "editor-formulated", "posed": null, "areaIds": [ "quantum-algorithm" ], "topicIds": [ "computational-complexity-and-computability", "quantum-circuit-complexity" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [ "01M2M9FBDRFBTD144AXBZE286J" ] }, "title": "Subquadratic total quantum gate complexity of RSA factoring", "status": "Unsolved", "fields": [ "Quantum algorithm" ], "topics": [ "Computational complexity and computability", "Quantum circuit complexity" ], "tags": [ "Quantum algorithm", "Computational complexity and computability", "Quantum circuit complexity" ], "statement": "Can every balanced RSA semiprime be factored with polynomially subquadratic total quantum gate complexity?\n\nLet $N=pq$ be an $n$-bit integer whose distinct odd prime factors have bit length $n/2+O(1)$. Does there exist a constant $\\delta>0$ and a uniform hybrid quantum–classical algorithm that factors every such $N$ with probability at least $2/3$ using\n\n \\begin{equation}\nG_{\\mathrm{total}}(n)=O(n^{2-\\delta})\n\\tag{1}\n\\end{equation} \nquantum gates and $n^{O(1)}$ classical computation? The cost in Eq. (1) includes every quantum execution, restart, state preparation, measurement, reset, and gate-synthesis cost over a fixed finite universal gate set; modular exponentiation is not a unit-cost oracle.", "url": "https://qiqc-op.com/problem/op_15f8ca2e9d5d272b/", "json": "https://qiqc-op.com/api/problems/op_15f8ca2e9d5d272b.json", "tex": "https://qiqc-op.com/problem/op_15f8ca2e9d5d272b/op_15f8ca2e9d5d272b.tex", "created": "2026-09-16", "updated": "2026-09-16", "createdAt": "2026-09-16T05:56:05.000Z", "updatedAt": "2026-09-16T06:51:51.000Z", "sha256": "fd29b94f09db675ca1379e0e7d829411732b90ed03c371837133f010e0ee6725" }, { "id": "op_25342dbb8d64e728", "ulid": "01M2M9FB96NK6Y6EMA2PM3M8DM", "aliases": [ "op_25342dbb8d64e728", "01M2M9FB96NK6Y6EMA2PM3M8DM", "op-25342dbb8d64e728" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 2, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-16T05:01:47.686Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "source-stated", "posed": null, "areaIds": [ "quantum-communication" ], "topicIds": [ "channel-simulation", "one-shot-and-finite-blocklength-bounds" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [] }, "title": "High-dimensional crossover of optimized port-based teleportation", "status": "Unsolved", "fields": [ "Quantum Communication" ], "topics": [ "Channel simulation", "One-shot and finite-blocklength bounds" ], "tags": [ "Quantum Communication", "Channel simulation", "One-shot and finite-blocklength bounds" ], "statement": "What is the high-dimensional crossover fidelity of optimized deterministic port-based teleportation?\n\nLet $F_d^*(N)$ be the maximum entanglement fidelity of teleporting an unknown $d$-dimensional state with $N$ ports, optimizing both the resource state and Alice’s measurement while Bob only selects a port. Use\n\n \\begin{equation}\nF_e(\\mathcal T)=\\operatorname{Tr}\\!\\left[\\Phi_d(\\operatorname{id}\\otimes\\mathcal T)(\\Phi_d)\\right],\n\\qquad\n\\Phi_d=\\frac1d\\sum_{a,b=1}^d|aa\\rangle\\langle bb|.\n\\tag{1}\n\\end{equation} \nFor each fixed $c>0$, define\n\n \\begin{equation}\nN_d=\\max\\{1,\\lfloor cd^2\\rfloor\\},\n\\qquad\n\\varphi(c)=\\lim_{d\\to\\infty}F_d^*(N_d).\n\\tag{2}\n\\end{equation} \nDoes the limit in Eq. (2) exist for every $c>0$, and if so, what is $\\varphi(c)$? Equation (1) fixes the fidelity convention.", "url": "https://qiqc-op.com/problem/op_25342dbb8d64e728/", "json": "https://qiqc-op.com/api/problems/op_25342dbb8d64e728.json", "tex": "https://qiqc-op.com/problem/op_25342dbb8d64e728/op_25342dbb8d64e728.tex", "created": "2026-09-16", "updated": "2026-09-16", "createdAt": "2026-09-16T05:56:05.000Z", "updatedAt": "2026-09-16T06:51:51.000Z", "sha256": "88be55c4e9467426a6a9f41ae02319fe149d1a2f61fa68cbe1a007d432e185a5" }, { "id": "op_278166a49baedfdd", "ulid": "01M2M9FAJ3CTQ7V1WMFR7X6VD7", "aliases": [ "op_278166a49baedfdd", "01M2M9FAJ3CTQ7V1WMFR7X6VD7", "op-278166a49baedfdd" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 2, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-16T05:01:46.947Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "source-stated", "posed": null, "areaIds": [ "quantum-communication" ], "topicIds": [ "matrix-and-entropy-inequalities" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [ "01M2M9FAFXNJ6DPB5Y3JNBB9PY" ] }, "title": "Quantum Mrs. Gerber lower bound for information combining", "status": "Unsolved", "fields": [ "Quantum Communication" ], "topics": [ "Matrix and entropy inequalities" ], "tags": [ "Quantum Communication", "Matrix and entropy inequalities" ], "statement": "Does the quantum Mrs. Gerber lower bound hold for two independent uniform bits with arbitrary quantum side information?\n\nLet\n\n \\begin{equation}\n\\rho_{X_iB_i}=\\frac12\\sum_{x=0}^1|x\\rangle\\langle x|\\otimes\\rho_x^{(i)},\n\\qquad\n\\rho_{X_1B_1X_2B_2}=\\rho_{X_1B_1}\\otimes\\rho_{X_2B_2},\n\\tag{1}\n\\end{equation} \nwhere $X_1,X_2$ are uniform bits. Put $Z=X_1\\oplus X_2$ and $s_i=H(X_i\\mid B_i)$, using base-two von Neumann entropy. Define\n\n \\begin{equation}\nh_2(p)=-p\\log_2p-(1-p)\\log_2(1-p),\\quad 0\\log_2 0=0,\n\\qquad\nF(s,t)=h_2\\!\\left(h_2^{-1}(s)*h_2^{-1}(t)\\right),\n\\tag{2}\n\\end{equation} \nwhere $p*q=p(1-q)+(1-p)q$ and $h_2^{-1}$ takes values in $[0,1/2]$. Is\n\n \\begin{equation}\nH(Z\\mid B_1B_2)\\ge\n\\begin{cases}\nF(s_1,s_2), & s_1+s_2\\leq1,\\\\\ns_1+s_2-1+F(1-s_1,1-s_2), & s_1+s_2\\geq1\n\\end{cases}\n\\tag{3}\n\\end{equation} \nfor every state in Eq. (1)? Equation (2) fixes the function used in the bound (3).", "url": "https://qiqc-op.com/problem/op_278166a49baedfdd/", "json": "https://qiqc-op.com/api/problems/op_278166a49baedfdd.json", "tex": "https://qiqc-op.com/problem/op_278166a49baedfdd/op_278166a49baedfdd.tex", "created": "2026-09-16", "updated": "2026-09-16", "createdAt": "2026-09-16T05:56:05.000Z", "updatedAt": "2026-09-16T06:51:51.000Z", "sha256": "1e4aac8d92a44575c03a09bb31567138f096d7e34188ebdaaa6eb67c8ae3a0c2" }, { "id": "op_30954594cf01ebb3", "ulid": "01M2M9FC484RZN7CN5FV721TKH", "aliases": [ "op_30954594cf01ebb3", "01M2M9FC484RZN7CN5FV721TKH", "op-30954594cf01ebb3" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 2, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-16T05:01:48.552Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "source-stated", "posed": null, "areaIds": [ "quantum-metrology", "quantum-algorithm" ], "topicIds": [ "quantum-estimation", "computational-complexity-and-computability" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [] }, "title": "Sparse Hamiltonian learning without short-time control", "status": "Unsolved", "fields": [ "Quantum metrology", "Quantum algorithm" ], "topics": [ "Quantum estimation", "Computational complexity and computability" ], "tags": [ "Quantum metrology", "Quantum algorithm", "Quantum estimation", "Computational complexity and computability" ], "statement": "Can every polynomially sparse Hamiltonian be learned with Heisenberg-limited total evolution time when every oracle call has a fixed minimum duration?\n\nLet\n\n \\begin{equation}\nH=\\sum_{P\\neq I^{\\otimes n}}a_PP,\n\\qquad\n|\\{P:a_P\\neq0\\}|\\leq m\\leq n^c,\n\\qquad\n\\|H\\|_{\\mathrm{op}}\\leq1,\n\\tag{1}\n\\end{equation} \nwhere $c>0$ is fixed, the sum ranges over nonidentity $n$-qubit Pauli strings, and the nonzero coefficients and their supports are unknown. An oracle supplies only forward evolution $e^{-iHt}$ for chosen times $t\\geq T$, where $T>0$ is fixed independently of $n,m,$ and $\\varepsilon$. Known controls and ancillas may be used between calls.\n\nCan a learner output $\\widehat a_P$ with $\\max_P|\\widehat a_P-a_P|\\leq\\varepsilon$ and success probability at least $2/3$, using total evolution time $\\widetilde O(\\operatorname{poly}(n,m)/\\varepsilon)$ and polynomial query, circuit, and classical-processing costs for every Hamiltonian in Eq. (1)?", "url": "https://qiqc-op.com/problem/op_30954594cf01ebb3/", "json": "https://qiqc-op.com/api/problems/op_30954594cf01ebb3.json", "tex": "https://qiqc-op.com/problem/op_30954594cf01ebb3/op_30954594cf01ebb3.tex", "created": "2026-09-16", "updated": "2026-09-16", "createdAt": "2026-09-16T05:56:05.000Z", "updatedAt": "2026-09-16T06:51:51.000Z", "sha256": "6e4e9b63fd1c033af26c9431f2721d37bf1172275b54652210157803917f9c21" }, { "id": "op_330054883db95e6b", "ulid": "01M2M9FBDRFBTD144AXBZE286J", "aliases": [ "op_330054883db95e6b", "01M2M9FBDRFBTD144AXBZE286J", "op-330054883db95e6b" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 2, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-16T05:01:47.832Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "editor-formulated", "posed": null, "areaIds": [ "quantum-algorithm" ], "topicIds": [ "computational-complexity-and-computability", "quantum-circuit-complexity" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [ "01M2M9FBBE2TAHMFT2MAQA57WB" ] }, "title": "Sublinear logical quantum space for RSA factoring", "status": "Unsolved", "fields": [ "Quantum algorithm" ], "topics": [ "Computational complexity and computability", "Quantum circuit complexity" ], "tags": [ "Quantum algorithm", "Computational complexity and computability", "Quantum circuit complexity" ], "statement": "Can every balanced RSA semiprime be factored in polynomial time using sublinear logical quantum space?\n\nLet $N=pq$ be an $n$-bit integer whose distinct odd prime factors have bit length $n/2+O(1)$. Does a uniform hybrid algorithm exist that factors every such $N$ with probability at least $2/3$ while satisfying\n\n \\begin{equation}\nq(n)=o(n),\n\\qquad\nT_{\\mathrm{total}}(n)=n^{O(1)}?\n\\tag{1}\n\\end{equation} \nHere $q(n)$ in Eq. (1) counts all simultaneously live logical qubits, including control, data, workspace, quantum memory, and resource states. Polynomial classical storage, intermediate measurements, and resets are allowed, and total runtime includes repetitions and classical processing.", "url": "https://qiqc-op.com/problem/op_330054883db95e6b/", "json": "https://qiqc-op.com/api/problems/op_330054883db95e6b.json", "tex": "https://qiqc-op.com/problem/op_330054883db95e6b/op_330054883db95e6b.tex", "created": "2026-09-16", "updated": "2026-09-16", "createdAt": "2026-09-16T05:56:05.000Z", "updatedAt": "2026-09-16T06:51:51.000Z", "sha256": "6ee02cb06baca564ada53426c2f845bd6e6693ca6d10b8edad62a195ea6fff8c" }, { "id": "op_3a7c93378a832c01", "ulid": "01M2M9FCFHH8YFPDM24122JFN1", "aliases": [ "op_3a7c93378a832c01", "01M2M9FCFHH8YFPDM24122JFN1", "op-3a7c93378a832c01" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 2, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-16T05:01:48.913Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "editor-formulated", "posed": null, "areaIds": [ "quantum-algorithm" ], "topicIds": [ "random-circuit-sampling", "computational-complexity-and-computability", "quantum-supremacy" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [ "01M20CXWDYD1RXVWDKFMFM675K" ] }, "title": "Sampling weakly depolarized constant-depth random circuits", "status": "Unsolved", "fields": [ "Quantum algorithm" ], "topics": [ "Random circuit sampling", "Computational complexity and computability", "Quantum supremacy" ], "tags": [ "Quantum algorithm", "Random circuit sampling", "Computational complexity and computability", "Quantum supremacy" ], "statement": "Can weak final-layer depolarizing noise make constant-depth two-dimensional Haar-random circuits classically samplable to inverse-polynomial total-variation error?\n\nLet $n=L^2$ qubits occupy an even-sided square grid. Apply a fixed-depth nearest-neighbor brickwork circuit $U$ of independent Haar-random two-qubit gates to $|0^n\\rangle$, followed on every qubit by\n\n \\begin{equation}\n\\mathcal D_\\gamma(X)=(1-\\gamma)X+\\gamma\\operatorname{Tr}(X)I/2.\n\\tag{1}\n\\end{equation} \nLet $P_{U,\\gamma}$ be the computational-basis output distribution. For fixed $a>0$, set $\\varepsilon=n^{-a}$ and $\\gamma=K\\log(n/\\varepsilon)/n$. Is there a constant $d_0$ such that, for every fixed depth $d>d_0$, some constant $K>0$ permits a polynomial-time classical sampler with output distribution $Q_U$ satisfying\n\n \\begin{equation}\n\\mathbb E_U\\|Q_U-P_{U,\\gamma}\\|_{\\mathrm{TV}}\\leq\\varepsilon,\n\\qquad\n\\|P-Q\\|_{\\mathrm{TV}}=\\frac12\\sum_x|P(x)-Q(x)|?\n\\tag{2}\n\\end{equation} \nEquation (2) is the target for the noise channel in Eq. (1).", "url": "https://qiqc-op.com/problem/op_3a7c93378a832c01/", "json": "https://qiqc-op.com/api/problems/op_3a7c93378a832c01.json", "tex": "https://qiqc-op.com/problem/op_3a7c93378a832c01/op_3a7c93378a832c01.tex", "created": "2026-09-16", "updated": "2026-09-16", "createdAt": "2026-09-16T05:56:05.000Z", "updatedAt": "2026-09-16T06:51:51.000Z", "sha256": "46c2f52a0d92776d2250d4755cd1d3d73eddff33aa71f8720fc6c0774c27f5cc" }, { "id": "op_428e6c37ba03149a", "ulid": "01M2M9FAM8V4XN25WJB1Q8H13Y", "aliases": [ "op_428e6c37ba03149a", "01M2M9FAM8V4XN25WJB1Q8H13Y", "op-428e6c37ba03149a" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 2, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-16T05:01:47.016Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "source-stated", "posed": null, "areaIds": [ "quantum-metrology" ], "topicIds": [ "quantum-estimation", "one-shot-and-finite-blocklength-bounds" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [] }, "title": "Optimal dimension-free copy complexity of shadow tomography", "status": "Unsolved", "fields": [ "Quantum metrology" ], "topics": [ "Quantum estimation", "One-shot and finite-blocklength bounds" ], "tags": [ "Quantum metrology", "Quantum estimation", "One-shot and finite-blocklength bounds" ], "statement": "Can list-dependent shadow tomography always achieve dimension-free copy complexity $O(\\log M/\\varepsilon^2)$?\n\nLet $0\\leq E_1,\\ldots,E_M\\leq I_d$ be a known list of effects and let $\\rho$ be an unknown $d$-dimensional state supplied as independent copies. Does a universal constant $C$ exist such that, for every $d$, $M\\geq2$, and $0<\\varepsilon<1/4$, a collective measurement on\n\n \\begin{equation}\nN\\leq\\left\\lceil C\\frac{\\log M}{\\varepsilon^2}\\right\\rceil\n\\quad\\text{copies yields}\\quad\n\\inf_\\rho\\Pr_\\rho\\!\\left[\n\\max_j|\\widehat\\mu_j-\\operatorname{Tr}(\\rho E_j)|\\leq\\varepsilon\n\\right]\\geq\\frac23?\n\\tag{1}\n\\end{equation} \nThe list is fixed before measurement, and no computational-efficiency requirement is imposed. Equation (1) concerns list-dependent shadow tomography rather than measurement-independent classical shadows.", "url": "https://qiqc-op.com/problem/op_428e6c37ba03149a/", "json": "https://qiqc-op.com/api/problems/op_428e6c37ba03149a.json", "tex": "https://qiqc-op.com/problem/op_428e6c37ba03149a/op_428e6c37ba03149a.tex", "created": "2026-09-16", "updated": "2026-09-16", "createdAt": "2026-09-16T05:56:05.000Z", "updatedAt": "2026-09-16T06:51:51.000Z", "sha256": "082c80c07f4e267187398a7e6e480860dec28d782148e35b003cf1e665c7d94f" }, { "id": "op_42d3766b44e3a89b", "ulid": "01M2M9FAB7KQ0FCTGXPQD2DRSJ", "aliases": [ "op_42d3766b44e3a89b", "01M2M9FAB7KQ0FCTGXPQD2DRSJ", "op-42d3766b44e3a89b" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 2, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-16T05:01:46.727Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "source-stated", "posed": null, "areaIds": [ "quantum-communication" ], "topicIds": [ "matrix-and-entropy-inequalities" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [] }, "title": "Quantum validity of the Zhang–Yeung inequality", "status": "Unsolved", "fields": [ "Quantum Communication" ], "topics": [ "Matrix and entropy inequalities" ], "tags": [ "Quantum Communication", "Matrix and entropy inequalities" ], "statement": "Does the four-party Zhang–Yeung inequality hold for the von Neumann entropies of every finite-dimensional quantum state?\n\nFor a state $\\rho_{ABCD}$, define\n\n \\begin{equation}\n\\begin{aligned}\nS(R)&=-\\operatorname{Tr}(\\rho_R\\log_2\\rho_R),\\\\\nI(A:B)&=S(A)+S(B)-S(AB),\\\\\nI(A:B\\mid C)&=S(AC)+S(BC)-S(C)-S(ABC).\n\\end{aligned}\n\\tag{1}\n\\end{equation} \nUsing the quantities in Eq. (1), the proposed inequality is\n\n \\begin{equation}\n2I(C:D)\\leq I(A:B)+I(A:CD)+3I(C:D\\mid A)+I(C:D\\mid B)\n\\tag{2}\n\\end{equation} \nDoes Eq. (2) hold for every $\\rho_{ABCD}$, without independence, separability, stabilizer, or holographic assumptions?", "url": "https://qiqc-op.com/problem/op_42d3766b44e3a89b/", "json": "https://qiqc-op.com/api/problems/op_42d3766b44e3a89b.json", "tex": "https://qiqc-op.com/problem/op_42d3766b44e3a89b/op_42d3766b44e3a89b.tex", "created": "2026-09-16", "updated": "2026-09-16", "createdAt": "2026-09-16T05:56:05.000Z", "updatedAt": "2026-09-16T06:51:51.000Z", "sha256": "9e44283e17c3a9431c8444f95841f76a419080cf879a781ea6c432468f9a4080" }, { "id": "op_4dba7a000cf4cd07", "ulid": "01M2M9FBJBZK4X6HQK8CWAC9F8", "aliases": [ "op_4dba7a000cf4cd07", "01M2M9FBJBZK4X6HQK8CWAC9F8", "op-4dba7a000cf4cd07" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 2, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-16T05:01:47.979Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "source-stated", "posed": null, "areaIds": [ "quantum-algorithm" ], "topicIds": [ "quantum-linear-algebra", "computational-complexity-and-computability" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [ "01M22P0HX43A7EQD9XKQ1G7QQJ" ] }, "title": "Multiplicative sparse-access lower bound for quantum linear systems", "status": "Unsolved", "fields": [ "Quantum algorithm" ], "topics": [ "Quantum linear algebra", "Computational complexity and computability" ], "tags": [ "Quantum algorithm", "Quantum linear algebra", "Computational complexity and computability" ], "statement": "Does sparse-oracle quantum linear-system solving require $\\Omega(\\kappa\\sqrt{s}\\log(1/\\varepsilon))$ queries in the worst case?\n\nLet $A$ be an invertible $D\\times D$ matrix with at most $s$ nonzero entries per row and column, $\\|A\\|\\leq1$, and smallest singular value at least $1/\\kappa$. Given standard sparse-location and entry-value oracles for $A$ and an efficiently prepared state $|b\\rangle$, the task is to prepare a state within Euclidean distance $\\varepsilon$ of\n\n \\begin{equation}\n|x\\rangle=\\frac{A^{-1}|b\\rangle}{\\|A^{-1}|b\\rangle\\|}.\n\\tag{1}\n\\end{equation} \nFor $s\\geq3$, $\\kappa\\geq2$, $0<\\varepsilon<1/10$, sufficiently large $D$, and success probability at least $2/3$, is the worst-case query complexity necessarily\n\n \\begin{equation}\nQ_{\\mathrm{sparse}}(\\kappa,s,\\varepsilon)\n=\\Omega\\!\\left(\\kappa\\sqrt{s}\\log\\frac1\\varepsilon\\right)?\n\\tag{2}\n\\end{equation} \nEquation (2) asks for a joint lower bound for the state-preparation task in Eq. (1).", "url": "https://qiqc-op.com/problem/op_4dba7a000cf4cd07/", "json": "https://qiqc-op.com/api/problems/op_4dba7a000cf4cd07.json", "tex": "https://qiqc-op.com/problem/op_4dba7a000cf4cd07/op_4dba7a000cf4cd07.tex", "created": "2026-09-16", "updated": "2026-09-16", "createdAt": "2026-09-16T05:56:05.000Z", "updatedAt": "2026-09-16T06:51:51.000Z", "sha256": "56db4d1773b46cc992e1a3261fa14a4f82979d5c13123116db45e6b409b0c6d3" }, { "id": "op_6d9a72b32070a900", "ulid": "01M2M9FB4MRH9G9THBXZ1QJF2M", "aliases": [ "op_6d9a72b32070a900", "01M2M9FB4MRH9G9THBXZ1QJF2M", "op-6d9a72b32070a900" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 2, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-16T05:01:47.540Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "source-stated", "posed": null, "areaIds": [ "quantum-resource-theory" ], "topicIds": [ "quantum-magic" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [ "01M1HME780N8J8JBTFX6CSPACD" ] }, "title": "Hilbert–Schmidt inradius of the multiqubit stabilizer polytope", "status": "Unsolved", "fields": [ "Quantum Resource Theory" ], "topics": [ "Quantum magic" ], "tags": [ "Quantum Resource Theory", "Quantum magic" ], "statement": "Is every $n$-qubit state of purity at most $1/(2^n-1/2)$ a convex mixture of stabilizer states?\n\nLet $d=2^n$ and\n\n \\begin{equation}\n\\mathrm{STAB}_n=\\operatorname{conv}\\{|s\\rangle\\langle s|:|s\\rangle\\text{ is an }n\\text{-qubit stabilizer state}\\}.\n\\tag{1}\n\\end{equation} \nIs the following implication valid for every $n\\geq1$?\n\n \\begin{equation}\n\\operatorname{Tr}(\\rho^2)\\leq\\frac1{d-1/2}\n\\quad\\Longrightarrow\\quad\n\\rho\\in\\mathrm{STAB}_n.\n\\tag{2}\n\\end{equation} \nEquivalently, does the Hilbert–Schmidt ball about $I/d$ contained in Eq. (1) have radius $1/\\sqrt{d(2d-1)}$? This is the threshold asserted by Eq. (2).", "url": "https://qiqc-op.com/problem/op_6d9a72b32070a900/", "json": "https://qiqc-op.com/api/problems/op_6d9a72b32070a900.json", "tex": "https://qiqc-op.com/problem/op_6d9a72b32070a900/op_6d9a72b32070a900.tex", "created": "2026-09-16", "updated": "2026-09-16", "createdAt": "2026-09-16T05:56:05.000Z", "updatedAt": "2026-09-16T06:51:51.000Z", "sha256": "4f79311e309fd2f63133bb567bea5115c595ca189ad0cf11808b50a2af394e52" }, { "id": "op_6f1b7742387af9e7", "ulid": "01M2M9FAXBSQS2W1SPQPBYXRSQ", "aliases": [ "op_6f1b7742387af9e7", "01M2M9FAXBSQS2W1SPQPBYXRSQ", "op-6f1b7742387af9e7" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 2, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-16T05:01:47.307Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "editor-formulated", "posed": null, "areaIds": [ "quantum-resource-theory" ], "topicIds": [ "quantum-magic", "resource-conversion" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [ "01M1Q787QRD6APNHX659G4CTEF", "01M2M9FBZT1N3HH3MV5SRK29N1" ] }, "title": "Distillation of every Wigner-negative depolarized Strange state", "status": "Unsolved", "fields": [ "Quantum Resource Theory" ], "topics": [ "Quantum magic", "Resource conversion" ], "tags": [ "Quantum Resource Theory", "Quantum magic", "Resource conversion" ], "statement": "Can every Wigner-negative state in the depolarized qutrit Strange-state family be distilled arbitrarily accurately by catalyst-free stabilizer operations?\n\nDefine\n\n \\begin{equation}\n|S\\rangle=\\frac{|1\\rangle-|2\\rangle}{\\sqrt2},\n\\qquad\n\\rho_\\varepsilon=(1-\\varepsilon)|S\\rangle\\langle S|+\\varepsilon\\frac{I_3}{3}.\n\\tag{1}\n\\end{equation} \nFor every $0\\leq\\varepsilon<3/4$ and $\\delta>0$, can finitely many copies of the state in Eq. (1) be converted, with positive success probability, to a state within trace distance $\\delta$ of $|S\\rangle\\langle S|$ using stabilizer ancillas, Clifford unitaries, Pauli measurements, classical feedforward, and discarding, but no magic catalyst?", "url": "https://qiqc-op.com/problem/op_6f1b7742387af9e7/", "json": "https://qiqc-op.com/api/problems/op_6f1b7742387af9e7.json", "tex": "https://qiqc-op.com/problem/op_6f1b7742387af9e7/op_6f1b7742387af9e7.tex", "created": "2026-09-16", "updated": "2026-09-16", "createdAt": "2026-09-16T05:56:05.000Z", "updatedAt": "2026-09-16T06:51:51.000Z", "sha256": "e645919045333a062aaba028ba5aa132fa573a0fa970904d77e252d2d75ed8e4" }, { "id": "op_72937f30974953ba", "ulid": "01M2M9FCD8BZKAGDQ46Z2H5KCX", "aliases": [ "op_72937f30974953ba", "01M2M9FCD8BZKAGDQ46Z2H5KCX", "op-72937f30974953ba" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 2, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-16T05:01:48.840Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "editor-formulated", "posed": null, "areaIds": [ "quantum-algorithm" ], "topicIds": [ "random-circuit-sampling", "computational-complexity-and-computability" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [] }, "title": "Conditional-correlation decay in amplitude-damped random circuits", "status": "Unsolved", "fields": [ "Quantum algorithm" ], "topics": [ "Random circuit sampling", "Computational complexity and computability" ], "tags": [ "Quantum algorithm", "Random circuit sampling", "Computational complexity and computability" ], "statement": "Do computational-basis outputs of one-dimensional Haar-random circuits with fixed amplitude damping have conditional mutual information that decays exponentially with separation, uniformly in circuit depth?\n\nConsider an $n$-qubit nearest-neighbor Haar-random brickwork circuit on a line, starting from $|0^n\\rangle$. After each layer, apply amplitude damping of fixed strength $\\gamma\\in(0,1)$ independently to every qubit, with\n\n \\begin{equation}\nK_0=|0\\rangle\\langle0|+\\sqrt{1-\\gamma}|1\\rangle\\langle1|,\n\\qquad\nK_1=\\sqrt\\gamma|0\\rangle\\langle1|.\n\\tag{1}\n\\end{equation} \nLet $P_U$ be the computational-basis output distribution. For disjoint $A,B,C$, define\n\n \\begin{equation}\nI_{P_U}(A:C\\mid B)=H_{P_U}(AB)+H_{P_U}(BC)-H_{P_U}(B)-H_{P_U}(ABC),\n\\tag{2}\n\\end{equation} \nand let $r(A,C)$ be the chain distance between nonempty $A$ and $C$. For every fixed $\\gamma$, do constants $a,b,c>0$ exist, independent of system size, depth, and the subsets, such that\n\n \\begin{equation}\n\\mathbb E_U I_{P_U}(A:C\\mid B)\\leq an^b e^{-c r(A,C)}?\n\\tag{3}\n\\end{equation} \nEquation (3) concerns the distribution produced using Eq. (1) and the classical conditional mutual information in Eq. (2).", "url": "https://qiqc-op.com/problem/op_72937f30974953ba/", "json": "https://qiqc-op.com/api/problems/op_72937f30974953ba.json", "tex": "https://qiqc-op.com/problem/op_72937f30974953ba/op_72937f30974953ba.tex", "created": "2026-09-16", "updated": "2026-09-16", "createdAt": "2026-09-16T05:56:05.000Z", "updatedAt": "2026-09-16T06:51:51.000Z", "sha256": "c119e250a0b2688f3209d8b1ed904ffae7eca19abebd49ed237c5f4363420e97" }, { "id": "op_845158213592821f", "ulid": "01M2M9FC22Z4C4YCDYT9XZCB59", "aliases": [ "op_845158213592821f", "01M2M9FC22Z4C4YCDYT9XZCB59", "op-845158213592821f" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 2, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-16T05:01:48.482Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "editor-formulated", "posed": null, "areaIds": [ "quantum-metrology", "quantum-algorithm" ], "topicIds": [ "quantum-estimation", "computational-complexity-and-computability", "quantum-thermodynamics" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [] }, "title": "Unknown-structure Hamiltonian learning from Gibbs states at all temperatures", "status": "Unsolved", "fields": [ "Quantum metrology", "Quantum algorithm" ], "topics": [ "Quantum estimation", "Computational complexity and computability", "Quantum thermodynamics" ], "tags": [ "Quantum metrology", "Quantum algorithm", "Quantum estimation", "Computational complexity and computability", "Quantum thermodynamics" ], "statement": "Can every bounded-degree local Hamiltonian with unknown interaction support be learned efficiently from copies of its Gibbs state at any fixed inverse temperature?\n\nFix constants $k$ and $g$. Let $\\mathcal P_{n,k}$ be the nonidentity $n$-qubit Pauli strings of weight at most $k$, and consider\n\n \\begin{equation}\nH=\\sum_{P\\in\\mathcal P_{n,k}}a_PP,\n\\quad |a_P|\\leq1,\n\\quad \\max_j|\\{P:a_P\\neq0,\\ j\\in\\operatorname{supp}(P)\\}|\\leq g,\n\\qquad\n\\rho_\\beta=\\frac{e^{-\\beta H}}{\\operatorname{Tr}(e^{-\\beta H})}.\n\\tag{1}\n\\end{equation} \nThe nonzero Pauli terms are unknown, while $\\beta>0$ is known and fixed independently of $n$. Can independent copies of $\\rho_\\beta$ in Eq. (1) be used to output coefficients satisfying $\\max_P|\\widehat a_P-a_P|\\leq\\varepsilon$ with probability at least $2/3$, using $\\operatorname{poly}(n,1/\\varepsilon)$ copies and classical time for every fixed $\\beta$?", "url": "https://qiqc-op.com/problem/op_845158213592821f/", "json": "https://qiqc-op.com/api/problems/op_845158213592821f.json", "tex": "https://qiqc-op.com/problem/op_845158213592821f/op_845158213592821f.tex", "created": "2026-09-16", "updated": "2026-09-16", "createdAt": "2026-09-16T05:56:05.000Z", "updatedAt": "2026-09-16T06:51:51.000Z", "sha256": "1628343facd0b4100077d56fd6e3a31fd24a8f6200b8a4e92f8c604defaefa73" }, { "id": "op_91c81ca77b99eebf", "ulid": "01M2M9FC6H8290J9PJ3S9MH1SW", "aliases": [ "op_91c81ca77b99eebf", "01M2M9FC6H8290J9PJ3S9MH1SW", "op-91c81ca77b99eebf" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 2, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-16T05:01:48.625Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "editor-formulated", "posed": null, "areaIds": [ "quantum-metrology" ], "topicIds": [ "quantum-estimation", "gaussian-quantum-information", "one-shot-and-finite-blocklength-bounds" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [] }, "title": "Optimal single-copy tomography of fermionic Gaussian states", "status": "Unsolved", "fields": [ "Quantum metrology" ], "topics": [ "Quantum estimation", "Gaussian quantum information", "One-shot and finite-blocklength bounds" ], "tags": [ "Quantum metrology", "Quantum estimation", "Gaussian quantum information", "One-shot and finite-blocklength bounds" ], "statement": "Can arbitrary mixed fermionic Gaussian states be learned to trace distance $\\varepsilon$ from $\\Theta(m^2/\\varepsilon^2)$ copies without measurements across copies?\n\nFor $m$ fermionic modes with Majorana operators satisfying $c_ac_b+c_bc_a=2\\delta_{ab}I$, let $\\mathcal G_m$ consist of the states\n\n \\begin{equation}\n\\rho=\\frac{e^{-K}}{\\operatorname{Tr}(e^{-K})},\n\\qquad\nK=\\frac{i}{4}\\sum_{a,b=1}^{2m}A_{ab}c_ac_b,\n\\qquad\nA^T=-A\\in\\mathbb R^{2m\\times2m},\n\\tag{1}\n\\end{equation} \nincluding their limiting pure states. Let $N_1(m,\\varepsilon)$ be the minimum worst-case number of copies needed to output $\\widehat\\rho\\in\\mathcal G_m$ with\n\n \\begin{equation}\n\\Pr\\!\\left[\\frac12\\|\\rho-\\widehat\\rho\\|_1\\leq\\varepsilon\\right]\\geq\\frac23.\n\\tag{2}\n\\end{equation} \nEach adaptive measurement may act on all modes of one copy and an ancilla, but no quantum memory may connect different copies. Is $N_1(m,\\varepsilon)=\\Theta(m^2/\\varepsilon^2)$ for the family in Eq. (1) under the success criterion in Eq. (2)?", "url": "https://qiqc-op.com/problem/op_91c81ca77b99eebf/", "json": "https://qiqc-op.com/api/problems/op_91c81ca77b99eebf.json", "tex": "https://qiqc-op.com/problem/op_91c81ca77b99eebf/op_91c81ca77b99eebf.tex", "created": "2026-09-16", "updated": "2026-09-16", "createdAt": "2026-09-16T05:56:05.000Z", "updatedAt": "2026-09-16T06:51:51.000Z", "sha256": "d14758b924d2fd0c17dc16d4e508d88a934fb11d4ea8b8e04877346faac9995f" }, { "id": "op_92fc12b81d55a704", "ulid": "01M2M9FAPEZ3SWR509F5FVMBWX", "aliases": [ "op_92fc12b81d55a704", "01M2M9FAPEZ3SWR509F5FVMBWX", "op-92fc12b81d55a704" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 2, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-16T05:01:47.086Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "source-stated", "posed": null, "areaIds": [ "quantum-resource-theory", "quantum-algorithm" ], "topicIds": [ "quantum-thermodynamics", "quantum-state-preparation", "quantum-relative-entropy" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [] }, "title": "Uniform modified log-Sobolev constant for one-dimensional Gibbs samplers", "status": "Unsolved", "fields": [ "Quantum Resource Theory", "Quantum algorithm" ], "topics": [ "Quantum thermodynamics", "Quantum state preparation", "Quantum relative entropy" ], "tags": [ "Quantum Resource Theory", "Quantum algorithm", "Quantum thermodynamics", "Quantum state preparation", "Quantum relative entropy" ], "statement": "Do normalized one-dimensional Chen–Kastoryano–Gilyén Gibbs samplers have a positive modified log-Sobolev constant uniformly in system size at every fixed finite temperature?\n\nFix $\\beta>0$, a finite on-site dimension, and a class of one-dimensional finite-range Hamiltonians with bounded local strength. Let $\\rho_\\beta(H)=e^{-\\beta H}/\\operatorname{Tr}(e^{-\\beta H})$, and let $\\mathcal L_{H,\\beta}$ be the Chen–Kastoryano–Gilyén generator using all single-site Pauli jumps, Gaussian-filter width $\\beta^{-1}$, and the unrescaled sum of local terms.\n\nDoes a constant $\\alpha_\\beta>0$ exist, independent of chain length and of $H$ in this class, such that for every state $\\sigma$ and every $t\\geq0$\n\n \\begin{equation}\nD\\!\\left(e^{t\\mathcal L_{H,\\beta,*}}(\\sigma)\\middle\\|\\rho_\\beta(H)\\right)\n\\leq e^{-2\\alpha_\\beta t}D\\!\\left(\\sigma\\middle\\|\\rho_\\beta(H)\\right)?\n\\tag{1}\n\\end{equation} \nRelative entropy in Eq. (1) uses natural logarithms; constants may depend on $\\beta$ but not on system size.", "url": "https://qiqc-op.com/problem/op_92fc12b81d55a704/", "json": "https://qiqc-op.com/api/problems/op_92fc12b81d55a704.json", "tex": "https://qiqc-op.com/problem/op_92fc12b81d55a704/op_92fc12b81d55a704.tex", "created": "2026-09-16", "updated": "2026-09-16", "createdAt": "2026-09-16T05:56:05.000Z", "updatedAt": "2026-09-16T06:51:51.000Z", "sha256": "8a5b8d5c733619975494a4bb5639c347f82f726671614a9df828db5e3a630583" }, { "id": "op_953280faa99bec1d", "ulid": "01M2M9FBZT1N3HH3MV5SRK29N1", "aliases": [ "op_953280faa99bec1d", "01M2M9FBZT1N3HH3MV5SRK29N1", "op-953280faa99bec1d" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 2, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-16T05:01:48.410Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "source-stated", "posed": null, "areaIds": [ "quantum-resource-theory", "quantum-error-correction" ], "topicIds": [ "quantum-magic", "quantum-coding-theory", "resource-conversion" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [ "01M2M9FAXBSQS2W1SPQPBYXRSQ" ] }, "title": "Qutrit magic-distillation thresholds beyond the ternary Golay code", "status": "Unsolved", "fields": [ "Quantum Resource Theory", "Quantum Error Correction" ], "topics": [ "Quantum magic", "Quantum coding theory", "Resource conversion" ], "tags": [ "Quantum Resource Theory", "Quantum Error Correction", "Quantum magic", "Quantum coding theory", "Resource conversion" ], "statement": "Does any finite qutrit stabilizer-code protocol distill depolarized Strange states above the ternary Golay code’s threshold?\n\nLet\n\n \\begin{equation}\n|S\\rangle=\\frac{|1\\rangle-|2\\rangle}{\\sqrt2},\n\\qquad\n\\rho_S(\\varepsilon)=(1-\\varepsilon)|S\\rangle\\langle S|+\\varepsilon\\frac{I_3}{3},\\qquad 0\\leq\\varepsilon\\leq1.\n\\tag{1}\n\\end{equation} \nFor a finite $n$-to-$1$ qutrit stabilizer-code protocol $\\mathcal P$, let $F_{\\mathcal P}(\\varepsilon)$ be the output mixing parameter after accepted syndrome projection, Clifford decoding and correction, and twirling back to the family in Eq. (1). Define\n\n \\begin{equation}\n\\varepsilon_*(\\mathcal P)=\\sup\\left\\{a\\in[0,1]:\n\\lim_{r\\to\\infty}F_{\\mathcal P}^{\\circ r}(\\varepsilon)=0\n\\text{ for every }0<\\varepsilon\\varepsilon_G$ as defined in Eq. (2)?", "url": "https://qiqc-op.com/problem/op_953280faa99bec1d/", "json": "https://qiqc-op.com/api/problems/op_953280faa99bec1d.json", "tex": "https://qiqc-op.com/problem/op_953280faa99bec1d/op_953280faa99bec1d.tex", "created": "2026-09-16", "updated": "2026-09-16", "createdAt": "2026-09-16T05:56:05.000Z", "updatedAt": "2026-09-16T06:51:51.000Z", "sha256": "d357182bb48c7abe3cb0dfa83a0eefbf63f5d5ba91bafd359e4085bb805dc4da" }, { "id": "op_9aba71db480b451a", "ulid": "01M2M9FBS21XYQCZ5HAH34D10C", "aliases": [ "op_9aba71db480b451a", "01M2M9FBS21XYQCZ5HAH34D10C", "op-9aba71db480b451a" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 2, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-16T05:01:48.194Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "source-stated", "posed": null, "areaIds": [ "quantum-cryptography", "quantum-algorithm" ], "topicIds": [ "quantum-circuit-complexity", "computational-complexity-and-computability" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [] }, "title": "Pseudorandom unitaries from ordinary random two-qubit circuits", "status": "Unsolved", "fields": [ "Quantum Cryptography", "Quantum algorithm" ], "topics": [ "Quantum circuit complexity", "Computational complexity and computability" ], "tags": [ "Quantum Cryptography", "Quantum algorithm", "Quantum circuit complexity", "Computational complexity and computability" ], "statement": "Does a polynomial-length random walk generated by ordinary two-qubit gates form a pseudorandom-unitary ensemble?\n\nFor an ensemble $\\mathcal U_n$ of efficiently implementable $n$-qubit unitaries, require that every quantum polynomial-time distinguisher with forward oracle access satisfy\n\n \\begin{equation}\n\\left|\\Pr_{U\\sim\\mathcal U_n}[\\mathcal D^U(1^n)=1]\n-\\Pr_{V\\sim\\operatorname{Haar}(2^n)}[\\mathcal D^V(1^n)=1]\\right|\n\\leq\\operatorname{negl}(n).\n\\tag{1}\n\\end{equation} \nHere $\\operatorname{negl}(n)0$. One hidden unitary is sampled and reused; inverse, transpose, conjugate, and controlled-unitary oracles are not supplied.\n\nFix a finite universal gate set $\\mathcal G\\subset SU(4)$. At each step choose a qubit pair and a gate from $\\mathcal G$ independently and uniformly, and let $U_{n,m}$ be the resulting circuit. Do constants $c,C_{\\mathcal G}>0$ exist such that\n\n \\begin{equation}\n\\mathcal U_n=\\{U_{n,m(n)}:m(n)=\\lceil C_{\\mathcal G}n^c\\rceil\\}\n\\tag{2}\n\\end{equation} \nsatisfies the pseudorandomness condition in Eq. (1)? Equation (2) fixes one polynomial-size ensemble against all polynomial-time distinguishers.", "url": "https://qiqc-op.com/problem/op_9aba71db480b451a/", "json": "https://qiqc-op.com/api/problems/op_9aba71db480b451a.json", "tex": "https://qiqc-op.com/problem/op_9aba71db480b451a/op_9aba71db480b451a.tex", "created": "2026-09-16", "updated": "2026-09-16", "createdAt": "2026-09-16T05:56:05.000Z", "updatedAt": "2026-09-16T06:51:51.000Z", "sha256": "846943032535dfde7c3aae836d8671071ba562c69fd1e9d2b74d7cb9c157d32b" }, { "id": "op_aaf9791beced84e4", "ulid": "01M2M9FCB1FHWBC06BEF4123WQ", "aliases": [ "op_aaf9791beced84e4", "01M2M9FCB1FHWBC06BEF4123WQ", "op-aaf9791beced84e4" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 2, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-16T05:01:48.769Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "source-stated", "posed": null, "areaIds": [ "quantum-algorithm" ], "topicIds": [ "random-circuit-sampling", "quantum-circuit-complexity" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [] }, "title": "Exact spectral gap of random Pauli rotations", "status": "Unsolved", "fields": [ "Quantum algorithm" ], "topics": [ "Random circuit sampling", "Quantum circuit complexity" ], "tags": [ "Quantum algorithm", "Random circuit sampling", "Quantum circuit complexity" ], "statement": "Is the exact Haar-mixing spectral gap of uniformly random Pauli rotations equal to $2^n(2^n-3)/(8(4^n-1))$ for every $n\\geq4$?\n\nLet $\\mathcal P_n=\\{I,X,Y,Z\\}^{\\otimes n}\\setminus\\{I^{\\otimes n}\\}$. One step chooses $P\\in\\mathcal P_n$ and $\\theta\\in[0,2\\pi)$ uniformly and applies $e^{i\\theta P}$. On $L^2(\\mathrm{SU}(2^n),\\mathrm{Haar})$, define\n\n \\begin{equation}\n(K_nf)(U)=\\mathbb E_{P,\\theta}f(e^{i\\theta P}U),\n\\qquad\n\\Delta_n=1-\\|K_n|_{L^2_0}\\|_{2\\to2},\n\\tag{1}\n\\end{equation} \nwhere $L^2_0$ is the zero-Haar-mean subspace. The conjectured value of the gap in Eq. (1) is\n\n \\begin{equation}\n\\Delta_n=\\frac{2^n(2^n-3)}{8(4^n-1)}\n\\tag{2}\n\\end{equation} \nDoes Eq. (2) hold for every $n\\geq4$?", "url": "https://qiqc-op.com/problem/op_aaf9791beced84e4/", "json": "https://qiqc-op.com/api/problems/op_aaf9791beced84e4.json", "tex": "https://qiqc-op.com/problem/op_aaf9791beced84e4/op_aaf9791beced84e4.tex", "created": "2026-09-16", "updated": "2026-09-16", "createdAt": "2026-09-16T05:56:05.000Z", "updatedAt": "2026-09-16T06:51:51.000Z", "sha256": "31f71b966ee73d7fdded0245feee56579324eb6b232b6643fc81257697afb23d" }, { "id": "op_ae968c5086f34fea", "ulid": "01M2M9FB28MYHRXE8ZS6ZE45TV", "aliases": [ "op_ae968c5086f34fea", "01M2M9FB28MYHRXE8ZS6ZE45TV", "op-ae968c5086f34fea" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 2, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-16T05:01:47.464Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "editor-formulated", "posed": null, "areaIds": [ "quantum-resource-theory" ], "topicIds": [ "quantum-magic", "resource-conversion", "additivity-and-regularization" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [] }, "title": "Optimal asymptotic distillation yield of dephased T states", "status": "Unsolved", "fields": [ "Quantum Resource Theory" ], "topics": [ "Quantum magic", "Resource conversion", "Additivity and regularization" ], "tags": [ "Quantum Resource Theory", "Quantum magic", "Resource conversion", "Additivity and regularization" ], "statement": "What is the optimal catalyst-free asymptotic $T$-state yield of a dephased single-qubit $T$ state?\n\nLet\n\n \\begin{equation}\n|T\\rangle=\\frac{|0\\rangle+e^{i\\pi/4}|1\\rangle}{\\sqrt2},\n\\qquad\n\\tau=|T\\rangle\\langle T|,\n\\qquad\n\\rho_p=(1-p)\\tau+pZ\\tau Z.\n\\tag{1}\n\\end{equation} \nFor $0\\leq p\\leq1$, define the catalyst-free stabilizer-operation yield\n\n \\begin{equation}\nD_T(\\rho_p)=\\sup\\left\\{R:\n\\exists\\ \\Lambda_n,\\\n\\frac12\\left\\|\\Lambda_n(\\rho_p^{\\otimes n})\n-\\tau^{\\otimes\\lfloor Rn\\rfloor}\\right\\|_1\\longrightarrow0\\right\\},\n\\tag{2}\n\\end{equation} \nwhere $\\Lambda_n$ uses stabilizer ancillas, Clifford unitaries, Pauli measurements, classical feedforward, and discarding, but no magic catalyst. Determine the quantity in Eq. (2), with matching achievable and converse bounds, already for a fixed noise level such as $p=0.01$ in Eq. (1).", "url": "https://qiqc-op.com/problem/op_ae968c5086f34fea/", "json": "https://qiqc-op.com/api/problems/op_ae968c5086f34fea.json", "tex": "https://qiqc-op.com/problem/op_ae968c5086f34fea/op_ae968c5086f34fea.tex", "created": "2026-09-16", "updated": "2026-09-16", "createdAt": "2026-09-16T05:56:05.000Z", "updatedAt": "2026-09-16T06:51:51.000Z", "sha256": "67af68110d3af968a1de980a6f42721835f3e682e18106fa52d5df436751b2df" }, { "id": "op_b17234613637b07e", "ulid": "01M2M9FARQFKAC8QQ5V36DHYP7", "aliases": [ "op_b17234613637b07e", "01M2M9FARQFKAC8QQ5V36DHYP7", "op-b17234613637b07e" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 2, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-16T05:01:47.159Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "editor-formulated", "posed": null, "areaIds": [ "quantum-metrology", "quantum-algorithm" ], "topicIds": [ "quantum-hypothesis-testing", "one-shot-and-finite-blocklength-bounds", "quantum-thermodynamics" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [] }, "title": "Single-copy identity testing of local Gibbs states", "status": "Unsolved", "fields": [ "Quantum metrology", "Quantum algorithm" ], "topics": [ "Quantum hypothesis testing", "One-shot and finite-blocklength bounds", "Quantum thermodynamics" ], "tags": [ "Quantum metrology", "Quantum algorithm", "Quantum hypothesis testing", "One-shot and finite-blocklength bounds", "Quantum thermodynamics" ], "statement": "What is the minimax copy complexity of identity testing two unknown $k$-local Gibbs states using only measurements on individual copies?\n\nFor fixed $k\\geq2$, let\n\n \\begin{equation}\n\\mathcal H_{n,k}=\\left\\{\\sum_{P:\\,1\\leq|P|\\leq k}h_PP:|h_P|\\leq1\\right\\},\n\\qquad\n\\mathcal G_{n,k,\\beta}=\\left\\{\\frac{e^{-\\beta H}}{\\operatorname{Tr}(e^{-\\beta H})}:H\\in\\mathcal H_{n,k}\\right\\},\n\\tag{1}\n\\end{equation} \nwhere $P$ ranges over nonidentity $n$-qubit Pauli strings and $|P|$ is Pauli weight. No bounded-degree or geometric promise is imposed. Independent copies of unknown $\\rho,\\sigma\\in\\mathcal G_{n,k,\\beta}$ are supplied under the promise\n\n \\begin{equation}\n\\rho=\\sigma\n\\qquad\\text{or}\\qquad\n\\|\\rho-\\sigma\\|_1\\geq\\varepsilon.\n\\tag{2}\n\\end{equation} \nLet $N^*_{\\mathrm{sc}}(n,k,\\beta,\\varepsilon)$ be the least worst-case total copy count for success probability at least $2/3$, allowing adaptive measurements on individual copies but no joint measurement across copies. Determine $N^*_{\\mathrm{sc}}$ up to logarithmic factors, including its dependence on $n$, $\\beta$, and $\\varepsilon$, for the class in Eq. (1) and the test in Eq. (2).", "url": "https://qiqc-op.com/problem/op_b17234613637b07e/", "json": "https://qiqc-op.com/api/problems/op_b17234613637b07e.json", "tex": "https://qiqc-op.com/problem/op_b17234613637b07e/op_b17234613637b07e.tex", "created": "2026-09-16", "updated": "2026-09-16", "createdAt": "2026-09-16T05:56:05.000Z", "updatedAt": "2026-09-16T06:51:51.000Z", "sha256": "debc83e6f20d2e9d94b702e37c4106370b9a9c86cd1d0291a69cb8e674f20ac6" }, { "id": "op_bc61eaba5c332fec", "ulid": "01M2M9FB6ZWP4KA207XNSVBXXB", "aliases": [ "op_bc61eaba5c332fec", "01M2M9FB6ZWP4KA207XNSVBXXB", "op-bc61eaba5c332fec" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 2, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-16T05:01:47.615Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "source-stated", "posed": null, "areaIds": [ "quantum-resource-theory" ], "topicIds": [ "quantum-magic", "resource-conversion", "one-shot-and-finite-blocklength-bounds" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [ "01M1Q787QRD6APNHX659G4CTEF" ] }, "title": "Optimal universal eight-copy concentration to a CCZ state", "status": "Unsolved", "fields": [ "Quantum Resource Theory" ], "topics": [ "Quantum magic", "Resource conversion", "One-shot and finite-blocklength bounds" ], "tags": [ "Quantum Resource Theory", "Quantum magic", "Resource conversion", "One-shot and finite-blocklength bounds" ], "statement": "What is the optimal success probability for a fixed catalyst-free stabilizer protocol that converts eight copies of an arbitrary unknown pure qubit into an exact $|CCZ\\rangle$ state?\n\nLet $\\mathcal U_8$ be the fixed protocols built from stabilizer ancillas, Clifford unitaries, Pauli measurements, classical feedforward, and discarding, chosen independently of the input, whose accepted branches map $\\psi^{\\otimes8}$ exactly to\n\n \\begin{equation}\n|CCZ\\rangle=2^{-3/2}\\sum_{x\\in\\{0,1\\}^3}(-1)^{x_1x_2x_3}|x\\rangle.\n\\tag{1}\n\\end{equation} \nAcceptance may be zero on stabilizer inputs, and no magic catalyst is allowed. For a pure-state Bloch vector $(x,y,z)$, define $p_8^*(\\psi)=\\sup_{\\Lambda\\in\\mathcal U_8}p_\\Lambda(\\psi)$ and $m_3(\\psi)=(1-x^6-y^6-z^6)/2$. The proposed optimum is\n\n \\begin{equation}\np_8^*(\\psi)=\\frac23m_3(\\psi)\n\\tag{2}\n\\end{equation} \nDoes Eq. (2) hold for every pure qubit $\\psi$, with the target state fixed by Eq. (1)?", "url": "https://qiqc-op.com/problem/op_bc61eaba5c332fec/", "json": "https://qiqc-op.com/api/problems/op_bc61eaba5c332fec.json", "tex": "https://qiqc-op.com/problem/op_bc61eaba5c332fec/op_bc61eaba5c332fec.tex", "created": "2026-09-16", "updated": "2026-09-16", "createdAt": "2026-09-16T05:56:05.000Z", "updatedAt": "2026-09-16T06:51:51.000Z", "sha256": "96f90da5b015674d207d92ad6b6148eb904484522654e3021bc2c7419e6f0d6f" }, { "id": "op_c847b635f53b0812", "ulid": "01M2M9FBXJJ9AW5EYY6M1R9RAX", "aliases": [ "op_c847b635f53b0812", "01M2M9FBXJJ9AW5EYY6M1R9RAX", "op-c847b635f53b0812" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 2, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-16T05:01:48.338Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "source-stated", "posed": null, "areaIds": [ "quantum-error-correction", "quantum-algorithm" ], "topicIds": [ "quantum-coding-theory", "quantum-state-preparation", "quantum-circuit-complexity" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [] }, "title": "CNOT-count and depth frontier for quantum Golay state preparation", "status": "Unsolved", "fields": [ "Quantum Error Correction", "Quantum algorithm" ], "topics": [ "Quantum coding theory", "Quantum state preparation", "Quantum circuit complexity" ], "tags": [ "Quantum Error Correction", "Quantum algorithm", "Quantum coding theory", "Quantum state preparation", "Quantum circuit complexity" ], "statement": "What is the exact CNOT-count and depth frontier for ancilla-free Clifford preparation of the logical zero state of the $[[23,1,7]]_2$ quantum Golay code?\n\nLet $C_{23}$ be the classical binary $[23,12,7]$ Golay code and\n\n \\begin{equation}\n|0_G\\rangle=2^{-11/2}\\sum_{c\\in C_{23}^{\\perp}}|c\\rangle.\n\\tag{1}\n\\end{equation} \nConsider Clifford circuits on exactly $23$ qubits initialized in $|0\\rangle^{\\otimes23}$, with arbitrary connectivity, free single-qubit Clifford gates, and unit-cost CNOT gates, but no ancillas, measurements, or postselection. If $N_{\\mathrm{CX}}(U)$ is the CNOT count and $d_2(U)$ the number of disjoint two-qubit layers, define\n\n \\begin{equation}\ng_G(D)=\\min\\left\\{N_{\\mathrm{CX}}(U):\nU|0\\rangle^{\\otimes23}=e^{i\\theta}|0_G\\rangle,\\ d_2(U)\\leq D\\right\\}.\n\\tag{2}\n\\end{equation} \nDetermine $g_G(\\infty)$ and the tradeoff $g_G(D)$ in Eq. (2), particularly for $D=7$ and $D=8$, for the state in Eq. (1).", "url": "https://qiqc-op.com/problem/op_c847b635f53b0812/", "json": "https://qiqc-op.com/api/problems/op_c847b635f53b0812.json", "tex": "https://qiqc-op.com/problem/op_c847b635f53b0812/op_c847b635f53b0812.tex", "created": "2026-09-16", "updated": "2026-09-16", "createdAt": "2026-09-16T05:56:05.000Z", "updatedAt": "2026-09-16T06:51:51.000Z", "sha256": "eb304c0e514bf3cc73240b74b2b14efe140083f04240e8cdc90c5f8b019c06dc" }, { "id": "op_d2e499b973f792e7", "ulid": "01M2M9FAFXNJ6DPB5Y3JNBB9PY", "aliases": [ "op_d2e499b973f792e7", "01M2M9FAFXNJ6DPB5Y3JNBB9PY", "op-d2e499b973f792e7" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 2, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-16T05:01:46.877Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "source-stated", "posed": null, "areaIds": [ "quantum-communication" ], "topicIds": [ "matrix-and-entropy-inequalities" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [ "01M2M9FAJ3CTQ7V1WMFR7X6VD7" ] }, "title": "Quantum erasure-channel upper bound for information combining", "status": "Unsolved", "fields": [ "Quantum Communication" ], "topics": [ "Matrix and entropy inequalities" ], "tags": [ "Quantum Communication", "Matrix and entropy inequalities" ], "statement": "Does the quantum erasure-channel upper information-combining bound hold for two independent uniform bits with arbitrary quantum side information?\n\nLet\n\n \\begin{equation}\n\\rho_{X_iB_i}=\\frac12\\sum_{x=0}^1|x\\rangle\\langle x|\\otimes\\rho_x^{(i)},\n\\qquad\n\\rho_{X_1B_1X_2B_2}=\\rho_{X_1B_1}\\otimes\\rho_{X_2B_2},\n\\tag{1}\n\\end{equation} \nwhere $X_1,X_2$ are uniform bits. Put $Z=X_1\\oplus X_2$ and $s_i=H(X_i\\mid B_i)$, using base-two von Neumann entropy. The proposed upper bound is\n\n \\begin{equation}\nH(Z\\mid B_1B_2)\\leq s_1+s_2-s_1s_2\n\\tag{2}\n\\end{equation} \nDoes Eq. (2) hold for every state in Eq. (1)?", "url": "https://qiqc-op.com/problem/op_d2e499b973f792e7/", "json": "https://qiqc-op.com/api/problems/op_d2e499b973f792e7.json", "tex": "https://qiqc-op.com/problem/op_d2e499b973f792e7/op_d2e499b973f792e7.tex", "created": "2026-09-16", "updated": "2026-09-16", "createdAt": "2026-09-16T05:56:05.000Z", "updatedAt": "2026-09-16T06:51:51.000Z", "sha256": "1af92d07fdd8bd3f8a490f63b34cb0f3b77049002a9f3a3fbf3c7eb90670f38e" }, { "id": "op_d3e38bc5b5b35af3", "ulid": "01M2M9FC8R2FX3MFDQX7HZ7ZPW", "aliases": [ "op_d3e38bc5b5b35af3", "01M2M9FC8R2FX3MFDQX7HZ7ZPW", "op-d3e38bc5b5b35af3" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 2, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-16T05:01:48.696Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "editor-formulated", "posed": null, "areaIds": [ "quantum-metrology" ], "topicIds": [ "quantum-estimation", "one-shot-and-finite-blocklength-bounds" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [] }, "title": "Optimal precision dependence of diamond-norm channel tomography", "status": "Unsolved", "fields": [ "Quantum metrology" ], "topics": [ "Quantum estimation", "One-shot and finite-blocklength bounds" ], "tags": [ "Quantum metrology", "Quantum estimation", "One-shot and finite-blocklength bounds" ], "statement": "Does tomography of an arbitrary $d$-dimensional quantum channel in diamond norm require and suffice with $\\Theta(d^4/\\varepsilon^2)$ channel uses?\n\nLet $\\Lambda:\\mathcal L(\\mathbb C^d)\\to\\mathcal L(\\mathbb C^d)$ be an unknown completely positive, trace-preserving channel with no Kraus-rank promise. Let $Q_\\diamond(d,\\varepsilon)$ be the minimum worst-case number of ordinary channel uses needed to output $\\widehat\\Lambda$ such that\n\n \\begin{equation}\n\\Pr[\\|\\widehat\\Lambda-\\Lambda\\|_\\diamond\\leq\\varepsilon]\\geq\\frac23,\n\\qquad\n\\|\\Phi\\|_\\diamond=\\sup_\\omega\\| (\\Phi\\otimes\\operatorname{id}_d)(\\omega)\\|_1.\n\\tag{1}\n\\end{equation} \nThe supremum in Eq. (1) is over states on $\\mathbb C^d\\otimes\\mathbb C^d$. Adaptive inputs, ancillas, quantum memory, and collective measurements are allowed, but no purification of the channel environment is supplied. Is $Q_\\diamond(d,\\varepsilon)=\\Theta(d^4/\\varepsilon^2)$ uniformly in $d$ and sufficiently small $\\varepsilon$?", "url": "https://qiqc-op.com/problem/op_d3e38bc5b5b35af3/", "json": "https://qiqc-op.com/api/problems/op_d3e38bc5b5b35af3.json", "tex": "https://qiqc-op.com/problem/op_d3e38bc5b5b35af3/op_d3e38bc5b5b35af3.tex", "created": "2026-09-16", "updated": "2026-09-16", "createdAt": "2026-09-16T05:56:05.000Z", "updatedAt": "2026-09-16T06:51:51.000Z", "sha256": "6363b6f9d25e9fe8814a241e59e4767602ce31e8a6526a479bcd1ad1e3222461" }, { "id": "op_dccbfd9860e08b89", "ulid": "01M2M9FADN2Y21FW8M2PS654T2", "aliases": [ "op_dccbfd9860e08b89", "01M2M9FADN2Y21FW8M2PS654T2", "op-dccbfd9860e08b89" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 2, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-16T05:01:46.805Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "source-stated", "posed": null, "areaIds": [ "quantum-algorithm" ], "topicIds": [ "computational-complexity-and-computability", "matrix-and-entropy-inequalities" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [ "01M21S9FW14RJDR83J8D64FKEM" ] }, "title": "Endpoint quantum KKL inequality for Boolean observables", "status": "Unsolved", "fields": [ "Quantum algorithm" ], "topics": [ "Computational complexity and computability", "Matrix and entropy inequalities" ], "tags": [ "Quantum algorithm", "Computational complexity and computability", "Matrix and entropy inequalities" ], "statement": "Does the Montanaro–Osborne $L^2$ quantum KKL inequality hold for every Boolean quantum observable?\n\nLet $A=A^\\dagger$ act on $n\\geq2$ qubits and satisfy $A^2=I$. Write $\\tau(A)=2^{-n}\\operatorname{Tr}A$, let $\\mathcal E_i$ completely depolarize qubit $i$, and define\n\n \\begin{equation}\n\\mathcal E_i(A)=\\frac{I_i}{2}\\otimes\\operatorname{Tr}_iA,\n\\qquad\nD_iA=A-\\mathcal E_i(A),\n\\qquad\n\\operatorname{Inf}^{(2)}_i(A)=\\tau[(D_iA)^\\dagger D_iA].\n\\tag{1}\n\\end{equation} \nDoes a universal constant $c>0$ exist such that every such $A$ satisfies\n\n \\begin{equation}\n\\max_i\\operatorname{Inf}^{(2)}_i(A)\n\\geq c\\bigl(1-\\tau(A)^2\\bigr)\\frac{\\log n}{n}?\n\\tag{2}\n\\end{equation} \nEquation (2) uses the squared influences defined in Eq. (1); the balanced case has $\\tau(A)=0$.", "url": "https://qiqc-op.com/problem/op_dccbfd9860e08b89/", "json": "https://qiqc-op.com/api/problems/op_dccbfd9860e08b89.json", "tex": "https://qiqc-op.com/problem/op_dccbfd9860e08b89/op_dccbfd9860e08b89.tex", "created": "2026-09-16", "updated": "2026-09-16", "createdAt": "2026-09-16T05:56:05.000Z", "updatedAt": "2026-09-16T06:51:51.000Z", "sha256": "2175acaf53ec434aa11e2dbd24c6eb5474c7f7844c9e6cccd7013125489693c1" }, { "id": "op_e8307b66d76af92a", "ulid": "01M2M9FBPVQVBEGZT5RKD2QNNZ", "aliases": [ "op_e8307b66d76af92a", "01M2M9FBPVQVBEGZT5RKD2QNNZ", "op-e8307b66d76af92a" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 2, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-16T05:01:48.123Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "editor-formulated", "posed": null, "areaIds": [ "quantum-cryptography" ], "topicIds": [ "computational-complexity-and-computability", "quantum-state-preparation" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [] }, "title": "One-way state generation from private Hamiltonian phase states", "status": "Unsolved", "fields": [ "Quantum Cryptography" ], "topics": [ "Computational complexity and computability", "Quantum state preparation" ], "tags": [ "Quantum Cryptography", "Computational complexity and computability", "Quantum state preparation" ], "statement": "Do private-architecture Hamiltonian phase states yield a one-way state generator for explicit polynomial parameters?\n\nFor uniformly random $A\\in\\mathbb F_2^{m\\times n}$ and independent uniform phases $\\theta_i\\in\\{2\\pi j/q:0\\leq j0$, and the architecture $A$ in Eq. (1) is hidden. In Eq. (2), the inverter must output a valid description in the same parameter space; invalid outputs fail verification.", "url": "https://qiqc-op.com/problem/op_e8307b66d76af92a/", "json": "https://qiqc-op.com/api/problems/op_e8307b66d76af92a.json", "tex": "https://qiqc-op.com/problem/op_e8307b66d76af92a/op_e8307b66d76af92a.tex", "created": "2026-09-16", "updated": "2026-09-16", "createdAt": "2026-09-16T05:56:05.000Z", "updatedAt": "2026-09-16T06:51:51.000Z", "sha256": "5bda7fc20a6ef569442495c9ace3f164f4ab59960258b99852fe8c07a8ad0274" }, { "id": "op_eff503a05675844e", "ulid": "01M2M9FATYE2RPEKTB8430B1WB", "aliases": [ "op_eff503a05675844e", "01M2M9FATYE2RPEKTB8430B1WB", "op-eff503a05675844e" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 2, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-16T05:01:47.230Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "editor-formulated", "posed": null, "areaIds": [ "quantum-resource-theory" ], "topicIds": [ "quantum-thermodynamics", "quantum-state-preparation" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [] }, "title": "Exact constant-bond PEPS mixtures for two-dimensional Gibbs states", "status": "Unsolved", "fields": [ "Quantum Resource Theory" ], "topics": [ "Quantum thermodynamics", "Quantum state preparation" ], "tags": [ "Quantum Resource Theory", "Quantum thermodynamics", "Quantum state preparation" ], "statement": "Is every fixed-temperature Gibbs state of a bounded finite-range qubit Hamiltonian on the square lattice an exact convex combination of PEPS with system-size-independent bond dimension?\n\nLet $\\Lambda$ be a finite rectangular square lattice and $H=\\sum_Xh_X$ satisfy $\\operatorname{diam}(X)\\leq R$ and $\\max_{v\\in\\Lambda}\\sum_{X\\ni v}\\|h_X\\|\\leq J$. Let $\\mathcal P_\\chi(\\Lambda)$ be the normalized pure PEPS on $\\Lambda$ with bond dimension at most $\\chi$.\n\nFor every fixed $0<\\beta<\\infty$, does a finite $\\chi(\\beta,R,J)$ exist, independent of $|\\Lambda|$ and $H$, such that\n\n \\begin{equation}\n\\frac{e^{-\\beta H}}{\\operatorname{Tr}(e^{-\\beta H})}\n\\in\\operatorname{conv}\\{|\\psi\\rangle\\langle\\psi|:\n\\psi\\in\\mathcal P_{\\chi(\\beta,R,J)}(\\Lambda)\\}?\n\\tag{1}\n\\end{equation} \nEquation (1) asks for exact finite convex membership, with no efficiency requirement.", "url": "https://qiqc-op.com/problem/op_eff503a05675844e/", "json": "https://qiqc-op.com/api/problems/op_eff503a05675844e.json", "tex": "https://qiqc-op.com/problem/op_eff503a05675844e/op_eff503a05675844e.tex", "created": "2026-09-16", "updated": "2026-09-16", "createdAt": "2026-09-16T05:56:05.000Z", "updatedAt": "2026-09-16T06:51:51.000Z", "sha256": "db11bb6a9609c0850f001d1def0bd39bfc3ebd9fce2478ffc455fca4a8f35367" }, { "id": "op_f271040c57d9a513", "ulid": "01M2M9FBVAPKNR5GMT7E2CK83C", "aliases": [ "op_f271040c57d9a513", "01M2M9FBVAPKNR5GMT7E2CK83C", "op-f271040c57d9a513" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 2, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-16T05:01:48.266Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "source-stated", "posed": null, "areaIds": [ "quantum-cryptography" ], "topicIds": [ "computational-complexity-and-computability", "quantum-circuit-complexity" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [] }, "title": "Scalable pseudorandom unitaries with an independent security parameter", "status": "Unsolved", "fields": [ "Quantum Cryptography" ], "topics": [ "Computational complexity and computability", "Quantum circuit complexity" ], "tags": [ "Quantum Cryptography", "Computational complexity and computability", "Quantum circuit complexity" ], "statement": "Can pseudorandom-unitary security scale independently of Hilbert-space dimension while construction uses only polynomially many oracle queries?\n\nFor $d=2^n$ and an independent security parameter $\\kappa$, seek a uniformly generated family $\\{U_{n,\\kappa,f}\\}_f$ indexed by Boolean functions $f:\\{0,1\\}^r\\to\\{0,1\\}$, with $r,q\\leq\\operatorname{poly}(n,\\kappa)$, and an algorithm $\\mathcal A^f$ using at most $q$ oracle queries such that\n\n \\begin{equation}\n\\left\\|\\mathcal A^f-U_{n,\\kappa,f}(\\,\\cdot\\,)U_{n,\\kappa,f}^\\dagger\\right\\|_\\diamond\n\\leq2^{-\\kappa}\n\\tag{1}\n\\end{equation} \nfor every $f$. For uniformly random $f$, require\n\n \\begin{equation}\n\\sup_{\\mathcal D:\\,\\#\\mathrm{queries}\\leq2^\\kappa}\n\\left|\\Pr_f[\\mathcal D^{U_{n,\\kappa,f}}=1]\n-\\Pr_{V\\sim\\operatorname{Haar}(d)}[\\mathcal D^V=1]\\right|\n\\leq2^{-\\kappa}.\n\\tag{2}\n\\end{equation} \nDoes a family satisfying Eqs. (1) and (2) exist with no imposed relation between $n$ and $\\kappa$? The distinguisher accesses the ideal unitary, not the function $f$.", "url": "https://qiqc-op.com/problem/op_f271040c57d9a513/", "json": "https://qiqc-op.com/api/problems/op_f271040c57d9a513.json", "tex": "https://qiqc-op.com/problem/op_f271040c57d9a513/op_f271040c57d9a513.tex", "created": "2026-09-16", "updated": "2026-09-16", "createdAt": "2026-09-16T05:56:05.000Z", "updatedAt": "2026-09-16T06:51:51.000Z", "sha256": "2bd4deaac0a0a3f760b2b2b5289139f1bfc0ba508bb0f9bcbf974ccc57951493" }, { "id": "op_a11fdfadc84e4379", "ulid": "01M21S9FW14RJDR83J8D64FKEM", "aliases": [ "op_a11fdfadc84e4379", "01M21S9FW14RJDR83J8D64FKEM", "op-a11fdfadc84e4379" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 2, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-09T00:32:38.785Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "source-stated", "posed": null, "areaIds": [ "quantum-algorithm" ], "topicIds": [ "computational-complexity-and-computability" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [] }, "title": "Aaronson–Ambainis conjecture", "status": "Unsolved", "fields": [ "Quantum algorithm" ], "topics": [ "Computational complexity and computability" ], "tags": [ "Quantum algorithm", "Computational complexity and computability" ], "statement": "Must every bounded low-degree real polynomial on the Boolean cube with non-negligible variance have a variable of inverse-polynomial influence?\n\nLet $p:\\mathbb R^N\\to\\mathbb R$ be a real multilinear polynomial of degree at most $d$ satisfying\n\n \\begin{equation}\n 0\\leq p(X)\\leq 1\n \\qquad\n \\text{for every }X\\in\\{0,1\\}^N .\n\\tag{1}\n\\end{equation} \nFor $i\\in[N]$, let $X^{(i)}$ denote $X$ with its $i$-th bit flipped, and define the influence of the $i$-th variable by\n\n \\begin{equation}\n \\operatorname{Inf}_i[p]\n =\n \\mathbb E_{X\\in\\{0,1\\}^N}\n \\left[\n \\left(\n p(X)-p(X^{(i)})\n \\right)^2\n \\right].\n\\tag{2}\n\\end{equation} \nDefine the variance of $p$ on the uniform Boolean cube by\n\n \\begin{equation}\n \\operatorname{Var}[p]\n =\n \\mathbb E_{X\\in\\{0,1\\}^N}\n \\left[\n \\left(\n p(X)-\\mathbb E[p]\n \\right)^2\n \\right].\n\\tag{3}\n\\end{equation} \nSuppose that for some $\\varepsilon>0$,\n\n \\begin{equation}\n \\operatorname{Var}[p]\\geq\\varepsilon .\n\\tag{4}\n\\end{equation} \nThe Aaronson–Ambainis conjecture asks whether there is a universal constant $C>0$ such that every polynomial satisfying (1) and (4) has some coordinate $i\\in[N]$ whose influence, as defined in (2), satisfies\n\n \\begin{equation}\n \\operatorname{Inf}_i[p]\n \\geq\n \\left(\\frac{\\varepsilon}{d}\\right)^C .\n\\tag{5}\n\\end{equation} \nEquivalently, the conjecture asks whether the maximum influence can always be bounded below by a fixed polynomial in $1/d$ and the variance (3), independently of the ambient dimension $N$.", "url": "https://qiqc-op.com/problem/op_a11fdfadc84e4379/", "json": "https://qiqc-op.com/api/problems/op_a11fdfadc84e4379.json", "tex": "https://qiqc-op.com/problem/op_a11fdfadc84e4379/op_a11fdfadc84e4379.tex", "created": "2026-09-09", "updated": "2026-09-16", "createdAt": "2026-09-09T00:35:07.000Z", "updatedAt": "2026-09-16T05:56:05.000Z", "sha256": "b71630f696fc7098916fd1e8bbc86afdaf43ba75ebb224b30ed318c0330ff3e1" }, { "id": "op_94fc1874fe23df16", "ulid": "01M20CXWDYD1RXVWDKFMFM675K", "aliases": [ "op_94fc1874fe23df16", "01M20CXWDYD1RXVWDKFMFM675K", "op-94fc1874fe23df16" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 2, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-08T11:37:21.086Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "source-stated", "posed": null, "areaIds": [ "quantum-algorithm" ], "topicIds": [ "computational-complexity-and-computability", "quantum-supremacy", "random-circuit-sampling" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [] }, "title": "Average-case approximation hardness of random-circuit output probabilities", "status": "Unsolved", "fields": [ "Quantum algorithm" ], "topics": [ "Computational complexity and computability", "Quantum supremacy", "Random circuit sampling" ], "tags": [ "Quantum algorithm", "Computational complexity and computability", "Quantum supremacy", "Random circuit sampling" ], "statement": "Does there exist a fixed family of $n$-qubit circuit layouts with $m=\\operatorname{poly}(n)$ one- and two-qubit gates for which the following task is $\\#\\mathrm{P}$-hard? Choose every gate independently from Haar measure, obtaining $C$. Gate entries use the standard sufficiently accurate finite-precision encoding of the source formulation. Given $C$ and $\\epsilon,\\delta\\in(0,1)$, estimate the all-zero output probability with\n\n \\begin{equation}\n p_0(C)=|\\langle0^n|C|0^n\\rangle|^2,\\qquad\n \\Pr_{C,\\mathcal A}[|\\widetilde p(C)-p_0(C)|\\leq\\epsilon 2^{-n}]\\geq1-\\delta.\n\\tag{1}\n\\end{equation} \nThe probability includes the estimator’s internal randomness. The guarantee in Eq. (1) must hold for arbitrary $\\epsilon,\\delta$, with running time measured in $n,1/\\epsilon,1/\\delta$.", "url": "https://qiqc-op.com/problem/op_94fc1874fe23df16/", "json": "https://qiqc-op.com/api/problems/op_94fc1874fe23df16.json", "tex": "https://qiqc-op.com/problem/op_94fc1874fe23df16/op_94fc1874fe23df16.tex", "created": "2026-09-08", "updated": "2026-09-16", "createdAt": "2026-09-08T12:16:40.000Z", "updatedAt": "2026-09-16T05:56:05.000Z", "sha256": "6136a39fd0e5494792ba0e46b96dc0e8c4d1a3e53b44d94b836b752c87d732de" }, { "id": "op_0c17d9fef967858d", "ulid": "01M20EG8DVMFZX9TWEXN602APF", "aliases": [ "op_0c17d9fef967858d", "01M20EG8DVMFZX9TWEXN602APF", "op-0c17d9fef967858d" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 2, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-08T12:04:51.771Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "editor-formulated", "posed": null, "areaIds": [ "quantum-algorithm" ], "topicIds": [ "computational-complexity-and-computability" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [] }, "title": "Quantum query complexity of Triangle Finding", "status": "Unsolved", "fields": [ "Quantum algorithm" ], "topics": [ "Computational complexity and computability" ], "tags": [ "Quantum algorithm", "Computational complexity and computability" ], "statement": "What is the bounded-error quantum query complexity of finding a triangle in an $n$-vertex graph given oracle access to its adjacency matrix?\n\nLet $G=(V,E)$ be a simple undirected graph with $V=[n]$, and let $A\\in\\{0,1\\}^{n\\times n}$ be its adjacency matrix. The input is accessed through a quantum oracle acting as\n\n \\begin{equation}\n O_A\\lvert u,v,z\\rangle\n =\\lvert u,v,z\\oplus A_{uv}\\rangle .\n\\tag{1}\n\\end{equation} \nThe task is to output three distinct vertices $u,v,w\\in[n]$ satisfying\n\n \\begin{equation}\n A_{uv}=A_{vw}=A_{wu}=1,\n\\tag{2}\n\\end{equation} \nif such vertices exist, and otherwise report that the graph is triangle-free. Let $Q_{\\triangle}(n)$ denote the minimum number of queries to the oracle in (1) required by a bounded-error quantum algorithm for this task.\n\nThe best known general bounds are\n\n \\begin{equation}\n Q_{\\triangle}(n)=O\\left(n^{5/4}\\right)\n \\qquad\\text{and}\\qquad\n Q_{\\triangle}(n)=\\Omega(n).\n\\tag{3}\n\\end{equation} \nThe open question is to determine the asymptotic behavior of $Q_{\\triangle}(n)$, equivalently to improve either the upper or lower bound in (3) until the bounds match.", "url": "https://qiqc-op.com/problem/op_0c17d9fef967858d/", "json": "https://qiqc-op.com/api/problems/op_0c17d9fef967858d.json", "tex": "https://qiqc-op.com/problem/op_0c17d9fef967858d/op_0c17d9fef967858d.tex", "created": "2026-09-08", "updated": "2026-09-16", "createdAt": "2026-09-08T12:10:01.000Z", "updatedAt": "2026-09-16T05:56:05.000Z", "sha256": "4c65eef4e14eb57d8242ad4c9e8fbcef3d066a6db5a604345370c21fdaadb0af" }, { "id": "op_37f7b003ec78a028", "ulid": "01M208CJ9MKVF4X0B83M9V7QZM", "aliases": [ "op_37f7b003ec78a028", "01M208CJ9MKVF4X0B83M9V7QZM", "op-37f7b003ec78a028" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 2, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-08T10:17:59.348Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "source-stated", "posed": "2016", "areaIds": [ "quantum-algorithm" ], "topicIds": [ "quantum-magic", "quantum-circuit-complexity", "computational-complexity-and-computability" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [ "01M1Q787QRD6APNHX659G4CTEF" ] }, "title": "Asymptotic growth of the stabilizer rank of T-state tensor powers", "status": "Unsolved", "fields": [ "Quantum algorithm" ], "topics": [ "Quantum magic", "Quantum circuit complexity", "Computational complexity and computability" ], "tags": [ "Quantum algorithm", "Quantum magic", "Quantum circuit complexity", "Computational complexity and computability" ], "statement": "Does the exact stabilizer rank of the tensor powers of the single-qubit magic state $|T\\rangle$ grow polynomially in the number of copies, or is it not polynomially bounded? The state is\n\n \\begin{equation}\n |T\\rangle := 2^{-1/2}\\bigl(|0\\rangle + e^{i\\pi/4}|1\\rangle\\bigr),\n\\tag{1}\n\\end{equation} \nand, for an $n$-qubit pure state $|\\psi\\rangle$, its exact stabilizer rank is\n\n \\begin{equation}\n \\chi(\\psi) := \\min\\Bigl\\{r : |\\psi\\rangle = \\sum_{j=1}^{r} c_j\\,|\\varphi_j\\rangle\n \\text{ for some } c_j \\in \\mathbb{C} \\text{ and } n\\text{-qubit stabilizer states }\n |\\varphi_j\\rangle\\Bigr\\},\n\\tag{2}\n\\end{equation} \nwhere an $n$-qubit stabilizer state is the common $+1$ eigenstate of $n$ independent commuting Hermitian Pauli operators. For a fixed approximation error $0<\\delta<1$, also define\n\n \\begin{equation}\n \\chi_\\delta(\\psi):=\\min\\left\\{\\chi(\\phi):\n \\left\\lVert |\\phi\\rangle-|\\psi\\rangle\\right\\rVert_2\\leq\\delta\\right\\},\n\\tag{3}\n\\end{equation} \nwhere the approximating vector need not be normalized. A second unresolved part of the problem is whether constant-error approximate rank is exponential: for example, do constants $c>0$ and $n_0$ exist such that\n\n \\begin{equation}\n \\chi_{0.01}\\bigl(|T\\rangle^{\\otimes n}\\bigr)\\geq 2^{cn}\n \\qquad\\text{for every }n\\geq n_0?\n\\tag{4}\n\\end{equation} \nEven an unconditional super-polynomial lower bound would be a major advance. This approximate-rank target is recorded with the exact-rank question because the two concern the same state family and decomposition measure; neither one currently has a super-polynomial lower bound [MT24][KS26].\n\nWhich of the two alternatives holds for $\\chi\\bigl(|T\\rangle^{\\otimes n}\\bigr)$: does there exist a constant $k$ with $\\chi\\bigl(|T\\rangle^{\\otimes n}\\bigr) \\le n^{k}$ for all sufficiently large $n$, or is $\\chi\\bigl(|T\\rangle^{\\otimes n}\\bigr)$ not bounded by any polynomial in $n$? The best current bounds are\n\n \\begin{equation}\n \\Omega\\!\\left(\\frac{n^{2}}{\\operatorname{polylog} n}\\right) \\le\n \\chi\\bigl(|T\\rangle^{\\otimes n}\\bigr) \\le O\\bigl(2^{\\alpha n}\\bigr),\n \\qquad \\alpha = \\tfrac{1}{4}\\log_{2} 3 \\le 0.3963,\n\\tag{5}\n\\end{equation} \nwhich leave both alternatives open. A sharper sub-question is whether the exponential rate \\(\\gamma = \\lim_{n\\to\\infty} \\frac{1}{n}\\log_{2}\n\\chi\\bigl(|T\\rangle^{\\otimes n}\\bigr)\\), which exists by sub-multiplicativity of the rank under tensor product and Fekete’s lemma, is strictly positive, since Eq. (5) only pins $\\gamma$ to the interval $[0, 0.3963]$. The small-copy frontier of the quantity in Eq. (2) on copies of the state in Eq. (1) is also open: the exact values are known at two, three, and four copies, while at five through seven copies only the following upper bounds are known, $\\chi\\bigl(|T\\rangle^{\\otimes 5}\\bigr) \\le 6$, $\\chi\\bigl(|T\\rangle^{\\otimes 6}\\bigr) \\le 6$, and $\\chi\\bigl(|T\\rangle^{\\otimes 7}\\bigr) \\le 12$; determine the exact values at $n = 5$, $6$, and $7$.", "url": "https://qiqc-op.com/problem/op_37f7b003ec78a028/", "json": "https://qiqc-op.com/api/problems/op_37f7b003ec78a028.json", "tex": "https://qiqc-op.com/problem/op_37f7b003ec78a028/op_37f7b003ec78a028.tex", "created": "2026-09-08", "updated": "2026-09-16", "createdAt": "2026-09-08T10:20:59.000Z", "updatedAt": "2026-09-16T05:56:05.000Z", "sha256": "56d90b8885d148dea536b46f36b1c0023158675a9e3ab4773f142485e2b7e0db" }, { "id": "op_64e58085fa92c1a8", "ulid": "01M27CPFREERH7V1EX15KMGMM9", "aliases": [ "op_64e58085fa92c1a8", "01M27CPFREERH7V1EX15KMGMM9", "op-64e58085fa92c1a8" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 1, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-11T04:47:59.758Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "derived", "posed": null, "areaIds": [ "quantum-resource-theory" ], "topicIds": [ "entanglement-measures", "bell-diagonal-states" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [] }, "title": "Squashed entanglement of qubit Bell-diagonal states", "status": "Unsolved", "fields": [ "Quantum Resource Theory" ], "topics": [ "Entanglement measures", "Bell-diagonal states" ], "tags": [ "Quantum Resource Theory", "Entanglement measures", "Bell-diagonal states" ], "statement": "Is there a closed-form formula for the squashed entanglement $E_{\\mathrm{sq}}$ of an arbitrary two-qubit Bell-diagonal state, or a certified approach to determining it numerically? Let $A$ and $B$ be qubits, let $I,X,Y,Z$ be the identity and Pauli matrices, and set $\\lvert\\Phi^+\\rangle:=(\\lvert00\\rangle+\\lvert11\\rangle)/\\sqrt2$ and $\\lvert\\Phi_P\\rangle:=(I\\otimes P)\\lvert\\Phi^+\\rangle$ for $P\\in\\{I,X,Y,Z\\}$. The states under consideration are\n\n \\begin{equation}\n \\rho_{\\mathbf p}:=\\sum_{P\\in\\{I,X,Y,Z\\}}p_P\n \\lvert\\Phi_P\\rangle\\!\\langle\\Phi_P\\rvert,\n \\qquad p_P\\geq0,\\qquad \\sum_Pp_P=1.\n\\tag{1}\n\\end{equation} \nWith $S(\\tau):=-\\operatorname{Tr}(\\tau\\log_2\\tau)$, define the squashed entanglement of the state in Eq. (1) by\n\n \\begin{equation}\n \\begin{aligned}\n E_{\\mathrm{sq}}(\\rho_{\\mathbf p})\n &:=\\frac12\\inf_{\\substack{\\omega_{ABE}\\geq0,\\ \\operatorname{Tr}\\omega_{ABE}=1\\\\\n \\operatorname{Tr}_E\\omega_{ABE}=\\rho_{\\mathbf p}}}\n I(A:B\\mid E)_\\omega,\\\\\n I(A:B\\mid E)_\\omega\n &:=S(\\omega_{AE})+S(\\omega_{BE})-S(\\omega_E)-S(\\omega_{ABE}).\n \\end{aligned}\n\\tag{2}\n\\end{equation} \nThe infimum in Eq. (2) ranges over all finite-dimensional quantum systems $E$, with no fixed bound on their dimension. Values are in ebits; this is the quantum squashed entanglement of [CW04].\n\nA certified numerical approach should, for any specified probabilities $\\mathbf p$ and tolerance $\\varepsilon>0$, produce an estimate of $E_{\\mathrm{sq}}(\\rho_{\\mathbf p})$ with a rigorously guaranteed absolute error at most $\\varepsilon$. Restricting $E$ to a classical register, or to a chosen finite dimension, suffices only if the restriction is proved optimal or its approximation error is controlled.", "url": "https://qiqc-op.com/problem/op_64e58085fa92c1a8/", "json": "https://qiqc-op.com/api/problems/op_64e58085fa92c1a8.json", "tex": "https://qiqc-op.com/problem/op_64e58085fa92c1a8/op_64e58085fa92c1a8.tex", "created": "2026-09-11", "updated": "2026-09-16", "createdAt": "2026-09-11T09:33:04.000Z", "updatedAt": "2026-09-16T01:49:51.000Z", "sha256": "a0d1569521997377562f6bdb48743e44f68b652ebdb8f78fcac8a7737fd47031" }, { "id": "op_8682c870bea8ab4e", "ulid": "01M26JZEVQ8NQEB21FMPX569QJ", "aliases": [ "op_8682c870bea8ab4e", "01M26JZEVQ8NQEB21FMPX569QJ", "op-8682c870bea8ab4e" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 1, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-10T21:18:30.775Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "source-stated", "posed": null, "areaIds": [ "quantum-communication" ], "topicIds": [ "matrix-and-entropy-inequalities", "gaussian-quantum-information" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [] }, "title": "Wigner entropy conjecture", "status": "Solved", "fields": [ "Quantum Communication" ], "topics": [ "Matrix and entropy inequalities", "Gaussian quantum information" ], "tags": [ "Quantum Communication", "Matrix and entropy inequalities", "Gaussian quantum information" ], "statement": "Does every finite-energy single-mode state with a nonnegative Wigner function have Wigner entropy at least $1+\\ln\\pi$? Let $\\rho$ be a density operator with $\\operatorname{Tr}(\\rho a^\\dagger a)<\\infty$, where $a=(q+ip)/\\sqrt2$ and $[q,p]=i$. Require $W_\\rho(q,p)\\geq0$. Use the normalization $\\int_{\\mathbb R^2}W_\\rho(q,p)\\,dq\\,dp=1$ and vacuum convention $W_{|0\\rangle}(q,p)=\\pi^{-1}e^{-q^2-p^2}$. With natural logarithms and $0\\ln0=0$, the proposed bound is\n\n \\begin{equation}\n h_W(\\rho):=-\\int_{\\mathbb R^2}W_\\rho(q,p)\\ln W_\\rho(q,p)\\,dq\\,dp\n \\geq1+\\ln\\pi.\n\\tag{1}\n\\end{equation} \nProve Eq. (1) for all such physical states, or exhibit a density operator violating it.", "url": "https://qiqc-op.com/problem/op_8682c870bea8ab4e/", "json": "https://qiqc-op.com/api/problems/op_8682c870bea8ab4e.json", "tex": "https://qiqc-op.com/problem/op_8682c870bea8ab4e/op_8682c870bea8ab4e.tex", "created": "2026-09-10", "updated": "2026-09-16", "createdAt": "2026-09-10T21:54:15.000Z", "updatedAt": "2026-09-16T01:47:46.000Z", "sha256": "04fd64bd8da054e850aec09783ed54a23e0c876e639209bbe16c11319a0a96e3" }, { "id": "op_d754e0d170c01f86", "ulid": "01M1Q787QR780435R6682GE26Y", "aliases": [ "op_d754e0d170c01f86", "01M1Q787QR780435R6682GE26Y", "op-d754e0d170c01f86" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 1, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-04T22:04:59Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "source-stated", "posed": null, "areaIds": [ "quantum-communication" ], "topicIds": [ "quantum-relative-entropy", "one-shot-and-finite-blocklength-bounds" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [] }, "title": "Diamond-smoothed max-relative-entropy AEP for quantum channels", "status": "Unsolved", "fields": [ "Quantum Communication" ], "topics": [ "Quantum relative entropy", "One-shot and finite-blocklength bounds" ], "tags": [ "Quantum Communication", "Quantum relative entropy", "One-shot and finite-blocklength bounds" ], "statement": "Does the max-relative entropy of finite-dimensional quantum channels satisfy an asymptotic equipartition property under uniform diamond-norm smoothing? Let $\\mathcal N,\\mathcal M:\\mathcal L(A)\\to\\mathcal L(B)$ be quantum channels with $D_{\\max}(\\mathcal N\\|\\mathcal M)<\\infty$. Using unnormalized Choi operators, define the channel max-relative entropy and its smoothed version by\n\n \\begin{equation}\n \\begin{aligned}\n D_{\\max}(\\mathcal N\\|\\mathcal M)\n &:=\\inf\\{\\lambda:J_{\\mathcal N}\\leq2^\\lambda J_{\\mathcal M}\\},\\\\\n D_{\\max}^{\\varepsilon}(\\mathcal N\\|\\mathcal M)\n &:=\\inf_{\\substack{\\widetilde{\\mathcal N}\\ {\\rm CPTP}:\\\\\n \\frac12\\|\\widetilde{\\mathcal N}-\\mathcal N\\|_\\diamond\n \\leq\\varepsilon}}\n D_{\\max}(\\widetilde{\\mathcal N}\\|\\mathcal M).\n \\end{aligned}\n\\tag{1}\n\\end{equation} \nThe smoothing in Eq. (1) requires one channel $\\widetilde{\\mathcal N}$ that approximates $\\mathcal N$ uniformly over all ancilla-assisted inputs. Define\n\n \\begin{equation}\n \\begin{aligned}\n D_{\\max}^{\\varepsilon,\\infty}(\\mathcal N\\|\\mathcal M)\n &:=\\limsup_{n\\to\\infty}\\frac1n\n D_{\\max}^{\\varepsilon}\n (\\mathcal N^{\\otimes n}\\|\\mathcal M^{\\otimes n}),\\\\\n D_{\\rm ch}^{\\infty}(\\mathcal N\\|\\mathcal M)\n &:=\\lim_{n\\to\\infty}\\frac1n\n \\sup_{\\psi_{R A^n}}\n D\\!\\left((\\operatorname{id}_R\\otimes\\mathcal N^{\\otimes n})(\\psi)\n \\middle\\|\n (\\operatorname{id}_R\\otimes\\mathcal M^{\\otimes n})(\\psi)\\right),\n \\end{aligned}\n\\tag{2}\n\\end{equation} \nwhere $R\\simeq A^{\\otimes n}$ suffices and $D$ is quantum relative entropy. Is the following identity valid for every such channel pair, and can the $\\limsup$ in Eq. (2) be replaced by a limit?\n\n \\begin{equation}\n \\sup_{\\varepsilon>0}\n D_{\\max}^{\\varepsilon,\\infty}(\\mathcal N\\|\\mathcal M)\n =D_{\\rm ch}^{\\infty}(\\mathcal N\\|\\mathcal M).\n\\tag{3}\n\\end{equation}", "url": "https://qiqc-op.com/problem/op_d754e0d170c01f86/", "json": "https://qiqc-op.com/api/problems/op_d754e0d170c01f86.json", "tex": "https://qiqc-op.com/problem/op_d754e0d170c01f86/op_d754e0d170c01f86.tex", "created": "2026-09-03", "updated": "2026-09-16", "createdAt": "2026-09-03T02:22:57.000Z", "updatedAt": "2026-09-16T01:44:56.000Z", "sha256": "4ab9ffe219518f5f935c49af046be7cfd8983c6d73eab8d80c6d2a6819f43038" }, { "id": "op_08387c140f552732", "ulid": "01M1HME7803DZWKPJRHYYKHX0C", "aliases": [ "op_08387c140f552732", "01M1HME7803DZWKPJRHYYKHX0C", "op-08387c140f552732", "v2-fixed-error-parallel-stein-lemma-for-quantum-channels", "open-problem-v2-problem-46" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 2, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-04T22:04:59Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "source-stated", "posed": null, "areaIds": [ "quantum-metrology", "quantum-communication" ], "topicIds": [ "channel-discrimination", "quantum-hypothesis-testing", "quantum-relative-entropy", "strong-converses" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [] }, "title": "Fixed-error parallel Stein lemma for quantum channels", "status": "Unsolved", "fields": [ "Quantum metrology", "Quantum Communication" ], "topics": [ "Channel discrimination", "Quantum hypothesis testing", "Quantum relative entropy", "Strong converses" ], "tags": [ "Quantum metrology", "Quantum Communication", "Channel discrimination", "Quantum hypothesis testing", "Quantum relative entropy", "Strong converses" ], "statement": "Let $\\mathcal N,\\mathcal M:\\mathcal L(A)\\to\\mathcal L(B)$ be quantum channels on finite-dimensional systems. For states $\\rho$ and $\\sigma$, define the relative entropy and the hypothesis-testing divergence by\n\n \\begin{equation}\n \\begin{aligned}\n D(\\rho\\|\\sigma)\n &:={\\rm Tr}\\!\\left[\\rho(\\log_2\\rho-\\log_2\\sigma)\\right],\\\\\n D_H^\\varepsilon(\\rho\\|\\sigma)\n &:=-\\log_2\\inf_{\\substack{0\\leq Q\\leq I\\\\\n {\\rm Tr}(Q\\rho)\\geq1-\\varepsilon}}\n {\\rm Tr}(Q\\sigma),\n \\end{aligned}\n\\tag{1}\n\\end{equation} \nwhere $D(\\rho\\|\\sigma)=+\\infty$ unless $\\operatorname{supp}\\rho\\subseteq\\operatorname{supp}\\sigma$, and $\\varepsilon\\in(0,1)$. For either divergence $\\mathbf D$ in Eq. (1), its stabilized channel extension is\n\n \\begin{equation}\n \\mathbf D_{\\rm ch}(\\mathcal N\\|\\mathcal M)\n :=\\sup_{\\psi_{RA}\\in\\mathcal D(R\\otimes A)}\n \\mathbf D\\!\\left(\n (\\operatorname{id}_R\\otimes\\mathcal N)(\\psi)\n \\middle\\|\n (\\operatorname{id}_R\\otimes\\mathcal M)(\\psi)\n \\right),\n \\qquad R\\simeq A.\n\\tag{2}\n\\end{equation} \nUsing Eq. (2), define the regularized channel relative entropy by\n\n \\begin{equation}\n D_{\\rm ch}^{\\infty}(\\mathcal N\\|\\mathcal M)\n :=\\lim_{n\\to\\infty}\\frac1n\n D_{\\rm ch}(\\mathcal N^{\\otimes n}\\|\\mathcal M^{\\otimes n}).\n\\tag{3}\n\\end{equation} \nWhenever Eq. (3) is finite, does the fixed-error parallel Stein limit exist and satisfy\n\n \\begin{equation}\n \\lim_{n\\to\\infty}\\frac1n\n D_{H,{\\rm ch}}^\\varepsilon\n (\\mathcal N^{\\otimes n}\\|\\mathcal M^{\\otimes n})\n =D_{\\rm ch}^{\\infty}(\\mathcal N\\|\\mathcal M)\n \\qquad\\text{for every }\\varepsilon\\in(0,1)?\n\\tag{4}\n\\end{equation}", "url": "https://qiqc-op.com/problem/op_08387c140f552732/", "json": "https://qiqc-op.com/api/problems/op_08387c140f552732.json", "tex": "https://qiqc-op.com/problem/op_08387c140f552732/op_08387c140f552732.tex", "created": "2026-09-01", "updated": "2026-09-16", "createdAt": "2026-09-01T05:13:32.000Z", "updatedAt": "2026-09-16T01:44:56.000Z", "sha256": "499a33df51228ae8d963f2e163afb1d595fffbcfe09b5ed9a41fa7fe5bbce6c7" }, { "id": "op_b08ad9d4371ed0cb", "ulid": "01M1HME780THG7M7MWH1AKVSRS", "aliases": [ "op_b08ad9d4371ed0cb", "01M1HME780THG7M7MWH1AKVSRS", "op-b08ad9d4371ed0cb", "v2-existence-of-an-eight-ququart-perfect-tensor", "open-problem-v2-problem-40" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 2, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-04T22:04:59Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "source-stated", "posed": null, "areaIds": [ "quantum-resource-theory", "quantum-error-correction" ], "topicIds": [ "absolutely-maximally-entangled-states", "quantum-coding-theory" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [] }, "title": "Eight-ququart AME state and the ((7,4,4)) ququart MDS code", "status": "Unsolved", "fields": [ "Quantum Resource Theory", "Quantum Error Correction" ], "topics": [ "Absolutely maximally entangled states", "Quantum coding theory" ], "tags": [ "Quantum Resource Theory", "Quantum Error Correction", "Absolutely maximally entangled states", "Quantum coding theory" ], "statement": "Does an absolutely maximally entangled state of eight ququarts exist? More precisely, with $[8]=\\{1,\\ldots,8\\}$, determine whether there is a unit vector $|\\psi\\rangle\\in(\\mathbb{C}^{4})^{\\otimes 8}$ such that\n\n \\begin{equation}\n \\operatorname{Tr}_{S^{c}}\\!\\left(|\\psi\\rangle\\!\\langle\\psi|\\right)\n =\\frac{I_{4^{4}}}{4^{4}}\n \\qquad\\text{for every }S\\subseteq[8]\\text{ with }|S|=4.\n\\tag{1}\n\\end{equation} \nCondition (1) is equivalently the existence of a rank-eight perfect tensor of bond dimension $4$: every flattening across a $4|4$ partition is proportional to a unitary [PYHP15]. Under the pure-code correspondence, it is also equivalent to a pure quantum MDS code with parameters $\\bigl[\\!\\bigl[8,0,5\\bigr]\\!\\bigr]_{4}$ [SZZL26]. Equivalently, the question asks whether there is a quantum maximum-distance-separable code with parameters $((7,4,4))_{4}$: a four-dimensional subspace $\\mathcal{Q}\\subseteq(\\mathbb{C}^{4})^{\\otimes 7}$ whose orthogonal projector $P$ satisfies\n\n \\begin{equation}\n PEP=c(E)\\,P\n \\quad\\text{for every operator }E\\text{ acting nontrivially on at most three ququarts},\n \\qquad\n \\dim\\mathcal{Q}=4=4^{\\,7-2(4-1)},\n\\tag{2}\n\\end{equation} \nwhere $c(E)\\in\\mathbb{C}$ and the last equality is saturation of the quantum Singleton bound [HG20].", "url": "https://qiqc-op.com/problem/op_b08ad9d4371ed0cb/", "json": "https://qiqc-op.com/api/problems/op_b08ad9d4371ed0cb.json", "tex": "https://qiqc-op.com/problem/op_b08ad9d4371ed0cb/op_b08ad9d4371ed0cb.tex", "created": "2026-09-01", "updated": "2026-09-15", "createdAt": "2026-09-01T05:13:32.000Z", "updatedAt": "2026-09-15T18:47:45.000Z", "sha256": "90124986a560c52bbbbfa0d94f51f23b4393de8668b16d597766532a10e41226" }, { "id": "op_0c091a6b5a55a8fd", "ulid": "01M2JD0QAYZMS56Z6TAHM15ERA", "aliases": [ "op_0c091a6b5a55a8fd", "01M2JD0QAYZMS56Z6TAHM15ERA", "op-0c091a6b5a55a8fd" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 1, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-15T11:25:13.950Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "source-stated", "posed": null, "areaIds": [ "quantum-error-correction" ], "topicIds": [ "quantum-ldpc-codes", "quantum-coding-theory" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [ "01M2JD0QDEDZ78QV24WQ3VS9BN" ] }, "title": "Asymptotically good quantum locally testable codes", "status": "Unsolved", "fields": [ "Quantum Error Correction" ], "topics": [ "Quantum LDPC codes", "Quantum coding theory" ], "tags": [ "Quantum Error Correction", "Quantum LDPC codes", "Quantum coding theory" ], "statement": "Does there exist an infinite family of qubit CSS codes with linear rate and linear distance whose check matrices have uniformly bounded row and column weights and constant soundness? Consider a family of qubit CSS codes $[[n,k_n,d_n]]$ with $n\\to\\infty$. Each code is specified by binary check matrices $H_X\\in\\mathbb{F}_2^{m_X\\times n}$ and $H_Z\\in\\mathbb{F}_2^{m_Z\\times n}$ with $m_X,m_Z>0$ and $H_XH_Z^{\\mathsf T}=0$. Write $|\\cdot|$ for Hamming weight.\n\nCall the family good and locally testable if there are constants $r,\\delta,s>0$ and $w\\in\\mathbb{N}$, independent of $n$, such that every code in the family satisfies three conditions. First, every row and every column of $H_X$ and $H_Z$ has weight at most $w$. Second, the rate and distance are linear:\n\n \\begin{equation}\n k_n\\ge rn,\\qquad d_n\\ge\\delta n.\n\\tag{1}\n\\end{equation} \nThird, both check matrices have constant soundness:\n\n \\begin{equation}\n \\frac{|H_\\sigma e|}{m_\\sigma}\n \\ge\n s\\,\\frac{\\min_{c\\in\\ker H_\\sigma}|e-c|}{n}\n \\qquad\n \\text{for every }\\sigma\\in\\{X,Z\\}\\text{ and }e\\in\\mathbb{F}_2^n.\n\\tag{2}\n\\end{equation} \nThe question is whether some family satisfies the weight bound, Eq. (1), and Eq. (2) simultaneously.", "url": "https://qiqc-op.com/problem/op_0c091a6b5a55a8fd/", "json": "https://qiqc-op.com/api/problems/op_0c091a6b5a55a8fd.json", "tex": "https://qiqc-op.com/problem/op_0c091a6b5a55a8fd/op_0c091a6b5a55a8fd.tex", "created": "2026-09-15", "updated": "2026-09-15", "createdAt": "2026-09-15T18:46:16.000Z", "updatedAt": "2026-09-15T18:46:16.000Z", "sha256": "564a25d4bbd7fe7096b42991ef93756f5c8883673c1e9bba741c20d1ad9b9f4b" }, { "id": "op_0c23167deb384894", "ulid": "01M2JDG8A02AY84NNNVZ52DD61", "aliases": [ "op_0c23167deb384894", "01M2JDG8A02AY84NNNVZ52DD61", "op-0c23167deb384894" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 1, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-15T11:33:42.848Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "source-stated", "posed": null, "areaIds": [ "quantum-resource-theory" ], "topicIds": [ "entanglement-cost", "local-operations-and-classical-communication" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [ "01M1HME780JH0D9Y0RQ750ZZPY", "01M2JDG8GMD6A9Z88MC2195VA6" ] }, "title": "LOCC entanglement cost of Werner states", "status": "Unsolved", "fields": [ "Quantum Resource Theory" ], "topics": [ "Entanglement cost", "Local operations and classical communication" ], "tags": [ "Quantum Resource Theory", "Entanglement cost", "Local operations and classical communication" ], "statement": "What is the entanglement cost of the Werner state $\\rho_{d,p}$ for every integer $d\\geq2$ and every antisymmetric weight $1/21/2$. The entanglement cost $E_C(\\rho)$ of a bipartite state $\\rho$ is the infimum of the rates $R$ for which local operations and classical communication (LOCC) transform $\\lceil nR\\rceil$ ebits into states whose trace distance from $\\rho^{\\otimes n}$ tends to zero as $n\\to\\infty$. Write $E_F$ for the entanglement of formation and $h_2(x):=-x\\log_2x-(1-x)\\log_2(1-x)$. For $1/20$ and $w\\in\\mathbb N$, and triples, with $n\\to\\infty$, of qubit CSS codes $\\mathcal Q_{a,n}$ with parameters $[[n,k_{a,n},d_{a,n}]]$, indexed by $a\\in\\{1,2,3\\}$, such that each code has $k_{a,n}\\ge1$ and $X$- and $Z$-check matrices whose row weights (check weights) and column weights (qubit degrees) are at most $w$,\n\n \\begin{equation}\n \\sum_{a=1}^{3}k_{a,n}\\ge rn,\\qquad\n \\min_{a\\in\\{1,2,3\\}}d_{a,n}\\ge\\delta n,\n\\tag{1}\n\\end{equation} \nand the transversal gate\n\n \\begin{equation}\n U_n:=\\prod_{j=1}^{n}\\operatorname{CCZ}_{(1,j),(2,j),(3,j)},\n \\qquad\n \\operatorname{CCZ}|x,y,z\\rangle=(-1)^{xyz}|x,y,z\\rangle\n \\quad\\text{for }x,y,z\\in\\{0,1\\},\n\\tag{2}\n\\end{equation} \npreserves $\\mathcal Q_{1,n}\\otimes\\mathcal Q_{2,n}\\otimes\\mathcal Q_{3,n}$ and induces a logical unitary that does not normalize its encoded Pauli group? Here $(a,j)$ denotes physical qubit $j$ of block $a$, so Eq. (2) applies one CCZ gate to each aligned triple of physical qubits.", "url": "https://qiqc-op.com/problem/op_15c6d5adf1796d86/", "json": "https://qiqc-op.com/api/problems/op_15c6d5adf1796d86.json", "tex": "https://qiqc-op.com/problem/op_15c6d5adf1796d86/op_15c6d5adf1796d86.tex", "created": "2026-09-15", "updated": "2026-09-15", "createdAt": "2026-09-15T18:46:16.000Z", "updatedAt": "2026-09-15T18:46:16.000Z", "sha256": "09b0b327cd3d6de5f840446e97e0277b3fd8ba53ecb9b5561c836ef370c72e11" }, { "id": "op_2778126c209a3d49", "ulid": "01M2JD9V4QDAJQZP7QRRNNHNX0", "aliases": [ "op_2778126c209a3d49", "01M2JD9V4QDAJQZP7QRRNNHNX0", "op-2778126c209a3d49" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 1, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-15T11:30:12.759Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "source-stated", "posed": null, "areaIds": [ "quantum-cryptography" ], "topicIds": [ "superactivation", "secret-key-distillation" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [] }, "title": "Superactivation of bipartite classical secret-key rates", "status": "Unsolved", "fields": [ "Quantum Cryptography" ], "topics": [ "Superactivation", "Secret-key distillation" ], "tags": [ "Quantum Cryptography", "Superactivation", "Secret-key distillation" ], "statement": "Can two classical sources, each with zero secret-key rate between two honest parties, yield a positive secret-key rate when used jointly? Let $p_{ABE}$ and $q_{A'B'E'}$ be probability distributions on finite classical alphabets. For such a source $p$, let $K_{\\mathrm{cl}}(p)$ denote its asymptotic secret-key rate, in bits per independent sample, under arbitrary local random processing and unlimited authenticated interactive public discussion. Eve holds exactly the specified classical variables and the entire public transcript; she is not supplied a quantum purification. Keys must become uniform, shared correctly, and independent of Eve’s information in total variation distance. The combined source is the independent product $p\\otimes q:=p_{ABE}\\,q_{A'B'E'}$, with Alice holding $AA'$, Bob holding $BB'$, and Eve holding $EE'$. The question is whether there is a pair satisfying\n\n \\begin{equation}\n K_{\\mathrm{cl}}(p)=K_{\\mathrm{cl}}(q)=0,\n \\qquad\n K_{\\mathrm{cl}}(p\\otimes q)>0.\n\\tag{1}\n\\end{equation}", "url": "https://qiqc-op.com/problem/op_2778126c209a3d49/", "json": "https://qiqc-op.com/api/problems/op_2778126c209a3d49.json", "tex": "https://qiqc-op.com/problem/op_2778126c209a3d49/op_2778126c209a3d49.tex", "created": "2026-09-15", "updated": "2026-09-15", "createdAt": "2026-09-15T18:46:16.000Z", "updatedAt": "2026-09-15T18:46:16.000Z", "sha256": "5820f44b0fbb96e17ff202a30c02aa64b316b33cda3320e12e82fa56befade4d" }, { "id": "op_27b7826ed57e893a", "ulid": "01M2JDG8GMD6A9Z88MC2195VA6", "aliases": [ "op_27b7826ed57e893a", "01M2JDG8GMD6A9Z88MC2195VA6", "op-27b7826ed57e893a" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 1, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-15T11:33:43.060Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "source-stated", "posed": null, "areaIds": [ "quantum-resource-theory" ], "topicIds": [ "entanglement-cost", "local-operations-and-classical-communication" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [ "01M1HME780JH0D9Y0RQ750ZZPY", "01M2JDG8A02AY84NNNVZ52DD61" ] }, "title": "LOCC entanglement cost of isotropic states", "status": "Unsolved", "fields": [ "Quantum Resource Theory" ], "topics": [ "Entanglement cost", "Local operations and classical communication" ], "tags": [ "Quantum Resource Theory", "Entanglement cost", "Local operations and classical communication" ], "statement": "What is the entanglement cost of the isotropic state $\\sigma_{d,F}$ for every integer $d\\geq2$ and every fidelity $1/d1/d$. The entanglement cost $E_C(\\rho)$ of a bipartite state $\\rho$ is the infimum of the rates $R$ for which local operations and classical communication (LOCC) transform $\\lceil nR\\rceil$ ebits into states whose trace distance from $\\rho^{\\otimes n}$ tends to zero as $n\\to\\infty$. Determine $E_C(\\sigma_{d,F})$ for all integers $d\\geq2$ and all $1/d0,\n\\tag{1}\n\\end{equation} \nwhere the joint rate in Eq. (1) is taken across the bipartition $AA':BB'$.", "url": "https://qiqc-op.com/problem/op_3be0df761e83d438/", "json": "https://qiqc-op.com/api/problems/op_3be0df761e83d438.json", "tex": "https://qiqc-op.com/problem/op_3be0df761e83d438/op_3be0df761e83d438.tex", "created": "2026-09-15", "updated": "2026-09-15", "createdAt": "2026-09-15T18:46:16.000Z", "updatedAt": "2026-09-15T18:46:16.000Z", "sha256": "5f8810a02b2c7dac8e1b245ade780dab3f99e83ced25737b80fef5d70b9fb243" }, { "id": "op_422d240a5f2cb7da", "ulid": "01M2JDGWMK4NYJB3K261A4ZCFK", "aliases": [ "op_422d240a5f2cb7da", "01M2JDGWMK4NYJB3K261A4ZCFK", "op-422d240a5f2cb7da" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 1, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-15T11:34:03.667Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "source-stated", "posed": null, "areaIds": [ "quantum-algorithm" ], "topicIds": [ "clifford-hierarchy", "computational-complexity-and-computability" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [ "01M2JDGWJBGZDXKT6QJ1D9HZWX" ] }, "title": "Efficient determination of the Clifford-hierarchy level of bounded-degree permutation gates", "status": "Unsolved", "fields": [ "Quantum algorithm" ], "topics": [ "Clifford hierarchy", "Computational complexity and computability" ], "tags": [ "Quantum algorithm", "Clifford hierarchy", "Computational complexity and computability" ], "statement": "For each fixed degree bound $d\\geq2$, is there a deterministic polynomial-time algorithm that computes the Clifford-hierarchy level of an $n$-qubit permutation gate from degree-at-most-$d$ algebraic normal forms of the permutation and its inverse, or reports that the gate lies outside the hierarchy? Let $n\\geq1$, and let $\\mathcal P_n$ be the $n$-qubit Pauli group, consisting of the operators $\\omega P_1\\otimes\\cdots\\otimes P_n$ with $\\omega\\in\\{\\pm1,\\pm i\\}$ and $P_j\\in\\{I,X,Y,Z\\}$. The Clifford hierarchy is defined by\n\n \\begin{equation}\n \\mathcal C_1(n):=\\mathcal P_n,\n \\qquad\n \\mathcal C_{k+1}(n):=\\bigl\\{U\\in U(2^n):UPU^\\dagger\\in\\mathcal C_k(n)\n \\ \\text{for every}\\ P\\in\\mathcal P_n\\bigr\\},\n \\qquad k\\geq1,\n\\tag{1}\n\\end{equation} \nand write $\\mathcal C_\\infty(n):=\\bigcup_{k\\geq1}\\mathcal C_k(n)$ and $\\ell_n(U):=\\min\\{k\\geq1:U\\in\\mathcal C_k(n)\\}$ for $U\\in\\mathcal C_\\infty(n)$.\n\nFor a bijection $\\pi:\\mathbb F_2^n\\to\\mathbb F_2^n$, let $U_\\pi|x\\rangle:=|\\pi(x)\\rangle$ be its permutation gate. Each coordinate $\\pi_i:\\mathbb F_2^n\\to\\mathbb F_2$ has a unique algebraic normal form, a multilinear polynomial over $\\mathbb F_2$. Fix an integer $d\\geq2$. An input consists of the algebraic normal forms of all coordinates of both $\\pi$ and $\\pi^{-1}$, each of total degree at most $d$, so it lists at most\n\n \\begin{equation}\n 2n\\sum_{j=0}^{d}\\binom{n}{j}\n\\tag{2}\n\\end{equation} \nmonomials. The required output is $\\ell_n(U_\\pi)$, for the levels of Eq. (1), if $U_\\pi\\in\\mathcal C_\\infty(n)$, and a report that $U_\\pi$ lies outside the hierarchy otherwise. No ancillary qubits are allowed. For each fixed $d$, in particular $d=2$, is there a deterministic algorithm that produces this output in time polynomial in $n$, and hence polynomial in the input length bounded by Eq. (2)?", "url": "https://qiqc-op.com/problem/op_422d240a5f2cb7da/", "json": "https://qiqc-op.com/api/problems/op_422d240a5f2cb7da.json", "tex": "https://qiqc-op.com/problem/op_422d240a5f2cb7da/op_422d240a5f2cb7da.tex", "created": "2026-09-15", "updated": "2026-09-15", "createdAt": "2026-09-15T18:46:16.000Z", "updatedAt": "2026-09-15T18:46:16.000Z", "sha256": "76416ad74fb0fb8d62122177f4d322d6717da82aca734fcfab4b12fa0240efa9" }, { "id": "op_458e9e86ccbddccb", "ulid": "01M2JD5FZC48CVYJ6FT6X31QKM", "aliases": [ "op_458e9e86ccbddccb", "01M2JD5FZC48CVYJ6FT6X31QKM", "op-458e9e86ccbddccb" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 1, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-15T11:27:50.252Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "derived", "posed": null, "areaIds": [ "quantum-error-correction" ], "topicIds": [ "quantum-coding-theory" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [ "01M2JD5FV21ZSE9KFGPTCH3BJ2" ] }, "title": "Weight-four generators for an [[11,3,3]] stabilizer code", "status": "Unsolved", "fields": [ "Quantum Error Correction" ], "topics": [ "Quantum coding theory" ], "tags": [ "Quantum Error Correction", "Quantum coding theory" ], "statement": "Does there exist a qubit stabilizer code with parameters $[[11,3,d]]$ and $d\\geq3$ whose stabilizer group is generated by eight independent commuting Pauli operators of weight at most four?\n\nWrite an $11$-qubit Pauli operator, up to a phase, as $X^{a}Z^{b}=\\bigotimes_{j=1}^{11}X^{a_j}Z^{b_j}$ with $v=(a,b)\\in\\mathbb F_2^{22}$. Its weight is $\\operatorname{wt}(v)=|\\{j:(a_j,b_j)\\neq(0,0)\\}|$, and two Pauli operators commute exactly when the symplectic form\n\n \\begin{equation}\n \\langle(a,b),(a',b')\\rangle:=a\\cdot b'+b\\cdot a'\\in\\mathbb F_2\n\\tag{1}\n\\end{equation} \nvanishes. Commuting Hermitian Pauli operators whose eight binary vectors are linearly independent generate a group not containing $-I$, and this group defines a code with $k=11-8=3$ logical qubits. Its distance is the least weight of a Pauli operator that commutes with all generators but is not a scalar multiple of an element of the stabilizer group; it does not depend on the signs of the generators. With the form in Eq. (1), the question therefore asks whether there are $g_1,\\dots,g_8\\in\\mathbb F_2^{22}$ such that\n\n \\begin{equation}\n \\begin{aligned}\n &g_1,\\dots,g_8\\ \\text{are linearly independent},\\qquad \\operatorname{wt}(g_i)\\leq4,\\qquad \\langle g_i,g_j\\rangle=0\\quad(1\\leq i,j\\leq8),\\\\\n &\\text{every } v\\in\\mathbb F_2^{22} \\text{ with } 1\\leq\\operatorname{wt}(v)\\leq2 \\text{ and } \\langle v,g_i\\rangle=0 \\text{ for all } i \\text{ lies in } \\operatorname{span}\\{g_1,\\dots,g_8\\}.\n \\end{aligned}\n\\tag{2}\n\\end{equation} \nThe second line of Eq. (2) is the condition $d\\geq3$. Degenerate codes are allowed, and no CSS structure is required.", "url": "https://qiqc-op.com/problem/op_458e9e86ccbddccb/", "json": "https://qiqc-op.com/api/problems/op_458e9e86ccbddccb.json", "tex": "https://qiqc-op.com/problem/op_458e9e86ccbddccb/op_458e9e86ccbddccb.tex", "created": "2026-09-15", "updated": "2026-09-15", "createdAt": "2026-09-15T18:46:16.000Z", "updatedAt": "2026-09-15T18:46:16.000Z", "sha256": "dfb0c8a48f9260fa6017a971f738dcb070965b3a54ff9389873cafab44f693f1" }, { "id": "op_59e9ede6e282a917", "ulid": "01M2JD9V2CZDMZ405QE7W8J0J2", "aliases": [ "op_59e9ede6e282a917", "01M2JD9V2CZDMZ405QE7W8J0J2", "op-59e9ede6e282a917" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 1, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-15T11:30:12.684Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "derived", "posed": null, "areaIds": [ "quantum-cryptography", "quantum-communication" ], "topicIds": [ "superactivation", "secret-key-distillation" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [ "01M1HME780SNG2DQVEGCDB0XSK", "01M2JD9V01MADK091VQM1GWX14", "01M2JD5FGZ7SN3V248P849GBQF" ] }, "title": "Superactivation of two-way secret-key capacity", "status": "Unsolved", "fields": [ "Quantum Cryptography", "Quantum Communication" ], "topics": [ "Superactivation", "Secret-key distillation" ], "tags": [ "Quantum Cryptography", "Quantum Communication", "Superactivation", "Secret-key distillation" ], "statement": "Can two quantum channels with zero two-way-assisted secret-key capacity have positive secret-key capacity when used jointly? Let $\\mathcal N:A_0\\to B_0$ and $\\mathcal M:A_1\\to B_1$ be independent finite-dimensional memoryless quantum channels. For a channel $\\mathcal C$, let $K_{\\leftrightarrow}(\\mathcal C)$ denote its secret-key capacity, in secret bits per channel use, when arbitrary adaptive local operations and unlimited authenticated two-way public communication are allowed between channel uses. Alice and Bob start without shared entanglement or secret key of positive rate. Eve receives every complementary-channel output and the full public transcript; the probability that Alice’s and Bob’s keys differ, and the trace distance between the key and a uniform key independent of Eve, must vanish asymptotically. One use of $\\mathcal N\\otimes\\mathcal M$ means one use of each channel, with independent environments and ordinary causal ordering. The question is whether there is a pair satisfying\n\n \\begin{equation}\n K_{\\leftrightarrow}(\\mathcal N)=K_{\\leftrightarrow}(\\mathcal M)=0,\n \\qquad\n K_{\\leftrightarrow}(\\mathcal N\\otimes\\mathcal M)>0.\n\\tag{1}\n\\end{equation}", "url": "https://qiqc-op.com/problem/op_59e9ede6e282a917/", "json": "https://qiqc-op.com/api/problems/op_59e9ede6e282a917.json", "tex": "https://qiqc-op.com/problem/op_59e9ede6e282a917/op_59e9ede6e282a917.tex", "created": "2026-09-15", "updated": "2026-09-15", "createdAt": "2026-09-15T18:46:16.000Z", "updatedAt": "2026-09-15T18:46:16.000Z", "sha256": "a93d9b1a7f97267debebfd88b2919cdbf5ca9d66580c4b59c747bab92076d002" }, { "id": "op_5a17b19b8adeb191", "ulid": "01M2JDA57BVAQNFNPGMHV6S9XQ", "aliases": [ "op_5a17b19b8adeb191", "01M2JDA57BVAQNFNPGMHV6S9XQ", "op-5a17b19b8adeb191" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 1, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-15T11:30:23.083Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "source-stated", "posed": null, "areaIds": [ "quantum-algorithm" ], "topicIds": [ "clifford-hierarchy" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [ "01M2JDA59FJ4PBBZ0VJ1QN5PN2" ] }, "title": "Smallest qubit number for a non-semi-Clifford third-level gate", "status": "Unsolved", "fields": [ "Quantum algorithm" ], "topics": [ "Clifford hierarchy" ], "tags": [ "Quantum algorithm", "Clifford hierarchy" ], "statement": "What is the smallest number of qubits on which the third level of the Clifford hierarchy contains a gate that is not semi-Clifford? For $n\\geq1$ qubits, let $\\mathcal{P}_n$ be the Pauli group with arbitrary global phases, consisting of the operators $e^{i\\theta}P_1\\otimes\\cdots\\otimes P_n$ with $\\theta\\in\\mathbb{R}$ and $P_j\\in\\{I,X,Y,Z\\}$, and define the Clifford hierarchy recursively by\n\n \\begin{equation}\n \\mathcal{C}_1(n):=\\mathcal{P}_n,\n \\qquad\n \\mathcal{C}_{k+1}(n):=\n \\bigl\\{U\\in U(2^n):UPU^\\dagger\\in\\mathcal{C}_k(n)\n \\ \\text{for every}\\ P\\in\\mathcal{P}_n\\bigr\\},\n \\qquad k\\geq1.\n\\tag{1}\n\\end{equation} \nThus $\\mathcal{C}_2(n)$ is the Clifford group. A unitary is semi-Clifford when it can be written as $C_LDC_R$ with $C_L,C_R\\in\\mathcal{C}_2(n)$ and $D$ diagonal in the computational basis; let $\\mathrm{SC}(n)$ denote the set of such unitaries. The quantity sought is\n\n \\begin{equation}\n n_{\\min}:=\\min\\bigl\\{n\\geq1:\n \\mathcal{C}_3(n)\\setminus\\mathrm{SC}(n)\\neq\\varnothing\\bigr\\},\n\\tag{2}\n\\end{equation} \nwhere $\\mathcal{C}_3(n)$ is the third level in Eq. (1). The known results give $n_{\\min}\\in\\{5,6,7\\}$. A complete answer determines the value of $n_{\\min}$ in Eq. (2); equivalently, it decides for $n=5$ and for $n=6$ whether every gate in $\\mathcal{C}_3(n)$ is semi-Clifford.", "url": "https://qiqc-op.com/problem/op_5a17b19b8adeb191/", "json": "https://qiqc-op.com/api/problems/op_5a17b19b8adeb191.json", "tex": "https://qiqc-op.com/problem/op_5a17b19b8adeb191/op_5a17b19b8adeb191.tex", "created": "2026-09-15", "updated": "2026-09-15", "createdAt": "2026-09-15T18:46:16.000Z", "updatedAt": "2026-09-15T18:46:16.000Z", "sha256": "72aa6f9109e5d3fce4cbb46cb5ed1a77201c2fdf4b90247d296a39bcb1564187" }, { "id": "op_5ea7cb9abb281829", "ulid": "01M2JD6D42ZF21JR0CXSY184XW", "aliases": [ "op_5ea7cb9abb281829", "01M2JD6D42ZF21JR0CXSY184XW", "op-5ea7cb9abb281829" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 1, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-15T11:28:20.098Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "derived", "posed": null, "areaIds": [ "quantum-error-correction" ], "topicIds": [ "quantum-coding-theory" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [ "01M2JD6D1YCESPTHQZDT5TZB77" ] }, "title": "Optimal dimension of seven-qubit distance-two codes", "status": "Unsolved", "fields": [ "Quantum Error Correction" ], "topics": [ "Quantum coding theory" ], "tags": [ "Quantum Error Correction", "Quantum coding theory" ], "statement": "For an exact seven-qubit code that detects every single-qubit error, what is the largest possible code-space dimension: is $K_{\\max}^{(2)}(7,2)$ equal to $24$, $25$, or $26$? Let $\\mathcal P_7:=\\{I,X,Y,Z\\}^{\\otimes7}$ be the Pauli basis, where the weight $\\operatorname{wt}(E)$ counts nonidentity tensor factors, and define\n\n \\begin{equation}\n K_{\\max}^{(2)}(7,2):=\\max\\Bigl\\{\\operatorname{rank}P:\\\n P=P^\\dagger=P^2\\ \\text{on}\\ (\\mathbb C^2)^{\\otimes7},\\\n PEP=c_EP\\ \\text{for every}\\ E\\in\\mathcal P_7\\ \\text{with}\\\n \\operatorname{wt}(E)=1\\Bigr\\},\n\\tag{1}\n\\end{equation} \nwhere each $c_E$ is a scalar depending only on $E$. The maximum in Eq. (1) is over all projectors, including degenerate, impure, and non-CWS codes. A distance-two code detects an arbitrary single-qubit error and corrects an erasure at a known location; it need not correct an arbitrary single-qubit error at an unknown location.", "url": "https://qiqc-op.com/problem/op_5ea7cb9abb281829/", "json": "https://qiqc-op.com/api/problems/op_5ea7cb9abb281829.json", "tex": "https://qiqc-op.com/problem/op_5ea7cb9abb281829/op_5ea7cb9abb281829.tex", "created": "2026-09-15", "updated": "2026-09-15", "createdAt": "2026-09-15T18:46:16.000Z", "updatedAt": "2026-09-15T18:46:16.000Z", "sha256": "525614603c73d2b5a1ddaffd95b616e7e3eb38f8e3fb2bf6dc03796179213c54" }, { "id": "op_62c9d73ced040f87", "ulid": "01M2JD0QFNXTMGCBQ405DNSW8Z", "aliases": [ "op_62c9d73ced040f87", "01M2JD0QFNXTMGCBQ405DNSW8Z", "op-62c9d73ced040f87" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 1, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-15T11:25:14.101Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "source-stated", "posed": null, "areaIds": [ "quantum-error-correction", "quantum-algorithm" ], "topicIds": [ "quantum-ldpc-codes", "quantum-circuit-complexity" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [] }, "title": "Linear-size logarithmic-depth encoders for good quantum LDPC codes", "status": "Unsolved", "fields": [ "Quantum Error Correction", "Quantum algorithm" ], "topics": [ "Quantum LDPC codes", "Quantum circuit complexity" ], "tags": [ "Quantum Error Correction", "Quantum algorithm", "Quantum LDPC codes", "Quantum circuit complexity" ], "statement": "Does some family of asymptotically good qubit CSS LDPC codes admit unitary encoding circuits with linearly many gates and logarithmic depth? Consider qubit CSS codes $[[n,k_n,d_n]]$ with $n\\to\\infty$, $k_n,d_n=\\Omega(n)$, and check weights and qubit degrees bounded uniformly in $n$. The question asks whether such a family has measurement-free unitary Clifford encoders $V_n$ on its $n$ physical qubits satisfying\n\n \\begin{equation}\n V_n\\bigl(|\\psi\\rangle\\otimes|0\\rangle^{\\otimes(n-k_n)}\\bigr)\n =\\operatorname{Enc}_n|\\psi\\rangle\n \\qquad\n \\text{for every }|\\psi\\rangle\\in(\\mathbb C^2)^{\\otimes k_n},\n\\tag{1}\n\\end{equation} \nwhere $\\operatorname{Enc}_n$ is an isometry onto the code space and $V_n$ is a circuit of $O(n)$ one- and two-qubit gates with depth $O(\\log n)$. The constants implicit in these bounds are independent of $n$, and gates may act on arbitrary pairs of qubits.", "url": "https://qiqc-op.com/problem/op_62c9d73ced040f87/", "json": "https://qiqc-op.com/api/problems/op_62c9d73ced040f87.json", "tex": "https://qiqc-op.com/problem/op_62c9d73ced040f87/op_62c9d73ced040f87.tex", "created": "2026-09-15", "updated": "2026-09-15", "createdAt": "2026-09-15T18:46:16.000Z", "updatedAt": "2026-09-15T18:46:16.000Z", "sha256": "e9d6e1d7f18e72a519cc2b407c343bea0dbe5552ae9a56e6eae5c9da236c9e65" }, { "id": "op_644f1aced78f36c9", "ulid": "01M2JDAQHG9DJJEKXYK3RW5GBG", "aliases": [ "op_644f1aced78f36c9", "01M2JDAQHG9DJJEKXYK3RW5GBG", "op-644f1aced78f36c9" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 1, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-15T11:30:41.840Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "derived", "posed": null, "areaIds": [ "quantum-error-correction" ], "topicIds": [ "quantum-coding-theory" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [] }, "title": "Two-error perfect quantum codes over non-prime-power local dimensions", "status": "Unsolved", "fields": [ "Quantum Error Correction" ], "topics": [ "Quantum coding theory" ], "tags": [ "Quantum Error Correction", "Quantum coding theory" ], "statement": "Does a nontrivial pure perfect quantum code correcting arbitrary errors on two qudits exist for some local dimension $q$ with at least two distinct prime factors? Let $q\\geq2$ and $n\\geq1$ be integers, and let $\\mathcal C\\subseteq(\\mathbb C^q)^{\\otimes n}$ be a code of integer dimension $K>1$ with orthogonal projector $P$. No stabilizer structure is assumed, and $K$ need not be a power of $q$. On one site let $X\\lvert j\\rangle=\\lvert j+1 \\bmod q\\rangle$ and $Z\\lvert j\\rangle=\\omega^j\\lvert j\\rangle$ with $\\omega=e^{2\\pi i/q}$, and use the phase-free Weyl basis $\\{X^aZ^b: 0\\leq a,b\\leq q-1\\}$ of $q^2$ operators, including the identity. Let $\\mathcal E_2$ be the set of $n$-fold tensor products of these operators with weight at most two, where weight counts non-identity sites. The code is a pure two-error-correcting code when\n\n \\begin{equation}\n PE^\\dagger FP=\\delta_{E,F}\\,P,\n \\qquad E,F\\in\\mathcal E_2,\n\\tag{1}\n\\end{equation} \nwhere $\\delta_{E,F}$ is the Kronecker delta, so that distinct correctable errors map the code space to mutually orthogonal subspaces. Put\n\n \\begin{equation}\n V_2(n,q):=\\lvert\\mathcal E_2\\rvert\n =1+n(q^2-1)+\\binom n2(q^2-1)^2 .\n\\tag{2}\n\\end{equation} \nEquation (1) implies $K\\,V_2(n,q)\\leq q^n$, and the code is Hamming-perfect when\n\n \\begin{equation}\n K\\,V_2(n,q)=q^n .\n\\tag{3}\n\\end{equation} \nThe question asks whether Eqs. (1) and (3) hold simultaneously for some $n$, some integer $K>1$, and some $q$ that is not a prime power. A complete answer either exhibits such a code or proves that none exists for any non-prime-power $q$.", "url": "https://qiqc-op.com/problem/op_644f1aced78f36c9/", "json": "https://qiqc-op.com/api/problems/op_644f1aced78f36c9.json", "tex": "https://qiqc-op.com/problem/op_644f1aced78f36c9/op_644f1aced78f36c9.tex", "created": "2026-09-15", "updated": "2026-09-15", "createdAt": "2026-09-15T18:46:16.000Z", "updatedAt": "2026-09-15T18:46:16.000Z", "sha256": "e08b1b61c339eb63b5b8def075a9f9ffd5c4bcf76cef73137bed2bfb6d135aa3" }, { "id": "op_7388956ad5c7eb27", "ulid": "01M2JD5FK9R2ZW46V6A18T2J1K", "aliases": [ "op_7388956ad5c7eb27", "01M2JD5FK9R2ZW46V6A18T2J1K", "op-7388956ad5c7eb27" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 1, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-15T11:27:49.865Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "derived", "posed": null, "areaIds": [ "quantum-communication" ], "topicIds": [ "superactivation", "quantum-capacity", "bosonic-channels" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [ "01M26JZET5QZ2FB0WSWESW77V6", "01M1Q787QR71ZZJVC3XC65A5F3" ] }, "title": "Finite-energy superactivation of Gaussian amplifiers and additive-noise channels", "status": "Unsolved", "fields": [ "Quantum Communication" ], "topics": [ "Superactivation", "Quantum capacity", "Bosonic channels" ], "tags": [ "Quantum Communication", "Superactivation", "Quantum capacity", "Bosonic channels" ], "statement": "Is there a single-mode phase-insensitive Gaussian amplifier or additive-noise channel of zero finite-energy quantum capacity that acquires positive finite-energy quantum capacity when used in parallel with a two-mode positive-partial-transpose Gaussian channel of zero finite-energy quantum capacity? Consider single-mode, phase-insensitive bosonic Gaussian channels $\\mathcal G_{\\tau,y}$ with finite real parameters $\\tau\\geq1$ and $y\\geq|1-\\tau|$. In the convention where the vacuum covariance matrix is $I_2$, their displacement vectors and covariance matrices transform as\n\n \\begin{equation}\n \\boldsymbol m\\longmapsto\\sqrt\\tau\\,\\boldsymbol m,\n \\qquad\n V\\longmapsto\\tau V+yI_2.\n\\tag{1}\n\\end{equation} \nHere $\\tau>1$ describes an amplifier, while $\\tau=1$ and $y>0$ describes an additive-noise channel. Restrict to the non-entanglement-breaking parameter range $y<1+\\tau$. Let $\\mathcal H$ be any Gaussian channel with two input modes and two output modes, acting on covariance matrices as $V\\mapsto X_HVX_H^T+Y_H$ for real $4\\times4$ matrices $X_H$ and $Y_H=Y_H^T$. Write\n\n \\begin{equation}\n \\Omega_2:=\\begin{pmatrix}0&1\\\\-1&0\\end{pmatrix}^{\\oplus2}.\n\\tag{2}\n\\end{equation} \nRequire both complete positivity and the positive-partial-transpose (PPT) property:\n\n \\begin{equation}\n Y_H+i(\\Omega_2-X_H\\Omega_2X_H^T)\\geq0,\n \\qquad\n Y_H+i(\\Omega_2+X_H\\Omega_2X_H^T)\\geq0.\n\\tag{3}\n\\end{equation} \nFor a bosonic channel $\\mathcal N$, let $Q_E(\\mathcal N)$ be its unassisted quantum capacity when the total mean input photon number over $n$ uses is at most $nE$, and define\n\n \\begin{equation}\n Q_{\\mathrm{fin}}(\\mathcal N):=\\sup_{00,\n\\tag{5}\n\\end{equation} \nwith capacities as in Eq. (4).", "url": "https://qiqc-op.com/problem/op_7388956ad5c7eb27/", "json": "https://qiqc-op.com/api/problems/op_7388956ad5c7eb27.json", "tex": "https://qiqc-op.com/problem/op_7388956ad5c7eb27/op_7388956ad5c7eb27.tex", "created": "2026-09-15", "updated": "2026-09-15", "createdAt": "2026-09-15T18:46:16.000Z", "updatedAt": "2026-09-15T18:46:16.000Z", "sha256": "f24744cb1f3461a930b84305d3e081c57a6bae1f88b72bc5174dd70a6ae5fba0" }, { "id": "op_78da2d4fc0735323", "ulid": "01M2JD5FX8BZEVY4SV12JCTBNH", "aliases": [ "op_78da2d4fc0735323", "01M2JD5FX8BZEVY4SV12JCTBNH", "op-78da2d4fc0735323" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 1, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-15T11:27:50.184Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "source-stated", "posed": null, "areaIds": [ "quantum-error-correction" ], "topicIds": [ "quantum-ldpc-codes", "quantum-coding-theory" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [] }, "title": "Distance of constant-rate quantum XYZ product codes", "status": "Unsolved", "fields": [ "Quantum Error Correction" ], "topics": [ "Quantum LDPC codes", "Quantum coding theory" ], "tags": [ "Quantum Error Correction", "Quantum LDPC codes", "Quantum coding theory" ], "statement": "Do there exist quantum XYZ product codes, built from classical parity-check matrices with uniformly bounded row and column weights, that have constant rate and minimum distance of order $N^{2/3}$ on $N$ physical qubits?\n\nFor $\\ell\\in\\{1,2,3\\}$, let $H_\\ell\\in\\mathbb F_2^{m_\\ell\\times n_\\ell}$ have rows indexed by a set $R_\\ell$ with $|R_\\ell|=m_\\ell$ and columns indexed by a disjoint set $B_\\ell$ with $|B_\\ell|=n_\\ell$. For a triple $x\\in\\prod_{\\ell=1}^{3}(R_\\ell\\sqcup B_\\ell)$, let $r(x)$ be the number of coordinates with $x_\\ell\\in R_\\ell$. The XYZ product code $\\mathcal Q(H_1,H_2,H_3)$ has one physical qubit for each triple $y$ with $r(y)$ even and one stabilizer generator $g_x$ for each triple $x$ with $r(x)$ odd. Call $x$ and $y$ adjacent in direction $\\ell$ if $x_{\\ell'}=y_{\\ell'}$ for all $\\ell'\\neq\\ell$ and the entry of $H_\\ell$ in row $\\{x_\\ell,y_\\ell\\}\\cap R_\\ell$ and column $\\{x_\\ell,y_\\ell\\}\\cap B_\\ell$ equals $1$; by parity, exactly one of $x_\\ell,y_\\ell$ lies in $R_\\ell$. The generators are\n\n \\begin{equation}\n g_x:=\\prod_{\\ell=1}^{3}\\ \\prod_{y\\ \\text{adjacent to}\\ x\\ \\text{in direction}\\ \\ell}\\sigma_\\ell^{(y)},\n \\qquad(\\sigma_1,\\sigma_2,\\sigma_3)=(X,Y,Z),\n\\tag{1}\n\\end{equation} \nwhere $\\sigma_\\ell^{(y)}$ acts on qubit $y$, and the number of qubits is\n\n \\begin{equation}\n N=n_1n_2n_3+n_1m_2m_3+m_1n_2m_3+m_1m_2n_3.\n\\tag{2}\n\\end{equation} \nThe generators in Eq. (1) commute. Let $G\\subseteq\\mathbb F_2^{2N}$ be the span of their binary symplectic representations and $G^{\\perp}$ its symplectic complement. The code has $K=N-\\dim G$ logical qubits and, when $K\\geq1$, distance $D=\\min\\{\\operatorname{wt}(v):v\\in G^{\\perp}\\setminus G\\}$, where $\\operatorname{wt}$ counts the qubits acted on nontrivially. These are the parameters of the stabilizer code obtained from any consistent choice of generator signs. The question asks for triples $(H_1^{(t)},H_2^{(t)},H_3^{(t)})$, $t\\in\\mathbb N$, all of whose row and column weights are at most a fixed $w$, and constants $c,c',C'>0$ such that\n\n \\begin{equation}\n N_t\\to\\infty,\\qquad K_t\\geq c\\,N_t,\\qquad c'N_t^{2/3}\\leq D_t\\leq C'N_t^{2/3}.\n\\tag{3}\n\\end{equation}", "url": "https://qiqc-op.com/problem/op_78da2d4fc0735323/", "json": "https://qiqc-op.com/api/problems/op_78da2d4fc0735323.json", "tex": "https://qiqc-op.com/problem/op_78da2d4fc0735323/op_78da2d4fc0735323.tex", "created": "2026-09-15", "updated": "2026-09-15", "createdAt": "2026-09-15T18:46:16.000Z", "updatedAt": "2026-09-15T18:46:16.000Z", "sha256": "5590d87ad5506e28d347ad2ee869067a6587eab8798fb316f09ae942eedc2194" }, { "id": "op_a72722e455d85d55", "ulid": "01M2JDAKND20N4AGZ4FDRNCFCG", "aliases": [ "op_a72722e455d85d55", "01M2JDAKND20N4AGZ4FDRNCFCG", "op-a72722e455d85d55" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 1, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-15T11:30:37.869Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "source-stated", "posed": null, "areaIds": [ "quantum-error-correction" ], "topicIds": [ "quantum-coding-theory" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [ "01M2JDAHKFFSTV9TZPGCX3QD7N" ] }, "title": "Minimal permutation-invariant qubit code for two errors", "status": "Unsolved", "fields": [ "Quantum Error Correction" ], "topics": [ "Quantum coding theory" ], "tags": [ "Quantum Error Correction", "Quantum coding theory" ], "statement": "Is nineteen the least number of physical qubits on which a two-dimensional code contained in the fully symmetric subspace corrects every error on at most two qubits? Let $\\operatorname{Sym}^n(\\mathbb C^2)\\subseteq(\\mathbb C^2)^{\\otimes n}$ be the subspace of vectors fixed by every permutation of the $n$ qubits, and let $d(\\mathcal Q)$ be the distance of a subspace $\\mathcal Q$, so that $\\mathcal Q$ corrects arbitrary errors on two qubits exactly when $d(\\mathcal Q)\\geq5$. Define\n\n \\begin{equation}\n n_{\\min}^{\\mathrm{PI}}(2):=\\min\\bigl\\{n\\in\\mathbb N:\\\n \\exists\\,\\mathcal Q\\subseteq\\operatorname{Sym}^n(\\mathbb C^2),\\\n \\dim\\mathcal Q=2,\\ d(\\mathcal Q)\\geq5\\bigr\\}.\n\\tag{1}\n\\end{equation} \nEquivalently, write an orthonormal logical basis in the Dicke states,\n\n \\begin{equation}\n \\lvert a_L\\rangle=\\sum_{w=0}^{n}c_{a,w}\\lvert D_w^n\\rangle,\n \\qquad\n \\lvert D_w^n\\rangle=\\binom nw^{-1/2}\n \\sum_{x\\in\\{0,1\\}^n:\\ \\lvert x\\rvert=w}\\lvert x\\rangle,\n \\qquad a\\in\\{0,1\\},\n\\tag{2}\n\\end{equation} \nwhere $\\lvert x\\rvert$ is the Hamming weight of the bit string $x$ and the coefficients $c_{a,w}$ are arbitrary complex numbers, and require\n\n \\begin{equation}\n \\langle a_L\\rvert E\\lvert b_L\\rangle=c_E\\,\\delta_{ab}\n \\qquad\\text{for every Pauli operator } E \\text{ with } \\operatorname{wt}(E)<5,\n\\tag{3}\n\\end{equation} \nwith $c_E\\in\\mathbb C$ independent of $a,b\\in\\{0,1\\}$ and $\\operatorname{wt}(E)$ the number of qubits on which $E$ acts nontrivially. Permutation invariance is pointwise invariance of every code vector, not merely invariance of the code projector under relabelling qubits. The question is whether the minimum in Eq. (1) equals $19$; equality means that Eq. (3) is solvable for $n=19$ and for no $n\\leq18$, over all complex coefficients and both parities of $n$.", "url": "https://qiqc-op.com/problem/op_a72722e455d85d55/", "json": "https://qiqc-op.com/api/problems/op_a72722e455d85d55.json", "tex": "https://qiqc-op.com/problem/op_a72722e455d85d55/op_a72722e455d85d55.tex", "created": "2026-09-15", "updated": "2026-09-15", "createdAt": "2026-09-15T18:46:16.000Z", "updatedAt": "2026-09-15T18:46:16.000Z", "sha256": "60a6d21e4816df0809af2445b855f478e9d0ed2d2190359176a60b03197dcf52" }, { "id": "op_ad474bf3083462b6", "ulid": "01M2JD5FV21ZSE9KFGPTCH3BJ2", "aliases": [ "op_ad474bf3083462b6", "01M2JD5FV21ZSE9KFGPTCH3BJ2", "op-ad474bf3083462b6" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 1, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-15T11:27:50.114Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "source-stated", "posed": null, "areaIds": [ "quantum-error-correction" ], "topicIds": [ "quantum-ldpc-codes", "quantum-coding-theory" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [ "01M2JD5FZC48CVYJ6FT6X31QKM" ] }, "title": "Square-root distance bound for weight-four stabilizer codes", "status": "Unsolved", "fields": [ "Quantum Error Correction" ], "topics": [ "Quantum LDPC codes", "Quantum coding theory" ], "tags": [ "Quantum Error Correction", "Quantum LDPC codes", "Quantum coding theory" ], "statement": "Is there a universal constant $C>0$ such that every qubit stabilizer code that encodes at least one logical qubit, and whose stabilizer group is generated by Pauli operators of weight at most four, has distance at most $C\\sqrt n$ on $n$ physical qubits?\n\nA stabilizer group $\\mathcal S$ on $n$ qubits is an abelian subgroup of the $n$-qubit Pauli group with $-I\\notin\\mathcal S$; if $\\mathcal S$ has $n-k$ independent generators, its common $+1$ eigenspace encodes $k$ logical qubits. The weight $\\operatorname{wt}(P)$ of a Pauli operator $P$ is the number of qubits on which $P$ acts nontrivially. For $k\\geq1$, the distance $d$ is the least weight of a Pauli operator that commutes with every element of $\\mathcal S$ but is not a scalar multiple of an element of $\\mathcal S$. The optimal generator weight of $\\mathcal S$ is\n\n \\begin{equation}\n W(\\mathcal S):=\\min\\Bigl\\{\\max_{P\\in T}\\operatorname{wt}(P):\\ T\\subseteq\\mathcal S,\\ \\langle T\\rangle=\\mathcal S\\Bigr\\}.\n\\tag{1}\n\\end{equation} \nWith $W(\\mathcal S)$ as in Eq. (1), the question asks whether\n\n \\begin{equation}\n \\exists\\,C>0\\quad\\forall\\,n\\geq1\\quad\\forall\\,\\mathcal S\\ \\text{with}\\ k\\geq1\\ \\text{and}\\ W(\\mathcal S)\\leq4:\\qquad d\\leq C\\sqrt n.\n\\tag{2}\n\\end{equation} \nEquation (2) assumes no CSS structure, no bound on the number of generators acting on a qubit, and no geometric locality. A negative answer requires stabilizer codes with $k\\geq1$ and $W(\\mathcal S)\\leq4$ whose ratios $d/\\sqrt n$ are unbounded.", "url": "https://qiqc-op.com/problem/op_ad474bf3083462b6/", "json": "https://qiqc-op.com/api/problems/op_ad474bf3083462b6.json", "tex": "https://qiqc-op.com/problem/op_ad474bf3083462b6/op_ad474bf3083462b6.tex", "created": "2026-09-15", "updated": "2026-09-15", "createdAt": "2026-09-15T18:46:16.000Z", "updatedAt": "2026-09-15T18:46:16.000Z", "sha256": "e528ac0c7ea7eca688ebb0b4aabb25eb6ce7b611d35fff6ea23ecba923d46424" }, { "id": "op_c45752aeb9a106bc", "ulid": "01M2JD5FGZ7SN3V248P849GBQF", "aliases": [ "op_c45752aeb9a106bc", "01M2JD5FGZ7SN3V248P849GBQF", "op-c45752aeb9a106bc" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 1, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-15T11:27:49.791Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "derived", "posed": null, "areaIds": [ "quantum-communication", "quantum-cryptography" ], "topicIds": [ "superactivation", "quantum-capacity", "private-capacity", "channel-degradability" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [ "01M1Q787QRG3HGYA0Y8F8JBPTE", "01M2JD9V2CZDMZ405QE7W8J0J2" ] }, "title": "Channels superactivated by antidegradable helpers", "status": "Unsolved", "fields": [ "Quantum Communication", "Quantum Cryptography" ], "topics": [ "Superactivation", "Quantum capacity", "Private capacity", "Channel degradability" ], "tags": [ "Quantum Communication", "Quantum Cryptography", "Superactivation", "Quantum capacity", "Private capacity", "Channel degradability" ], "statement": "What necessary and sufficient structural conditions characterize the finite-dimensional channels of zero quantum capacity, and separately those of zero private capacity, that acquire a positive capacity of the same kind when used in parallel with some antidegradable channel? Let $Q(\\mathcal N)$ and $P(\\mathcal N)$ denote the fully regularized unassisted quantum capacity and private classical capacity of a finite-dimensional memoryless quantum channel $\\mathcal N$. Rates are measured in qubits and private bits per channel use, respectively; the eavesdropper receives the full complementary-channel output. For an isometric dilation $V:A\\to B\\otimes E$, define the channel and its complementary channel by\n\n \\begin{equation}\n \\mathcal N(\\rho):=\\operatorname{Tr}_E(V\\rho V^\\dagger),\n \\qquad\n \\mathcal N^c(\\rho):=\\operatorname{Tr}_B(V\\rho V^\\dagger).\n\\tag{1}\n\\end{equation} \nA channel $\\mathcal A$ is antidegradable when $\\mathcal A=\\mathcal D\\circ\\mathcal A^c$ for some completely positive trace-preserving map $\\mathcal D$, with $\\mathcal A^c$ defined as in Eq. (1). Let $\\mathfrak A$ denote the set of all finite-dimensional antidegradable channels, with no fixed dimension bound. For $R\\in\\{Q,P\\}$, define the activation set\n\n \\begin{equation}\n \\mathsf{Act}_R:=\\bigl\\{\\mathcal N:\\ R(\\mathcal N)=0\\ \\text{and}\\ R(\\mathcal N\\otimes\\mathcal A)>0\\ \\text{for some}\\ \\mathcal A\\in\\mathfrak A\\bigr\\}.\n\\tag{2}\n\\end{equation} \nThe product $\\mathcal N\\otimes\\mathcal A$ uses independent channel environments and allows arbitrary joint encoding and decoding across uses, but no public communication assistance. The problem asks, for $R=Q$ and for $R=P$, for a necessary and sufficient structural condition on $\\mathcal N$ for membership in the set defined in Eq. (2).", "url": "https://qiqc-op.com/problem/op_c45752aeb9a106bc/", "json": "https://qiqc-op.com/api/problems/op_c45752aeb9a106bc.json", "tex": "https://qiqc-op.com/problem/op_c45752aeb9a106bc/op_c45752aeb9a106bc.tex", "created": "2026-09-15", "updated": "2026-09-15", "createdAt": "2026-09-15T18:46:16.000Z", "updatedAt": "2026-09-15T18:46:16.000Z", "sha256": "16cbb100554a51eb0d66596bdc565e4c43147ac3ab4ee41048f726ae2724964d" }, { "id": "op_cdd1f718ccfbccf6", "ulid": "01M2JDA59FJ4PBBZ0VJ1QN5PN2", "aliases": [ "op_cdd1f718ccfbccf6", "01M2JDA59FJ4PBBZ0VJ1QN5PN2", "op-cdd1f718ccfbccf6" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 1, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-15T11:30:23.151Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "source-stated", "posed": null, "areaIds": [ "quantum-algorithm" ], "topicIds": [ "clifford-hierarchy" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [ "01M2JDA57BVAQNFNPGMHV6S9XQ" ] }, "title": "Semi-Clifford structure of two-qudit hierarchy gates above the third level", "status": "Unsolved", "fields": [ "Quantum algorithm" ], "topics": [ "Clifford hierarchy" ], "tags": [ "Quantum algorithm", "Clifford hierarchy" ], "statement": "Is every two-qudit gate in every level $k\\geq4$ of the Clifford hierarchy semi-Clifford when the local dimension is an odd prime? Let $p$ be an odd prime, let $\\omega:=e^{2\\pi i/p}$, and define on $\\mathbb{C}^p$ the operators $X|j\\rangle=|j+1\\bmod p\\rangle$ and $Z|j\\rangle=\\omega^j|j\\rangle$ for $j\\in\\{0,\\ldots,p-1\\}$. For $n$ qudits, let $\\mathcal{P}_{n,p}$ be the group generated by arbitrary global phases and the single-site operators $X_i,Z_i$ ($1\\leq i\\leq n$), each acting as the identity on the other sites, and define\n\n \\begin{equation}\n \\mathcal{C}_1(n,p):=\\mathcal{P}_{n,p},\n \\qquad\n \\mathcal{C}_{\\ell+1}(n,p):=\n \\bigl\\{U\\in U(p^n):UPU^\\dagger\\in\\mathcal{C}_\\ell(n,p)\n \\ \\text{for every}\\ P\\in\\mathcal{P}_{n,p}\\bigr\\},\n \\qquad \\ell\\geq1.\n\\tag{1}\n\\end{equation} \nA unitary is semi-Clifford when it equals $C_LDC_R$ with $C_L,C_R\\in\\mathcal{C}_2(n,p)$ and $D$ diagonal in the computational basis; write $\\mathrm{SC}(n,p)$ for this set. The question asks whether, with the levels of Eq. (1),\n\n \\begin{equation}\n \\mathcal{C}_k(2,p)\\subseteq\\mathrm{SC}(2,p)\n \\qquad\\text{for every odd prime }p\\text{ and every integer }k\\geq4.\n\\tag{2}\n\\end{equation} \nA complete answer either proves Eq. (2) or exhibits an odd prime $p$, a level $k\\geq4$, and a gate in $\\mathcal{C}_k(2,p)$ that is not semi-Clifford.", "url": "https://qiqc-op.com/problem/op_cdd1f718ccfbccf6/", "json": "https://qiqc-op.com/api/problems/op_cdd1f718ccfbccf6.json", "tex": "https://qiqc-op.com/problem/op_cdd1f718ccfbccf6/op_cdd1f718ccfbccf6.tex", "created": "2026-09-15", "updated": "2026-09-15", "createdAt": "2026-09-15T18:46:16.000Z", "updatedAt": "2026-09-15T18:46:16.000Z", "sha256": "774a9cff7fd0a93198ff1a12e2aa56a9a3da65a3c02b08d2a3de3f69084572b3" }, { "id": "op_ddfbbe7ca0ead0c8", "ulid": "01M2JD6CZP7FXP1FE70S72FA9J", "aliases": [ "op_ddfbbe7ca0ead0c8", "01M2JD6CZP7FXP1FE70S72FA9J", "op-ddfbbe7ca0ead0c8" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 1, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-15T11:28:19.958Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "derived", "posed": null, "areaIds": [ "quantum-error-correction" ], "topicIds": [ "quantum-coding-theory" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [] }, "title": "Non-additive ((6,2,3)) qubit code with transversal T", "status": "Unsolved", "fields": [ "Quantum Error Correction" ], "topics": [ "Quantum coding theory" ], "tags": [ "Quantum Error Correction", "Quantum coding theory" ], "statement": "Does there exist a genuinely non-additive two-dimensional six-qubit code of distance three on which a tensor product of single-qubit unitaries implements the logical $T$ gate exactly? Write $\\mathcal P_n:=\\{I,X,Y,Z\\}^{\\otimes n}$ for the phase-free Hermitian Pauli basis, where the weight $\\operatorname{wt}(E)$ of $E\\in\\mathcal P_n$ counts its nonidentity tensor factors. An $((n,K,d))_2$ code is a $K$-dimensional subspace of $(\\mathbb C^2)^{\\otimes n}$ that detects every Pauli operator of weight less than $d$; degenerate codes are allowed. A $((6,2,3))_2$ code is thus the range of an isometry $V:\\mathbb C^2\\to(\\mathbb C^2)^{\\otimes6}$, with code projector $P:=VV^\\dagger$, satisfying\n\n \\begin{equation}\n V^\\dagger V=I_2,\n \\qquad\n V^\\dagger EV=c_EI_2\n \\quad\\text{for all }E\\in\\mathcal P_6\\text{ with }\\operatorname{wt}(E)<3,\n \\qquad c_E\\in\\mathbb R.\n\\tag{1}\n\\end{equation} \nThe code is genuinely non-additive if no tensor product of single-qubit unitaries, even combined with a permutation of the qubits, maps it onto a Pauli stabilizer code, that is, onto the joint $+1$ eigenspace of an abelian subgroup of the Pauli group. The logical gate and its transversal realization are\n\n \\begin{equation}\n T:=\\operatorname{diag}(1,e^{i\\pi/4}),\n \\qquad\n \\Bigl(\\bigotimes_{j=1}^{6}U_j\\Bigr)V=e^{i\\phi}VT,\n \\qquad U_1,\\ldots,U_6\\in U(2),\\quad \\phi\\in\\mathbb R.\n\\tag{2}\n\\end{equation} \nThe local factors $U_j$ may differ from one another and need not be physical $T$ gates. No ancillary qubits, measurements, code switching, or additional transversal logical gates are assumed. The question asks whether some isometry satisfying Eq. (1) has a genuinely non-additive range and admits unitaries $U_1,\\ldots,U_6$ and a phase $\\phi$ satisfying Eq. (2).", "url": "https://qiqc-op.com/problem/op_ddfbbe7ca0ead0c8/", "json": "https://qiqc-op.com/api/problems/op_ddfbbe7ca0ead0c8.json", "tex": "https://qiqc-op.com/problem/op_ddfbbe7ca0ead0c8/op_ddfbbe7ca0ead0c8.tex", "created": "2026-09-15", "updated": "2026-09-15", "createdAt": "2026-09-15T18:46:16.000Z", "updatedAt": "2026-09-15T18:46:16.000Z", "sha256": "e520004d927154f64682523d15d04cc5a01cbacfc78050ab91634dfb4d4ce5df" }, { "id": "op_e432fc5ff73e63be", "ulid": "01M2JDGWJBGZDXKT6QJ1D9HZWX", "aliases": [ "op_e432fc5ff73e63be", "01M2JDGWJBGZDXKT6QJ1D9HZWX", "op-e432fc5ff73e63be" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 1, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-15T11:34:03.595Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "source-stated", "posed": null, "areaIds": [ "quantum-algorithm" ], "topicIds": [ "clifford-hierarchy" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [ "01M2JDGWMK4NYJB3K261A4ZCFK" ] }, "title": "Highest Clifford-hierarchy level of controlled gates with higher-level targets", "status": "Unsolved", "fields": [ "Quantum algorithm" ], "topics": [ "Clifford hierarchy" ], "tags": [ "Quantum algorithm", "Clifford hierarchy" ], "statement": "What is the highest Clifford-hierarchy level that a controlled gate can reach when its target lies in a given level $k\\geq3$ and has a given Pauli periodicity, and is that level bounded independently of the number of target qubits? Let $n\\geq1$, and let $\\mathcal P_n$ be the $n$-qubit Pauli group, consisting of the operators $\\omega P_1\\otimes\\cdots\\otimes P_n$ with $\\omega\\in\\{\\pm1,\\pm i\\}$ and $P_j\\in\\{I,X,Y,Z\\}$. The Clifford hierarchy is defined by\n\n \\begin{equation}\n \\mathcal C_1(n):=\\mathcal P_n,\n \\qquad\n \\mathcal C_{k+1}(n):=\\bigl\\{U\\in U(2^n):UPU^\\dagger\\in\\mathcal C_k(n)\n \\ \\text{for every}\\ P\\in\\mathcal P_n\\bigr\\},\n \\qquad k\\geq1.\n\\tag{1}\n\\end{equation} \nWrite $\\mathcal C_\\infty(n):=\\bigcup_{k\\geq1}\\mathcal C_k(n)$, and let $\\ell_n(V):=\\min\\{k\\geq1:V\\in\\mathcal C_k(n)\\}$ be the exact level of $V\\in\\mathcal C_\\infty(n)$. For $U\\in U(2^n)$, define the controlled gate and the Pauli periodicity\n\n \\begin{equation}\n CU:=|0\\rangle\\langle0|\\otimes I+|1\\rangle\\langle1|\\otimes U\\in U(2^{n+1}),\n \\qquad\n m(U):=\\min\\bigl\\{t\\in\\mathbb Z_{\\geq0}:U^{2^t}\\in\\mathcal P_n\\bigr\\},\n\\tag{2}\n\\end{equation} \nwith $m(U):=\\infty$ when no such $t$ exists. Only the four phases $\\pm1,\\pm i$ count as Pauli phases in Eqs. (1) and (2). This convention is essential: replacing $U$ by $e^{i\\theta}U$ multiplies $CU$ by a phase gate on the control qubit, and it can change both $m(U)$ and the level of $CU$.\n\nFor integers $k\\geq2$ and $m\\geq0$, the highest level reachable by controlling a $k$th-level target with Pauli periodicity $m$ is\n\n \\begin{equation}\n L(n,k,m):=\\max\\bigl\\{\\ell_{n+1}(CU):\n U\\in\\mathcal C_k(n),\\ m(U)=m,\\ CU\\in\\mathcal C_\\infty(n+1)\\bigr\\}.\n\\tag{3}\n\\end{equation} \nThe set in Eq. (3) is nonempty because $U=e^{i\\pi/2^{m+1}}I$ qualifies, and the maximum is attained because $\\mathcal C_k(n)$ is finite modulo global phase while fixing $m(U)$ leaves finitely many admissible phases. Membership $U\\in\\mathcal C_k(n)$ includes the lower levels, so $L(n,k,m)$ is nondecreasing in $k$. Determine $L(n,k,m)$ for all $n\\geq1$, $k\\geq3$, and $m\\geq0$. In particular, is $L(n,k,m)$ bounded by a function of $k$ and $m$ alone, independently of the number $n$ of target qubits?", "url": "https://qiqc-op.com/problem/op_e432fc5ff73e63be/", "json": "https://qiqc-op.com/api/problems/op_e432fc5ff73e63be.json", "tex": "https://qiqc-op.com/problem/op_e432fc5ff73e63be/op_e432fc5ff73e63be.tex", "created": "2026-09-15", "updated": "2026-09-15", "createdAt": "2026-09-15T18:46:16.000Z", "updatedAt": "2026-09-15T18:46:16.000Z", "sha256": "e7e0c4c9a61a2b8c12fbda6a454a46705c1b9809d81ecb8dc47efb5f7ab8c5e8" }, { "id": "op_fc663252f5664f40", "ulid": "01M2JDG8C75QPM012SF6713NZW", "aliases": [ "op_fc663252f5664f40", "01M2JDG8C75QPM012SF6713NZW", "op-fc663252f5664f40" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 1, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-15T11:33:42.919Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "source-stated", "posed": null, "areaIds": [ "quantum-communication", "quantum-resource-theory" ], "topicIds": [ "quantum-capacity", "bound-entanglement", "entanglement-distillation" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [ "01M1HME780V1V27MDXXVA3SCW5" ] }, "title": "Zero two-way quantum capacity of endpoint Holevo–Werner channels", "status": "Unsolved", "fields": [ "Quantum Communication", "Quantum Resource Theory" ], "topics": [ "Quantum capacity", "Bound entanglement", "Entanglement distillation" ], "tags": [ "Quantum Communication", "Quantum Resource Theory", "Quantum capacity", "Bound entanglement", "Entanglement distillation" ], "statement": "Is the two-way assisted quantum capacity of the Holevo–Werner channel $\\mathcal W_d$ zero for every integer $d\\geq3$? For $d\\geq3$, define $\\mathcal W_d:\\mathcal L(\\mathbb C^d)\\to\\mathcal L(\\mathbb C^d)$ by\n\n \\begin{equation}\n \\mathcal W_d(X):=\\frac{\\operatorname{Tr}(X)I_d-\\frac12X^{\\mathsf T}}{d-\\frac12},\n\\tag{1}\n\\end{equation} \nwhere $\\mathsf T$ denotes the transpose in the computational basis. Let $S_d$ be the swap operator on $\\mathbb C^d\\otimes\\mathbb C^d$, and let $\\phi_d:=\\lvert\\phi_d\\rangle\\!\\langle\\phi_d\\rvert$ with $\\lvert\\phi_d\\rangle:=d^{-1/2}\\sum_{j=0}^{d-1}\\lvert jj\\rangle$. The channel in Eq. (1) has the normalized Choi state\n\n \\begin{equation}\n \\rho_d:=(\\operatorname{id}\\otimes\\mathcal W_d)(\\phi_d)\n =\\frac{I_{d^2}-\\frac12S_d}{d^2-\\frac d2}.\n\\tag{2}\n\\end{equation} \nLet $Q_{\\leftrightarrow}(\\mathcal W_d)$ be the supremum of the rates, in qubits per channel use, that can be transmitted with vanishing error by protocols that interleave uses of $\\mathcal W_d$ with adaptive local operations and unlimited two-way classical communication, starting without shared entanglement. The question is whether\n\n \\begin{equation}\n Q_{\\leftrightarrow}(\\mathcal W_d)=0\n \\qquad\\text{for every integer }d\\geq3 .\n\\tag{3}\n\\end{equation}", "url": "https://qiqc-op.com/problem/op_fc663252f5664f40/", "json": "https://qiqc-op.com/api/problems/op_fc663252f5664f40.json", "tex": "https://qiqc-op.com/problem/op_fc663252f5664f40/op_fc663252f5664f40.tex", "created": "2026-09-15", "updated": "2026-09-15", "createdAt": "2026-09-15T18:46:16.000Z", "updatedAt": "2026-09-15T18:46:16.000Z", "sha256": "eca97bd4095e41ef22adf779b7c53317bc9996fb8b7b7a0367672974f509de6b" }, { "id": "op_1726fa212d725bc8", "ulid": "01M22C443FNCYPRHCEFQK17404", "aliases": [ "op_1726fa212d725bc8", "01M22C443FNCYPRHCEFQK17404", "op-1726fa212d725bc8" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 1, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-09T06:01:45.839Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "source-stated", "posed": null, "areaIds": [ "quantum-algorithm" ], "topicIds": [ "computational-complexity-and-computability" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [] }, "title": "Geelen’s simulation conjecture for vertex-minor-closed graph classes", "status": "Unsolved", "fields": [ "Quantum algorithm" ], "topics": [ "Computational complexity and computability" ], "tags": [ "Quantum algorithm", "Computational complexity and computability" ], "statement": "Does Geelen’s simulation conjecture hold for every nonempty proper class $\\mathcal C\\subsetneq\\mathcal G_{\\mathrm{fin}}$ of finite simple graphs that is closed under isomorphism, local complementation, and vertex deletion? Here $\\mathcal G_{\\mathrm{fin}}$ denotes all finite simple graphs, and local complementation toggles the edges between distinct neighbors of one vertex. The resource state of $G=(V,E)\\in\\mathcal C$, with $n:=|V|$, is defined in Eq. (1):\n\n \\begin{equation}\n\\begin{aligned}\n|G\\rangle&:=\\prod_{\\{u,v\\}\\in E}CZ_{uv}|+\\rangle^{\\otimes n},\\\\\n|+\\rangle&:=\\frac{|0\\rangle+|1\\rangle}{\\sqrt2},\n\\qquad CZ:=\\operatorname{diag}(1,1,1,-1).\n\\end{aligned}\n\\tag{1}\n\\end{equation} \nLet $\\mathsf{BQP}_{\\mathcal C}$ denote the languages decided with error at most $1/3$ by uniform polynomial-time measurement-based quantum computations whose classical controller generates a graph in $\\mathcal C$ and adaptively specifies single-qubit projective measurements with polynomial-size, efficiently evaluable algebraic descriptions, allowing randomized polynomial-time classical processing. Let $\\mathsf{BPP}$ denote randomized classical polynomial-time decision with error at most $1/3$. The conjectured equality is Eq. (2):\n\n \\begin{equation}\n\\mathsf{BQP}_{\\mathcal C}=\\mathsf{BPP}.\n\\tag{2}\n\\end{equation}", "url": "https://qiqc-op.com/problem/op_1726fa212d725bc8/", "json": "https://qiqc-op.com/api/problems/op_1726fa212d725bc8.json", "tex": "https://qiqc-op.com/problem/op_1726fa212d725bc8/op_1726fa212d725bc8.tex", "created": "2026-09-09", "updated": "2026-09-15", "createdAt": "2026-09-09T09:07:10.000Z", "updatedAt": "2026-09-15T01:28:24.000Z", "sha256": "d2f536a7be8e4ee2ea8ab289ccdda4b2079c225ccf0f18708366edd156f1ce8c" }, { "id": "op_38f9dbb6c3f92b0c", "ulid": "01M22N8KM3DY3A60ZMT6VG3STQ", "aliases": [ "op_38f9dbb6c3f92b0c", "01M22N8KM3DY3A60ZMT6VG3STQ", "op-38f9dbb6c3f92b0c" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 1, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-09T08:41:29.987Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "source-stated", "posed": null, "areaIds": [ "quantum-communication" ], "topicIds": [ "quantum-capacity", "strong-converses" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [ "01M1Q787QRY0AKF1CQTFSQME12", "01M1HME780WGEQBEXGETXMBSCQ" ] }, "title": "Strong converse for the quantum capacity of a general channel", "status": "Solved", "fields": [ "Quantum Communication" ], "topics": [ "Quantum capacity", "Strong converses" ], "tags": [ "Quantum Communication", "Quantum capacity", "Strong converses" ], "statement": "Does every finite-dimensional memoryless quantum channel obey the strong converse at its unassisted quantum capacity? Let $\\mathcal N:\\mathcal L(A)\\to\\mathcal L(B)$ be a completely positive trace-preserving map. An $n$-use entanglement-transmission code consists of arbitrary quantum channels $\\mathcal E_n:\\mathcal L(S_n)\\to\\mathcal L(A^{\\otimes n})$ and $\\mathcal D_n:\\mathcal L(B^{\\otimes n})\\to\\mathcal L(\\widehat S_n)$, where $\\dim S_n=\\dim\\widehat S_n=M_n$. The parties share no assisting resource and exchange no classical messages. With $\\Phi_{M_n}$ the normalized maximally entangled state between a reference $R_n$ and the message system, define\n\n \\begin{equation}\n F_n:=\\operatorname{Tr}\\!\\left[\\Phi_{M_n}^{R_n\\widehat S_n}(\\operatorname{id}_{R_n}\\otimes\\mathcal D_n\\circ\\mathcal N^{\\otimes n}\\circ\\mathcal E_n)(\\Phi_{M_n}^{R_n S_n})\\right],\\qquad r_n:=\\frac1n\\log_2 M_n.\n\\tag{1}\n\\end{equation} \nThe quantum capacity $Q(\\mathcal N)$ is the supremum of $\\liminf_n r_n$ over sequences of codes in Eq. (1) with $F_n\\to1$. Is it true that every code sequence satisfies\n\n \\begin{equation}\n \\liminf_{n\\to\\infty}r_n>Q(\\mathcal N)\\quad\\Longrightarrow\\quad F_n\\longrightarrow0?\n\\tag{2}\n\\end{equation} \nA proof must cover arbitrary encoders and collective decoders. A counterexample must give one fixed finite-dimensional channel and a sequence violating Eq. (2).", "url": "https://qiqc-op.com/problem/op_38f9dbb6c3f92b0c/", "json": "https://qiqc-op.com/api/problems/op_38f9dbb6c3f92b0c.json", "tex": "https://qiqc-op.com/problem/op_38f9dbb6c3f92b0c/op_38f9dbb6c3f92b0c.tex", "created": "2026-09-09", "updated": "2026-09-15", "createdAt": "2026-09-09T09:07:10.000Z", "updatedAt": "2026-09-15T01:28:24.000Z", "sha256": "8eea93a5634cea74dd2831b6ad969beb061126798c1742702f4602bc92c0e063" }, { "id": "op_7eefb7506b175200", "ulid": "01M22N25PDE5R1X95GHQBCFTCB", "aliases": [ "op_7eefb7506b175200", "01M22N25PDE5R1X95GHQBCFTCB", "op-7eefb7506b175200" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 1, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-09T08:37:59.117Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "source-stated", "posed": null, "areaIds": [ "quantum-communication" ], "topicIds": [ "classical-capacity", "quantum-coding-theory" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [] }, "title": "Holevo random-coding exponent for classical–quantum channels", "status": "Solved", "fields": [ "Quantum Communication" ], "topics": [ "Classical capacity", "Quantum coding theory" ], "tags": [ "Quantum Communication", "Classical capacity", "Quantum coding theory" ], "statement": "Does every memoryless classical–quantum channel with a finite input alphabet and finite-dimensional output satisfy the random-coding lower bound in Eq. (5), including channels with mixed, noncommuting output states?\n\nLet $W:x\\mapsto W_x$ map a finite alphabet $\\mathcal X$, with $|\\mathcal X|\\geq2$, to density operators on a finite-dimensional Hilbert space $\\mathcal H_B$. All logarithms have base two. An $n$-use code consists of $M$ distinct words $x^n(m)\\in\\mathcal X^n$ and a decoding POVM $\\{\\Lambda_m\\}_{m=1}^M$ on $\\mathcal H_B^{\\otimes n}$. For equiprobable messages, its average error is Eq. (1):\n\n \\begin{equation}\np_e(\\mathcal C_n,\\Lambda):=1-\\frac1M\\sum_{m=1}^M\\operatorname{Tr}\\!\\left[\\Lambda_m\\bigotimes_{i=1}^nW_{x_i(m)}\\right].\n\\tag{1}\n\\end{equation} \nFor $0Q(\\mathcal N)$, there are constants $\\gamma_R>0$ and $n_R$ such that every code in Eq. (4) obeys\n\n \\begin{equation}\n r_n\\geq R,\\qquad n\\geq n_R\n \\quad\\Longrightarrow\\quad\n F_n\\leq 2^{-\\gamma_R n}.\n\\tag{5}\n\\end{equation} \nEquation (5) requires a bound for every encoder–decoder pair, rather than for almost every member of a random code ensemble.", "url": "https://qiqc-op.com/problem/op_cdad869fc8c3c0ea/", "json": "https://qiqc-op.com/api/problems/op_cdad869fc8c3c0ea.json", "tex": "https://qiqc-op.com/problem/op_cdad869fc8c3c0ea/op_cdad869fc8c3c0ea.tex", "created": "2026-09-03", "updated": "2026-09-15", "createdAt": "2026-09-03T02:22:57.000Z", "updatedAt": "2026-09-15T01:28:24.000Z", "sha256": "bf5156f216fd44c01299dfb30f8e57bf3e94a4515750d862087377396533c745" }, { "id": "op_414fbcc7d5dc0c64", "ulid": "01M1HME780WGEQBEXGETXMBSCQ", "aliases": [ "op_414fbcc7d5dc0c64", "01M1HME780WGEQBEXGETXMBSCQ", "op-414fbcc7d5dc0c64", "v2-exponential-strong-converse-for-transpose-degradable-channels", "open-problem-v2-problem-55" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 2, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-04T22:04:59Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "source-stated", "posed": null, "areaIds": [ "quantum-communication" ], "topicIds": [ "quantum-capacity", "channel-degradability", "strong-converses" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [] }, "title": "Exponential strong converse for transpose-degradable channels", "status": "Solved", "fields": [ "Quantum Communication" ], "topics": [ "Quantum capacity", "Channel degradability", "Strong converses" ], "tags": [ "Quantum Communication", "Quantum capacity", "Channel degradability", "Strong converses" ], "statement": "Does every finite-dimensional transpose-degradable channel satisfy an exponential strong converse for quantum communication at its single-letter quantum capacity? Let $V:A\\to B\\otimes E$ be an isometry and suppose that\n\n \\begin{equation}\n \\Phi(X):=\\operatorname{Tr}_E(VXV^\\dagger),\n \\qquad\n \\Phi^c(X):=\\operatorname{Tr}_B(VXV^\\dagger),\n \\qquad\n \\mathsf T_E\\circ\\Phi^c=\\mathcal G\\circ\\Phi,\n\\tag{1}\n\\end{equation} \nwhere $\\mathsf T_E$ is transpose in a fixed basis of $E$ and $\\mathcal G:\\mathcal L(B)\\to\\mathcal L(E)$ is completely positive and trace preserving. For the channel in Eq. (1), transpose degradability gives\n\n \\begin{equation}\n Q(\\Phi)=Q^{(1)}(\\Phi)\n :=\\max_{\\rho_A}\n \\left[S(\\Phi(\\rho_A))-S(\\Phi^c(\\rho_A))\\right],\n \\qquad\n S(\\sigma):=-\\operatorname{Tr}(\\sigma\\log_2\\sigma).\n\\tag{2}\n\\end{equation} \nEquation (2) fixes the rate threshold.\n\nAt blocklength $n$, let $\\mathcal E_n:\\mathcal L(S_n)\\to\\mathcal L(A^{\\otimes n})$ and $\\mathcal R_n:\\mathcal L(B^{\\otimes n})\\to\\mathcal L(\\widehat S_n)$ be arbitrary encoder and decoder channels, with $\\dim R_n=\\dim S_n=\\dim\\widehat S_n=M_n$. Define the maximally entangled target by\n\n \\begin{equation}\n \\varphi_{M_n}:=\n |\\varphi_{M_n}\\rangle\\!\\langle\\varphi_{M_n}|,\n \\qquad\n |\\varphi_{M_n}\\rangle\n :=\\frac1{\\sqrt{M_n}}\\sum_{i=1}^{M_n}|i\\rangle_{R_n}|i\\rangle_{S_n}.\n\\tag{3}\n\\end{equation} \nUsing the target in Eq. (3), let\n\n \\begin{equation}\n \\omega_n:=\n \\left(\\operatorname{id}_{R_n}\\otimes\n \\mathcal R_n\\circ\\Phi^{\\otimes n}\\circ\\mathcal E_n\\right)(\\varphi_{M_n}),\n \\qquad\n r_n:=\\frac1n\\log_2M_n,\n \\qquad\n F_n:=\\operatorname{Tr}(\\varphi_{M_n}^{R_n\\widehat S_n}\\omega_n).\n\\tag{4}\n\\end{equation} \nThe problem is whether, for every channel in Eq. (1), the quantities in Eq. (4) satisfy\n\n \\begin{equation}\n \\forall R>Q(\\Phi)\\ \\exists\\,\\gamma_R>0,\\ n_R\\in\\mathbb N:\n \\quad\n r_n\\geq R,\\ n\\geq n_R\n \\ \\Longrightarrow\\\n F_n\\leq2^{-\\gamma_R n}\n\\tag{5}\n\\end{equation} \nfor every encoder–decoder sequence. Equation (5) is the all-code exponential strong-converse property at the threshold in Eq. (2).", "url": "https://qiqc-op.com/problem/op_414fbcc7d5dc0c64/", "json": "https://qiqc-op.com/api/problems/op_414fbcc7d5dc0c64.json", "tex": "https://qiqc-op.com/problem/op_414fbcc7d5dc0c64/op_414fbcc7d5dc0c64.tex", "created": "2026-09-02", "updated": "2026-09-15", "createdAt": "2026-09-02T00:20:51.000Z", "updatedAt": "2026-09-15T01:28:24.000Z", "sha256": "34d19965e6131918cf7ef50f49b40b684a24c96f796c5f024b04eedc1bba5c3f" }, { "id": "op_2e91f2de9c41103f", "ulid": "01M26JZEN93BAESCCMMTRG84M0", "aliases": [ "op_2e91f2de9c41103f", "01M26JZEN93BAESCCMMTRG84M0", "op-2e91f2de9c41103f" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 1, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-10T21:18:30.569Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "source-stated", "posed": null, "areaIds": [ "quantum-cryptography" ], "topicIds": [ "secret-key-distillation", "bosonic-channels" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [ "01M1HME780AZ2GCKS76GDX97RQ", "01M26JZEPTR8YAH8EKFW9D0Q24", "01M26JZET5QZ2FB0WSWESW77V6" ] }, "title": "Secret-key capacity of thermal attenuators and amplifiers", "status": "Unsolved", "fields": [ "Quantum Cryptography" ], "topics": [ "Secret-key distillation", "Bosonic channels" ], "tags": [ "Quantum Cryptography", "Secret-key distillation", "Bosonic channels" ], "statement": "What is the secret-key capacity of a thermal attenuator for every transmissivity and thermal noise level? Fix a transmissivity $0<\\eta<1$ and $b\\geq0$. The single-mode thermal attenuator $\\mathcal L_{\\eta,b}$ is realized by a beam splitter with output annihilation operator\n\n \\begin{equation}\n a_{\\mathrm{out}}=\\sqrt\\eta\\,a+\\sqrt{1-\\eta}\\,e.\n\\tag{1}\n\\end{equation} \nThe independent input and environment modes obey $[a,a^\\dagger]=[e,e^\\dagger]=1$. The environment is a fresh independent thermal mode at each use, with mean photon number $b$ and density operator\n\n \\begin{equation}\n \\tau_b=\\frac1{b+1}\\sum_{n=0}^{\\infty}\\left(\\frac b{b+1}\\right)^n|n\\rangle\\langle n|\\quad(b>0),\\qquad \\tau_0=|0\\rangle\\langle0|.\n\\tag{2}\n\\end{equation} \nHere $|n\\rangle$ is the $n$-photon Fock state. The unused output mode is discarded. Equations (1) and (2) specify the channel on arbitrary input states. Protocols may use arbitrary adaptive local quantum operations and unlimited two-way public classical communication. The parties initially share no entanglement or secret key. There is no input-energy bound; the capacity is the supremum over finite mean input-energy budgets. The secret-key capacity $K$ is the supremum of asymptotic secret bits per channel use. Correctness and secrecy errors must vanish. Secrecy is against an adversary holding a purification of the channel environment and the public transcript. An equivalent formulation asks for the secret-key capacity of a thermal amplifier. With the same thermal environment, define\n\n \\begin{equation}\n\\begin{aligned}\n a_{\\mathrm{out}}&=\\sqrt\\kappa\\,a+\\sqrt{\\kappa-1}\\,e^\\dagger,\n \\qquad \\kappa>1,\\\\\n K(\\mathcal A_{\\kappa,b})&=K(\\mathcal L_{1/\\kappa,b}).\n\\end{aligned}\n\\tag{3}\n\\end{equation} \nThus Eq. (3) identifies the two capacity functions after changing parameters. This identity uses unlimited two-way communication and the supremum over finite energy budgets.", "url": "https://qiqc-op.com/problem/op_2e91f2de9c41103f/", "json": "https://qiqc-op.com/api/problems/op_2e91f2de9c41103f.json", "tex": "https://qiqc-op.com/problem/op_2e91f2de9c41103f/op_2e91f2de9c41103f.tex", "created": "2026-09-10", "updated": "2026-09-14", "createdAt": "2026-09-10T21:54:15.000Z", "updatedAt": "2026-09-14T21:26:47.000Z", "sha256": "b445f5ed5bc8adbcb3aa58b66b1dcd1c41e72d992d089fa40cf76eb4c4ba64e3" }, { "id": "op_45127bf1f4f35d7a", "ulid": "01M26KH5XKRWK5QH0SYAG510BK", "aliases": [ "op_45127bf1f4f35d7a", "01M26KH5XKRWK5QH0SYAG510BK", "op-45127bf1f4f35d7a" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 1, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-10T21:28:11.443Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "source-stated", "posed": null, "areaIds": [ "quantum-resource-theory" ], "topicIds": [ "quantum-separability" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [] }, "title": "Bounded witnesses for every continuous-variable entangled state", "status": "Solved", "fields": [ "Quantum Resource Theory" ], "topics": [ "Quantum separability" ], "tags": [ "Quantum Resource Theory", "Quantum separability" ], "statement": "Does every entangled bosonic state admit a bounded entanglement witness?\n\nLet $m,n\\geq1$ be arbitrary integers, with $\\mathcal H_A:=L^2(\\mathbb R^m)$ and $\\mathcal H_B:=L^2(\\mathbb R^n)$. Let $\\rho$ be an entangled density operator on $\\mathcal H_A\\otimes\\mathcal H_B$. Separability means membership in the trace-norm closed convex hull of product density operators. States with positive partial transpose $\\rho^{T_B}$ in a fixed product Fock basis are included.\n\nThe desired operator $L$ is bounded and self-adjoint and satisfies\n\n \\begin{equation}\n\\operatorname{Tr}(\\rho L)>\\sup_{\\|a\\|=\\|b\\|=1}\\langle a\\otimes b|L|a\\otimes b\\rangle.\n\\tag{1}\n\\end{equation} \nIn Eq. (1), $a\\in\\mathcal H_A$ and $b\\in\\mathcal H_B$. No finite-energy or moment-existence assumption is imposed.", "url": "https://qiqc-op.com/problem/op_45127bf1f4f35d7a/", "json": "https://qiqc-op.com/api/problems/op_45127bf1f4f35d7a.json", "tex": "https://qiqc-op.com/problem/op_45127bf1f4f35d7a/op_45127bf1f4f35d7a.tex", "created": "2026-09-10", "updated": "2026-09-14", "createdAt": "2026-09-10T21:54:15.000Z", "updatedAt": "2026-09-14T21:26:47.000Z", "sha256": "51846617a856aa2d02a536cb30efc7d9ce3bb6e0f851a5add5395a63b5a6ea02" }, { "id": "op_4b453163f7f8c748", "ulid": "01M26KH5T8PTXGWXV0GP1JW6WC", "aliases": [ "op_4b453163f7f8c748", "01M26KH5T8PTXGWXV0GP1JW6WC", "op-4b453163f7f8c748" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 1, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-10T21:28:11.336Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "derived", "posed": null, "areaIds": [ "quantum-resource-theory" ], "topicIds": [ "gaussian-quantum-information", "bound-entanglement", "entanglement-cost" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [] }, "title": "LOCC entanglement cost of PPT-entangled Gaussian states", "status": "Unsolved", "fields": [ "Quantum Resource Theory" ], "topics": [ "Gaussian quantum information", "Bound entanglement", "Entanglement cost" ], "tags": [ "Quantum Resource Theory", "Gaussian quantum information", "Bound entanglement", "Entanglement cost" ], "statement": "What is the exact LOCC entanglement cost of a finite-energy PPT-entangled Gaussian state?\n\nLet $\\rho_V$ be a zero-mean Gaussian state of $m$ modes held by Alice and $n$ modes held by Bob, with integers $m,n\\geq2$. Assume that $\\rho_V$ is entangled and its partial transpose $\\rho_V^{T_B}$ is positive. Its finite covariance matrix and canonical quadratures satisfy\n\n \\begin{equation}\n\\begin{gathered}\nV_{jk}:=\\operatorname{Tr}\\rho_V\\{R_j,R_k\\},\\qquad\n[R_j,R_k]=i(\\Omega_{m+n})_{jk},\\\\\n\\Omega_k:=\\bigoplus_{j=1}^k\\begin{pmatrix}0&1\\\\-1&0\\end{pmatrix}.\n\\end{gathered}\n\\tag{1}\n\\end{equation} \nEquation (1) uses vacuum covariance $I$. The allowed channels $\\Lambda_N$ use unrestricted local operations and classical communication (LOCC). Define the Bell-pair density operator by\n\n \\begin{equation}\n\\Phi_2:=|\\phi_2\\rangle\\langle\\phi_2|,\\qquad\n|\\phi_2\\rangle:=(|00\\rangle+|11\\rangle)/\\sqrt2.\n\\tag{2}\n\\end{equation} \nUsing Eq. (2), the entanglement cost $E_C(\\rho_V)$ is the infimum of rates $r\\geq0$ for which LOCC channels $\\Lambda_N$ satisfy\n\n \\begin{equation}\n\\lim_{N\\to\\infty}\\|\\Lambda_N(\\Phi_2^{\\otimes\\lceil rN\\rceil})-\\rho_V^{\\otimes N}\\|_1=0.\n\\tag{3}\n\\end{equation} \nDetermine the cost specified by Eq. (3) as a function of $V$. Separability means the trace-norm closed convex hull of product states. Here $\\|X\\|_1:=\\operatorname{Tr}\\sqrt{X^\\dagger X}$.", "url": "https://qiqc-op.com/problem/op_4b453163f7f8c748/", "json": "https://qiqc-op.com/api/problems/op_4b453163f7f8c748.json", "tex": "https://qiqc-op.com/problem/op_4b453163f7f8c748/op_4b453163f7f8c748.tex", "created": "2026-09-10", "updated": "2026-09-14", "createdAt": "2026-09-10T21:54:15.000Z", "updatedAt": "2026-09-14T21:26:47.000Z", "sha256": "20bed7a1402bb035874a9e0327ee1b8682a2999a858f329e69090bd2760e4da4" }, { "id": "op_c1499a0073ee800c", "ulid": "01M26KH5Q01RBBB2QYFHZ8D5V7", "aliases": [ "op_c1499a0073ee800c", "01M26KH5Q01RBBB2QYFHZ8D5V7", "op-c1499a0073ee800c" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 1, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-10T21:28:11.232Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "source-stated", "posed": null, "areaIds": [ "quantum-resource-theory" ], "topicIds": [ "gaussian-quantum-information", "entanglement-cost", "additivity-and-regularization" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [ "01M26KH5NF45HBHXTPG8ZBD75H" ] }, "title": "Additivity of entanglement of formation for two-mode Gaussian states", "status": "Unsolved", "fields": [ "Quantum Resource Theory" ], "topics": [ "Gaussian quantum information", "Entanglement cost", "Additivity and regularization" ], "tags": [ "Quantum Resource Theory", "Gaussian quantum information", "Entanglement cost", "Additivity and regularization" ], "statement": "Is unrestricted entanglement of formation additive on tensor powers of every finite-energy two-mode Gaussian state?\n\nLet $\\rho_{AB}$ be any finite-energy two-mode Gaussian density operator, shared between one mode at each party. Finite energy means finite total mean photon number. Define the unrestricted pure-state convex roof\n\n \\begin{equation}\n\\begin{gathered}\nE_F(\\omega):=\\inf_\\mu\\int S(\\operatorname{Tr}_B|\\psi\\rangle\\langle\\psi|)\\,\\mu(d\\psi),\n\\\\\nS(\\tau):=-\\operatorname{Tr}(\\tau\\log_2\\tau),\n\\end{gathered}\n\\tag{1}\n\\end{equation} \nIn Eq. (1), $\\mu$ ranges over pure-state probability measures with barycentre $\\omega$.\n\nThe target is $E_F(\\rho_{AB}^{\\otimes k})=kE_F(\\rho_{AB})$ for every integer $k\\geq1$. The bipartition for the tensor power is $A^k:B^k$. Non-Gaussian vectors are allowed in all decompositions.", "url": "https://qiqc-op.com/problem/op_c1499a0073ee800c/", "json": "https://qiqc-op.com/api/problems/op_c1499a0073ee800c.json", "tex": "https://qiqc-op.com/problem/op_c1499a0073ee800c/op_c1499a0073ee800c.tex", "created": "2026-09-10", "updated": "2026-09-14", "createdAt": "2026-09-10T21:54:15.000Z", "updatedAt": "2026-09-14T21:26:47.000Z", "sha256": "08f472679d7043c4f477fae9e2fd21ba6fbf8eef5840d8b52b83495226d7f0f2" }, { "id": "op_cb7cdf37ec9b9ec5", "ulid": "01M26KH5NF45HBHXTPG8ZBD75H", "aliases": [ "op_cb7cdf37ec9b9ec5", "01M26KH5NF45HBHXTPG8ZBD75H", "op-cb7cdf37ec9b9ec5" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 1, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-10T21:28:11.183Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "source-stated", "posed": null, "areaIds": [ "quantum-resource-theory" ], "topicIds": [ "gaussian-quantum-information", "entanglement-measures" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [ "01M1HME78068MQY7E9KA81B7WX" ] }, "title": "Gaussian optimality of two-mode entanglement of formation", "status": "Solved", "fields": [ "Quantum Resource Theory" ], "topics": [ "Gaussian quantum information", "Entanglement measures" ], "tags": [ "Quantum Resource Theory", "Gaussian quantum information", "Entanglement measures" ], "statement": "Does Gaussian entanglement of formation equal unrestricted entanglement of formation for every finite-energy two-mode Gaussian state?\n\nLet $\\rho_{AB}$ be any finite-energy two-mode Gaussian density operator, with one bosonic mode held by each party. Finite energy means finite total mean photon number. Define\n\n \\begin{equation}\n\\begin{gathered}\nE_F(\\rho_{AB}):=\\inf_\\mu\\int S(\\operatorname{Tr}_B|\\psi\\rangle\\langle\\psi|)\\,\\mu(d\\psi),\n\\\\\nS(\\omega):=-\\operatorname{Tr}(\\omega\\log_2\\omega).\n\\end{gathered}\n\\tag{1}\n\\end{equation} \nThe infimum in Eq. (1) is over probability measures on normalised pure vectors with barycentre $\\rho_{AB}$. The Gaussian entanglement of formation $E_F^{\\mathrm G}$ restricts the same infimum to pure Gaussian vectors.\n\nThe proposed equality is $E_F(\\rho_{AB})=E_F^{\\mathrm G}(\\rho_{AB})$, without an exchange-symmetry assumption.", "url": "https://qiqc-op.com/problem/op_cb7cdf37ec9b9ec5/", "json": "https://qiqc-op.com/api/problems/op_cb7cdf37ec9b9ec5.json", "tex": "https://qiqc-op.com/problem/op_cb7cdf37ec9b9ec5/op_cb7cdf37ec9b9ec5.tex", "created": "2026-09-10", "updated": "2026-09-14", "createdAt": "2026-09-10T21:54:15.000Z", "updatedAt": "2026-09-14T21:26:47.000Z", "sha256": "c3ab5482bae50ab46bb90144856dd26a7bfbfb355d2fc6f2aa723d004fc0f3eb" }, { "id": "op_1ab8b10386bddd66", "ulid": "01M22MXHAJ50EYBHXWXX635JQK", "aliases": [ "op_1ab8b10386bddd66", "01M22MXHAJ50EYBHXWXX635JQK", "op-1ab8b10386bddd66" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 1, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-09T08:35:27.186Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "derived", "posed": null, "areaIds": [ "quantum-communication" ], "topicIds": [ "quantum-source-coding", "channel-simulation" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [ "01M208AZBKSZTSQ6KS8YV38RFK" ] }, "title": "Visible quantum compression with free shared randomness at the Holevo rate", "status": "Unsolved", "fields": [ "Quantum Communication" ], "topics": [ "Quantum source coding", "Channel simulation" ], "tags": [ "Quantum Communication", "Quantum source coding", "Channel simulation" ], "statement": "Does free shared classical randomness make the Holevo information achievable for visible compression of every finite mixed-state ensemble? Let $p$ be a probability distribution with full support on a finite alphabet $\\mathcal X$, and let $W_x$ be density operators on a finite-dimensional system $B$. Alice knows the independent and identically distributed label sequence $x^n$, with probability $p^n(x^n)$; the target is $W_{x^n}:=\\bigotimes_{t=1}^nW_{x_t}$. Alice and Bob share a finite random variable $S_n$, independent of $x^n$, with arbitrary distribution $q_n$ and no bound on its size. For each seed $s$, Alice prepares a density operator $\\tau_{x^n,s}$ on a quantum message system $M_n$ and sends it through a noiseless quantum channel. Bob applies a completely positive trace-preserving map $\\mathcal D_{n,s}:\\mathcal L(M_n)\\to\\mathcal L(B^{\\otimes n})$. No initially shared entanglement or additional communication is available. The reproduced state and block error are\n\n \\begin{equation}\n \\widehat W_{x^n}:=\\sum_s q_n(s)\\mathcal D_{n,s}(\\tau_{x^n,s}),\\qquad\n \\varepsilon_n:=\\frac12\\sum_{x^n}p^n(x^n)\\|W_{x^n}-\\widehat W_{x^n}\\|_1.\n\\tag{1}\n\\end{equation} \nThus the shared seed is averaged before taking the trace norm in Eq. (1). Define the rate, in qubits per signal, and the ensemble Holevo information by\n\n \\begin{equation}\n R_{\\mathrm{vis},q,\\mathrm{cr}}(p,W):=\\inf_{\\{\\text{codes}:\\varepsilon_n\\to0\\}}\\limsup_{n\\to\\infty}\\frac1n\\log_2\\dim M_n,\n \\qquad\n \\chi(p,W):=S\\!\\left(\\sum_xp_xW_x\\right)-\\sum_xp_xS(W_x),\n\\tag{2}\n\\end{equation} \nwhere $S$ is the von Neumann entropy in bits. Is $R_{\\mathrm{vis},q,\\mathrm{cr}}(p,W)=\\chi(p,W)$ in Eq. (2) for every ensemble, or is there a finite ensemble with a strict gap?", "url": "https://qiqc-op.com/problem/op_1ab8b10386bddd66/", "json": "https://qiqc-op.com/api/problems/op_1ab8b10386bddd66.json", "tex": "https://qiqc-op.com/problem/op_1ab8b10386bddd66/op_1ab8b10386bddd66.tex", "created": "2026-09-09", "updated": "2026-09-14", "createdAt": "2026-09-09T09:07:10.000Z", "updatedAt": "2026-09-14T21:26:47.000Z", "sha256": "042b6b81c8d681d1f6bbd04bde765e1c565c1795f420b1cfaa09d252de80e61c" }, { "id": "op_4154bf4cbe0d288e", "ulid": "01M22C4460W3FYGAFEKQT833BJ", "aliases": [ "op_4154bf4cbe0d288e", "01M22C4460W3FYGAFEKQT833BJ", "op-4154bf4cbe0d288e" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 1, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-09T06:01:45.920Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "derived", "posed": null, "areaIds": [ "quantum-resource-theory", "quantum-algorithm" ], "topicIds": [ "quantum-state-preparation", "resource-conversion", "computational-complexity-and-computability" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [ "01M22C448YAVGY33AV04GHTFGC" ] }, "title": "Deterministic quadratic-size vertex-minor-universal graphs", "status": "Unsolved", "fields": [ "Quantum Resource Theory", "Quantum algorithm" ], "topics": [ "Quantum state preparation", "Resource conversion", "Computational complexity and computability" ], "tags": [ "Quantum Resource Theory", "Quantum algorithm", "Quantum state preparation", "Resource conversion", "Computational complexity and computability" ], "statement": "Does there exist an absolute constant $C>0$ and a deterministic algorithm that, for each integer $k\\geq2$ supplied in unary, runs in time $k^{O(1)}$ and outputs a $k$-vertex-minor-universal graph with at most $Ck^2$ vertices? The output is a finite simple labelled graph $G_k=(V_k,E_k)$. Write $H\\leq_{\\mathrm{vm}}G_k$ when $H$ is obtainable by local complementations and vertex deletions that preserve the labels of surviving vertices; local complementation toggles edges between distinct neighbors of a vertex. Universality requires every graph on every prescribed $k$-element subset of $V_k$ to be obtainable, as in Eq. (1):\n\n \\begin{equation}\n\\begin{gathered}\nk\\leq|V_k|\\leq Ck^2,\\\\\n\\forall S\\subseteq V_k\\text{ with }|S|=k,\\quad\n\\forall E_H\\subseteq\\{\\{u,v\\}:u,v\\in S,\\ u\\neq v\\},\\\\\n(S,E_H)\\leq_{\\mathrm{vm}}G_k.\n\\end{gathered}\n\\tag{1}\n\\end{equation}", "url": "https://qiqc-op.com/problem/op_4154bf4cbe0d288e/", "json": "https://qiqc-op.com/api/problems/op_4154bf4cbe0d288e.json", "tex": "https://qiqc-op.com/problem/op_4154bf4cbe0d288e/op_4154bf4cbe0d288e.tex", "created": "2026-09-09", "updated": "2026-09-14", "createdAt": "2026-09-09T09:07:10.000Z", "updatedAt": "2026-09-14T21:26:47.000Z", "sha256": "d2a67dab9bf97fb09b81e277d33b07317362e76dc2f1a4b2cdd5ded2ae5422a2" }, { "id": "op_722706a9205dcff2", "ulid": "01M22MSWB2Y2FYAA13A3SJEVZ6", "aliases": [ "op_722706a9205dcff2", "01M22MSWB2Y2FYAA13A3SJEVZ6", "op-722706a9205dcff2" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 1, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-09T08:33:27.394Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "source-stated", "posed": "2025-12-29", "areaIds": [ "quantum-communication" ], "topicIds": [ "quantum-channel-structure" ], "keywords": [ "random-unitary decomposition", "mixed unitarity", "quantum dynamical semigroup", "first mixed-unitary time", "eventual mixed-unitary time" ], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [] }, "title": "Loss of mixed unitarity after a positive time in a quantum dynamical semigroup", "status": "Unsolved", "fields": [ "Quantum Communication" ], "topics": [ "Quantum channel structure" ], "tags": [ "Quantum Communication", "Quantum channel structure" ], "statement": "Does there exist a norm-continuous semigroup $(\\Phi_t)_{t\\geq0}$ of unital completely positive trace-preserving maps on $\\mathcal L(\\mathbb C^d)$, for some finite integer $d\\geq3$, such that $\\Phi_s$ is mixed unitary and $\\Phi_t$ is not mixed unitary for some $0b(q,d).\n\\tag{3}\n\\end{equation} \nA complete answer is either an explicit stabilizer code satisfying Eqs. (1) and (3), or a proof that Eq. (3) fails for every prime power $q$, every $d\\geq3$, and every $k\\geq1$, which establishes the quantum MDS conjecture for codes with at least one logical qudit.", "url": "https://qiqc-op.com/problem/op_2a6fc2b70a85f846/", "json": "https://qiqc-op.com/api/problems/op_2a6fc2b70a85f846.json", "tex": "https://qiqc-op.com/problem/op_2a6fc2b70a85f846/op_2a6fc2b70a85f846.tex", "created": "2026-09-13", "updated": "2026-09-14", "createdAt": "2026-09-13T09:05:31.000Z", "updatedAt": "2026-09-14T19:34:29.000Z", "sha256": "0f6aded75ab2f0c31290602967e266d4908ba9c0078005438ecd91007b41a8f5" }, { "id": "op_4765bc6f2ac91d46", "ulid": "01M2CZEFG3XB292BG36ZND889S", "aliases": [ "op_4765bc6f2ac91d46", "01M2CZEFG3XB292BG36ZND889S", "op-4765bc6f2ac91d46" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 1, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-13T08:51:52.451Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "source-stated", "posed": null, "areaIds": [ "quantum-error-correction" ], "topicIds": [ "quantum-coding-theory" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [] }, "title": "Existence of a ((7,3,3)) qubit code", "status": "Unsolved", "fields": [ "Quantum Error Correction" ], "topics": [ "Quantum coding theory" ], "tags": [ "Quantum Error Correction", "Quantum coding theory" ], "statement": "Does a qubit quantum error-correcting code with parameters $((7,3,3))_2$ exist, that is, a three-dimensional subspace of seven qubits with minimum distance at least three? Let $\\mathcal{P}_7:=\\{I,X,Y,Z\\}^{\\otimes7}$, where $I$ is the single-qubit identity and $X,Y,Z$ are the Pauli matrices, and for $E\\in\\mathcal{P}_7$ let $\\operatorname{wt}(E)$ be the number of non-identity tensor factors. A $((7,3,3))_2$ code is the range of an operator $P$ on $(\\mathbb{C}^{2})^{\\otimes7}$ with\n\n \\begin{equation}\n P=P^{\\dagger}=P^{2},\n \\qquad\n \\operatorname{Tr}P=3,\n\\tag{1}\n\\end{equation} \nsatisfying the Knill–Laflamme conditions\n\n \\begin{equation}\n PEP=\\frac{\\operatorname{Tr}(PE)}{3}\\,P\n \\qquad\\text{for every }E\\in\\mathcal{P}_7\\text{ with }\n 1\\leq\\operatorname{wt}(E)\\leq2.\n\\tag{2}\n\\end{equation} \nEquation (2) says that the code detects every Pauli error of weight at most two and hence, by linearity, exactly corrects an arbitrary error on any single qubit. No stabilizer structure, entanglement assistance, or other side resource is assumed; the codewords may be arbitrary vectors of $(\\mathbb{C}^{2})^{\\otimes7}$. Writing $K_{\\max}(7,3)$ for the largest dimension of a seven-qubit code of minimum distance at least three, the question is whether\n\n \\begin{equation}\n K_{\\max}(7,3)\\geq3.\n\\tag{3}\n\\end{equation} \nSuch a code would encode one logical qutrit. A complete answer is either an explicit rank-three projector satisfying Eqs. (1) and (2), or a proof that none exists.", "url": "https://qiqc-op.com/problem/op_4765bc6f2ac91d46/", "json": "https://qiqc-op.com/api/problems/op_4765bc6f2ac91d46.json", "tex": "https://qiqc-op.com/problem/op_4765bc6f2ac91d46/op_4765bc6f2ac91d46.tex", "created": "2026-09-13", "updated": "2026-09-13", "createdAt": "2026-09-13T09:05:31.000Z", "updatedAt": "2026-09-13T09:05:31.000Z", "sha256": "285dde3c2d149ed72aca11c807908f09d5c4d37f4dbde5fa500bfbb12945fb65" }, { "id": "op_87c77263c8bab523", "ulid": "01M1Q787QRJ5ASJACQYA9R7N35", "aliases": [ "op_87c77263c8bab523", "01M1Q787QRJ5ASJACQYA9R7N35", "op-87c77263c8bab523" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 1, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-04T22:04:59Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "source-stated", "posed": null, "areaIds": [ "quantum-communication" ], "topicIds": [ "quantum-recovery", "matrix-and-entropy-inequalities" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [] }, "title": "Ordinary-Petz recovery bound for conditional mutual information", "status": "Solved", "fields": [ "Quantum Communication" ], "topics": [ "Quantum recovery", "Matrix and entropy inequalities" ], "tags": [ "Quantum Communication", "Quantum recovery", "Matrix and entropy inequalities" ], "statement": "Does the ordinary, unrotated Petz map universally recover a tripartite state with fidelity controlled by its conditional mutual information? For a finite-dimensional state $\\rho_{ABC}$, define\n\n \\begin{equation}\n I(A;B\\mid C)_\\rho\n :=S(AC)_\\rho+S(BC)_\\rho-S(C)_\\rho-S(ABC)_\\rho,\n \\qquad\n S(\\tau):=-\\operatorname{Tr}(\\tau\\log_2\\tau).\n\\tag{1}\n\\end{equation} \nThe ordinary Petz map associated with $\\rho_{AC}$ and the channel $\\operatorname{Tr}_A:AC\\to C$ is\n\n \\begin{equation}\n \\mathcal P^{\\rho}_{C\\to AC}(X_C)\n :=\\rho_{AC}^{1/2}\\!\\left[\n I_A\\otimes\\rho_C^{-1/2}X_C\\rho_C^{-1/2}\n \\right]\\rho_{AC}^{1/2},\n\\tag{2}\n\\end{equation} \nwhere the inverse is taken on $\\operatorname{supp}\\rho_C$. With squared fidelity $F(\\tau,\\omega):=\\lVert\\sqrt\\tau\\sqrt\\omega\\rVert_1^2$, determine whether the quantity in Eq. (1) always satisfies\n\n \\begin{equation}\n I(A;B\\mid C)_\\rho\n \\stackrel{?}{\\geq}\n -\\log_2 F\\!\\left(\n \\rho_{ABC},\n (\\operatorname{id}_B\\otimes\\mathcal P^{\\rho}_{C\\to AC})(\\rho_{BC})\n \\right),\n\\tag{3}\n\\end{equation} \nwith the recovered systems ordered canonically as $ABC$.", "url": "https://qiqc-op.com/problem/op_87c77263c8bab523/", "json": "https://qiqc-op.com/api/problems/op_87c77263c8bab523.json", "tex": "https://qiqc-op.com/problem/op_87c77263c8bab523/op_87c77263c8bab523.tex", "created": "2026-09-03", "updated": "2026-09-13", "createdAt": "2026-09-03T02:22:57.000Z", "updatedAt": "2026-09-13T08:05:13.000Z", "sha256": "74ac69f62afcea215c3d4e701c20329377b6c266214fb69fde2078d74ba5b8c1" }, { "id": "op_06e9f0c7b3b62f3b", "ulid": "01M1Q787QRCCSDNVA159Y6S261", "aliases": [ "op_06e9f0c7b3b62f3b", "01M1Q787QRCCSDNVA159Y6S261", "op-06e9f0c7b3b62f3b" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 1, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-04T22:04:59Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "source-stated", "posed": null, "areaIds": [ "quantum-algorithm", "quantum-resource-theory" ], "topicIds": [ "channel-simulation", "superchannels-and-quantum-combs", "quantum-error-mitigation" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [] }, "title": "Multi-slot overhead of virtual channel conjugation", "status": "Unsolved", "fields": [ "Quantum algorithm", "Quantum Resource Theory" ], "topics": [ "Channel simulation", "Superchannels and quantum combs", "Quantum error mitigation" ], "tags": [ "Quantum algorithm", "Quantum Resource Theory", "Channel simulation", "Superchannels and quantum combs", "Quantum error mitigation" ], "statement": "What is the optimal quasiprobability overhead of implementing the complex conjugate of an unknown quantum channel from $n$ queries? Let $\\mathcal N:\\mathcal L(A)\\to\\mathcal L(B)$ be an unknown channel with $d_A:=\\dim A$ and $d_B:=\\dim B$, and fix orthonormal bases of $A$ and $B$. The complex conjugate of $\\mathcal N$ is the channel\n\n \\begin{equation}\n \\mathcal N^{*}(X):=\\overline{\\mathcal N(\\overline X)},\n\\tag{1}\n\\end{equation} \nwhere the bar is entrywise complex conjugation in the fixed bases, so the Choi operator of $\\mathcal N^{*}$ is the entrywise conjugate of that of $\\mathcal N$. An $n$-slot quantum comb is a physically realizable circuit with $n$ open slots, each receiving one use of the unknown channel, whose overall action is again a channel from $A$ to $B$; write $\\mathrm{Comb}_n$ for the set of such combs. An $n$-slot virtual comb is a real linear combination $\\widetilde{\\mathcal C}=\\sum_i c_i\\mathcal C_i$ with $\\mathcal C_i\\in\\mathrm{Comb}_n$, and its base norm\n\n \\begin{equation}\n \\|\\widetilde{\\mathcal C}\\|_{\\mathrm{base}}\n :=\\min\\Bigl\\{\\sum_i|c_i|:\n \\widetilde{\\mathcal C}=\\sum_ic_i\\mathcal C_i,\\\n c_i\\in\\mathbb R,\\ \\mathcal C_i\\in\\mathrm{Comb}_n\\Bigr\\}\n\\tag{2}\n\\end{equation} \nis the sampling overhead: estimating an expectation value of the output of $\\widetilde{\\mathcal C}$ to additive error $\\varepsilon$ by Monte Carlo sampling of the $\\mathcal C_i$ costs $O(\\|\\widetilde{\\mathcal C}\\|_{\\mathrm{base}}^{2}\\varepsilon^{-2})$ runs. Define the optimal $n$-query overhead of universal conjugation by\n\n \\begin{equation}\n g_n(d_A,d_B)\n :=\\inf\\Bigl\\{\\|\\widetilde{\\mathcal C}\\|_{\\mathrm{base}}:\n \\widetilde{\\mathcal C}\\text{ is an $n$-slot virtual comb with }\n \\widetilde{\\mathcal C}(\\mathcal N^{\\otimes n})=\\mathcal N^{*}\n \\text{ for every channel }\\mathcal N\\Bigr\\}.\n\\tag{3}\n\\end{equation} \nSince the extra slots may be discarded, $g_n\\leq g_1$. Determine $g_n(d_A,d_B)$ in Eq. (3) for $n\\geq2$: is $g_n(d_A,d_B)0$, decide whether every such family satisfies\n\n \\begin{equation}\n \\limsup_{N\\to\\infty}\n \\frac{\\mathcal F_{\\mathrm{QCQC}}^{(N)}(\\theta)}\n {\\mathcal F_{\\mathrm{PAR}}^{(N)}(\\theta)}\n =1.\n\\tag{1}\n\\end{equation} \nEquivalently, either prove Eq. (1) or construct a noisy family in this regular domain and physical QC-QC strategies with a persistent constant-factor advantage or a larger scaling exponent.", "url": "https://qiqc-op.com/problem/op_09b9fa91a1ac1a76/", "json": "https://qiqc-op.com/api/problems/op_09b9fa91a1ac1a76.json", "tex": "https://qiqc-op.com/problem/op_09b9fa91a1ac1a76/op_09b9fa91a1ac1a76.tex", "created": "2026-09-02", "updated": "2026-09-11", "createdAt": "2026-09-02T21:32:33.000Z", "updatedAt": "2026-09-11T01:54:51.000Z", "sha256": "a10fff1d327ba2349fc76d9713ff7ae2f6d1c6ffd997ac030c4e8b811f3aeb30" }, { "id": "op_7920f48995bc8511", "ulid": "01M1HME780EK3RBP2STMGR8JS5", "aliases": [ "op_7920f48995bc8511", "01M1HME780EK3RBP2STMGR8JS5", "op-7920f48995bc8511", "ruskai-2007-mutually-degradable-channels" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 2, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-04T22:04:59Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "source-stated", "posed": null, "areaIds": [ "quantum-communication" ], "topicIds": [ "channel-degradability", "quantum-channel-structure" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [] }, "title": "Nontrivial mutually degradable channel pairs", "status": "Solved", "fields": [ "Quantum Communication" ], "topics": [ "Channel degradability", "Quantum channel structure" ], "tags": [ "Quantum Communication", "Channel degradability", "Quantum channel structure" ], "statement": "Does there exist an integer $d\\geq2$ and a pair of distinct channels $\\mathcal M,\\mathcal N:\\mathcal L(A)\\to\\mathcal L(B)$, with $A\\simeq B\\simeq\\mathbb C^d$, that both have Choi rank exactly $d$, are mutually degradable, and are each nondegradable? Let $E\\simeq\\mathbb C^d$ and choose minimal Stinespring isometries $V_{\\mathcal M},V_{\\mathcal N}:A\\to B\\otimes E$ defining the channels and their complements by\n\n \\begin{equation}\n \\begin{aligned}\n \\mathcal M(\\rho)&=\\operatorname{Tr}_E\n (V_{\\mathcal M}\\rho V_{\\mathcal M}^{\\dagger}),\n &\\mathcal M^c(\\rho)&=\\operatorname{Tr}_B\n (V_{\\mathcal M}\\rho V_{\\mathcal M}^{\\dagger}),\\\\\n \\mathcal N(\\rho)&=\\operatorname{Tr}_E\n (V_{\\mathcal N}\\rho V_{\\mathcal N}^{\\dagger}),\n &\\mathcal N^c(\\rho)&=\\operatorname{Tr}_B\n (V_{\\mathcal N}\\rho V_{\\mathcal N}^{\\dagger}).\n \\end{aligned}\n\\tag{1}\n\\end{equation} \nEquation (1) fixes representatives of the complementary channels; changing a minimal dilation only applies an output unitary to a complement.\n\nFor $\\lvert\\Omega_d\\rangle:=\\sum_{j=1}^d\\lvert j\\rangle_{A'}\\lvert j\\rangle_A$, the required Choi-rank condition is\n\n \\begin{equation}\n J(\\mathcal T):=(\\operatorname{id}_{A'}\\otimes\\mathcal T)\n (\\lvert\\Omega_d\\rangle\\!\\langle\\Omega_d\\rvert),\n \\qquad\n \\operatorname{rank}J(\\mathcal M)\n =\\operatorname{rank}J(\\mathcal N)=d.\n\\tag{2}\n\\end{equation} \nThe equality in Eq. (2) makes the environment dimension in Eq. (1) minimal.\n\nMutual degradability requires channels $\\mathcal X,\\mathcal Y:\\mathcal L(B)\\to\\mathcal L(E)$ such that\n\n \\begin{equation}\n \\mathcal X\\circ\\mathcal M=\\mathcal N^c,\n \\qquad\n \\mathcal Y\\circ\\mathcal N=\\mathcal M^c.\n\\tag{3}\n\\end{equation} \nIn addition to Eq. (3), neither channel may admit its own degrading map:\n\n \\begin{equation}\n \\begin{aligned}\n &\\nexists\\ \\mathcal D_{\\mathcal M}:\\mathcal L(B)\\to\\mathcal L(E)\n \\quad\\text{CPTP with}\\quad\n \\mathcal M^c=\\mathcal D_{\\mathcal M}\\circ\\mathcal M,\\\\\n &\\nexists\\ \\mathcal D_{\\mathcal N}:\\mathcal L(B)\\to\\mathcal L(E)\n \\quad\\text{CPTP with}\\quad\n \\mathcal N^c=\\mathcal D_{\\mathcal N}\\circ\\mathcal N.\n \\end{aligned}\n\\tag{4}\n\\end{equation} \nEquation (4), together with $\\mathcal M\\neq\\mathcal N$, excludes the identity-channel and self-pair constructions described below. It does not exclude complementary pairs.", "url": "https://qiqc-op.com/problem/op_7920f48995bc8511/", "json": "https://qiqc-op.com/api/problems/op_7920f48995bc8511.json", "tex": "https://qiqc-op.com/problem/op_7920f48995bc8511/op_7920f48995bc8511.tex", "created": "2026-09-02", "updated": "2026-09-11", "createdAt": "2026-09-02T21:32:33.000Z", "updatedAt": "2026-09-11T01:54:51.000Z", "sha256": "972b27e26448387d68bb73a9dfc602a543e5a00530cc155a653cba29139c8608" }, { "id": "op_4830398d0c2feb5f", "ulid": "01M1HME7800GAYCEBF3MNS33JF", "aliases": [ "op_4830398d0c2feb5f", "01M1HME7800GAYCEBF3MNS33JF", "op-4830398d0c2feb5f", "v2-quantum-ldpc-codes-at-the-pauli-hashing-bound", "open-problem-v2-problem-6" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 2, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-04T22:04:59Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "editor-formulated", "posed": null, "areaIds": [ "quantum-error-correction" ], "topicIds": [ "quantum-ldpc-codes", "quantum-coding-theory", "decoding-algorithms" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [] }, "title": "Quantum LDPC codes at the Pauli hashing bound", "status": "Unsolved", "fields": [ "Quantum Error Correction" ], "topics": [ "Quantum LDPC codes", "Quantum coding theory", "Decoding algorithms" ], "tags": [ "Quantum Error Correction", "Quantum LDPC codes", "Quantum coding theory", "Decoding algorithms" ], "statement": "For which probability vectors $\\mathbf p=(p_I,p_X,p_Y,p_Z)$ can quantum low-density parity-check (LDPC) stabilizer codes reliably attain the exact hashing rate over the Pauli channel?\n\nThe channel and target rate are\n\n \\begin{equation}\n \\Lambda_{\\mathbf p}(\\rho)\n :=\\sum_{P\\in\\{I,X,Y,Z\\}}p_P P\\rho P,\n\\tag{1}\n\\end{equation} \n \\begin{equation}\n R_{\\mathrm{hash}}(\\mathbf p):=\\max\\{0,1-H(\\mathbf p)\\},\n \\qquad H(\\mathbf p):=-\\sum_{P\\in\\{I,X,Y,Z\\}}p_P\\log_2p_P,\n \\qquad 0\\log_2 0:=0.\n\\tag{2}\n\\end{equation} \nFor Eq. (1), seek a family of $[[n,k_n]]$ stabilizer codes with generator weight and qubit degree bounded independently of $n$, isometric encoding channels $\\mathcal E_n$ into the code spaces, and CPTP decoders $\\mathcal D_n$ back to the logical registers, such that\n\n \\begin{equation}\n \\liminf_{n\\to\\infty}\\frac{k_n}{n}\\ge R_{\\mathrm{hash}}(\\mathbf p),\n \\qquad\n F_e\\!\\left(2^{-k_n}I,\n \\mathcal D_n\\circ\\Lambda_{\\mathbf p}^{\\otimes n}\\circ\\mathcal E_n\n \\right)\\longrightarrow1.\n\\tag{3}\n\\end{equation} \nHere $F_e$ is entanglement fidelity for the maximally mixed logical input; Eq. (3) requires reliable transmission at the rate in Eq. (2). Transmission is unassisted, and the codes, decoders, and LDPC bounds may depend on $\\mathbf p$. A zero hashing rate permits $k_n=0$. No CSS restriction is imposed. The question is parameter-dependent: a negative answer for one noise subfamily does not classify all Pauli channels.", "url": "https://qiqc-op.com/problem/op_4830398d0c2feb5f/", "json": "https://qiqc-op.com/api/problems/op_4830398d0c2feb5f.json", "tex": "https://qiqc-op.com/problem/op_4830398d0c2feb5f/op_4830398d0c2feb5f.tex", "created": "2026-09-01", "updated": "2026-09-11", "createdAt": "2026-09-01T05:13:32.000Z", "updatedAt": "2026-09-11T01:54:51.000Z", "sha256": "2371bafaef579b2c43411b2526b7569c89b0da1356db45b092bbb1460729b076" }, { "id": "op_4cf3e7b8663b1d41", "ulid": "01M1HME78004TME758T7JBWF1D", "aliases": [ "op_4cf3e7b8663b1d41", "01M1HME78004TME758T7JBWF1D", "op-4cf3e7b8663b1d41", "v2-achievability-of-the-rains-bound-under-ppt-preserving-channels", "open-problem-v2-problem-4" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 2, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-04T22:04:59Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "source-stated", "posed": null, "areaIds": [ "quantum-resource-theory" ], "topicIds": [ "entanglement-distillation", "ppt-preserving-operations", "bell-diagonal-states" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [] }, "title": "Achievability of the Rains bound under completely PPT-preserving channels", "status": "Unsolved", "fields": [ "Quantum Resource Theory" ], "topics": [ "Entanglement distillation", "PPT-preserving operations", "Bell-diagonal states" ], "tags": [ "Quantum Resource Theory", "Entanglement distillation", "PPT-preserving operations", "Bell-diagonal states" ], "statement": "Is the Rains bound asymptotically achievable by completely positive-partial-transpose-preserving (completely PPT-preserving) channels for every full-rank two-qubit Bell-diagonal state, and what channel family achieves it? Consider\n\n \\begin{equation}\n \\rho_{\\mathbf p}\n =p_I\\lvert\\Phi^+\\rangle\\!\\langle\\Phi^+\\rvert\n +p_X\\lvert\\Psi^+\\rangle\\!\\langle\\Psi^+\\rvert\n +p_Y\\lvert\\Psi^-\\rangle\\!\\langle\\Psi^-\\rvert\n +p_Z\\lvert\\Phi^-\\rangle\\!\\langle\\Phi^-\\rvert,\n\\tag{1}\n\\end{equation} \nwhere $p_i>0$ and $p_I+p_X+p_Y+p_Z=1$, and $\\lvert\\Phi^\\pm\\rangle:=(\\lvert00\\rangle\\pm\\lvert11\\rangle)/\\sqrt2$ and $\\lvert\\Psi^\\pm\\rangle:=(\\lvert01\\rangle\\pm\\lvert10\\rangle)/\\sqrt2$.\n\nA channel $\\Lambda_n:A^nB^n\\to A'_nB'_n$ is completely PPT-preserving if\n\n \\begin{equation}\n \\Lambda_n\\text{ is CPTP},\\qquad\n \\Gamma_{B'_n}\\circ\\Lambda_n\\circ\\Gamma_{B^n}\n \\text{ is completely positive},\n\\tag{2}\n\\end{equation} \nwhere $\\Gamma$ is partial transposition on the indicated subsystem. Equation (2) requires PPT preservation also with arbitrary local ancillas. Write $D(\\rho\\|\\tau)=\\operatorname{Tr}\\rho(\\log_2\\rho-\\log_2\\tau)$ with the standard support convention, and define\n\n \\begin{equation}\n R(\\rho):=\\inf_{\\tau\\geq0,\\,\\|\\tau^{\\Gamma_B}\\|_1\\leq1}\n D(\\rho\\|\\tau).\n\\tag{3}\n\\end{equation} \nFor the state in Eq. (1), achievability of the Rains bound in Eq. (3) means a sequence satisfying Eq. (2) and\n\n \\begin{equation}\n \\liminf_{n\\to\\infty}\\frac{\\log_2 M_n}{n}\n \\geq R(\\rho_{\\mathbf p}),\\qquad\n \\big\\|\\Lambda_n(\\rho_{\\mathbf p}^{\\otimes n})\n -\\Phi_{M_n}\\big\\|_1\\longrightarrow0,\n\\tag{4}\n\\end{equation} \nwhere $M_n$ is a positive integer and $\\Phi_M:=M^{-1}\\sum_{i,j=1}^M|ii\\rangle\\langle jj|$. Equation (4) imposes no efficiency requirement on the construction.", "url": "https://qiqc-op.com/problem/op_4cf3e7b8663b1d41/", "json": "https://qiqc-op.com/api/problems/op_4cf3e7b8663b1d41.json", "tex": "https://qiqc-op.com/problem/op_4cf3e7b8663b1d41/op_4cf3e7b8663b1d41.tex", "created": "2026-09-01", "updated": "2026-09-11", "createdAt": "2026-09-01T05:13:32.000Z", "updatedAt": "2026-09-11T01:54:51.000Z", "sha256": "ebec822ce662b6a619930f653299dc4f050cf0da88629b0eea44728ed29f6696" }, { "id": "op_e490c9462b37a548", "ulid": "01M1HME780DDSDKPH6BERTWRWB", "aliases": [ "op_e490c9462b37a548", "01M1HME780DDSDKPH6BERTWRWB", "op-e490c9462b37a548", "v2-two-way-quantum-capacity-amplitude-damping-channel", "open-problem-v2-problem-5" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 2, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-04T22:04:59Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "source-stated", "posed": null, "areaIds": [ "quantum-communication" ], "topicIds": [ "quantum-capacity", "local-operations-and-classical-communication" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [] }, "title": "Two-way quantum capacity: amplitude-damping channel", "status": "Unsolved", "fields": [ "Quantum Communication" ], "topics": [ "Quantum capacity", "Local operations and classical communication" ], "tags": [ "Quantum Communication", "Quantum capacity", "Local operations and classical communication" ], "statement": "What is the two-way quantum capacity $\\mathcal{Q}_2(\\mathcal A_p)$ of the qubit amplitude-damping channel\n\n \\begin{equation}\n \\mathcal A_p(\\rho)=A_0\\rho A_0^\\dagger+A_1\\rho A_1^\\dagger,\n \\qquad 0\\le p\\le1?\n\\tag{1}\n\\end{equation} \nThe operators in Eq. (1) are\n\n \\begin{equation}\n \\begin{aligned}\n A_0&=\\lvert0\\rangle\\!\\langle0\\rvert\n +\\sqrt{1-p}\\,\\lvert1\\rangle\\!\\langle1\\rvert\n =\\begin{pmatrix}1&0\\\\0&\\sqrt{1-p}\\end{pmatrix},\\\\\n A_1&=\\sqrt p\\,\\lvert0\\rangle\\!\\langle1\\rvert\n =\\begin{pmatrix}0&\\sqrt p\\\\0&0\\end{pmatrix}.\n \\end{aligned}\n\\tag{2}\n\\end{equation} \nEquation (2) uses $p$ as the decay probability of the excited state. Here $\\mathcal{Q}_2$ permits adaptive local operations and unlimited two-way classical communication between uses of the channel.", "url": "https://qiqc-op.com/problem/op_e490c9462b37a548/", "json": "https://qiqc-op.com/api/problems/op_e490c9462b37a548.json", "tex": "https://qiqc-op.com/problem/op_e490c9462b37a548/op_e490c9462b37a548.tex", "created": "2026-09-01", "updated": "2026-09-11", "createdAt": "2026-09-01T05:13:32.000Z", "updatedAt": "2026-09-11T01:54:51.000Z", "sha256": "2c6d22a0e769d1483e3cbcf8aac3c7f7b98fc48c448ec45f0b01ad7ced371c9c" }, { "id": "op_e4a8ae208470f288", "ulid": "01M1HME780CWYHZHQFF0KN89BM", "aliases": [ "op_e4a8ae208470f288", "01M1HME780CWYHZHQFF0KN89BM", "op-e4a8ae208470f288", "v2-three-dimensional-self-correcting-quantum-memory", "open-problem-v2-problem-9" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 2, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-04T22:04:59Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "source-stated", "posed": null, "areaIds": [ "quantum-error-correction" ], "topicIds": [ "self-correcting-quantum-memories", "quantum-thermodynamics", "quantum-coding-theory" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [] }, "title": "Three-dimensional self-correcting quantum memory", "status": "Unsolved", "fields": [ "Quantum Error Correction" ], "topics": [ "Self-correcting quantum memories", "Quantum thermodynamics", "Quantum coding theory" ], "tags": [ "Quantum Error Correction", "Self-correcting quantum memories", "Quantum thermodynamics", "Quantum coding theory" ], "statement": "Does there exist a passive self-correcting quantum memory in three spatial dimensions? Consider finite-dimensional spins on a three-dimensional lattice $\\Lambda_L$ of linear size $L$ and a Hamiltonian $H_L=\\sum_{X\\subseteq\\Lambda_L}h_X$ whose interaction range, local strength $\\lVert h_X\\rVert$, and number of terms incident on each spin are bounded independently of $L$. Let its ground space $\\mathcal C_L$ encode a logical space $\\mathcal Q_L$ with $\\dim\\mathcal Q_L\\ge2$, and let $\\mathcal V_L:\\mathcal S(\\mathcal Q_L)\\to\\mathcal S(\\mathcal C_L)$ be the channel induced by an isometric encoding into $\\mathcal C_L$.\n\nDuring storage no active control or error correction is permitted. The encoded state evolves under a local thermal channel $\\mathcal E_{t,\\beta}^{(L)}$ at inverse temperature $\\beta$, such as a Davies semigroup, followed only at readout by a decoder $\\mathcal D_L$. For a fixed $0<\\varepsilon<1$, define the worst-case storage time by\n\n \\begin{equation}\n \\tau_L(\\beta,\\varepsilon)\n :=\\sup\\!\\left\\{t\\ge0:\n \\sup_{0\\le s\\le t}\\ \\sup_{\\rho\\in\\mathcal S(\\mathcal Q_L)}\n \\left\\|\n \\bigl(\\mathcal D_L\\circ\\mathcal E_{s,\\beta}^{(L)}\n \\circ\\mathcal V_L\\bigr)(\\rho)-\\rho\n \\right\\|_1\n \\le\\varepsilon\\right\\},\n\\tag{1}\n\\end{equation} \nwhere $\\mathcal S(\\mathcal Q_L)$ is the set of logical states. The problem is to construct such a Hamiltonian family with efficient decoders and a finite critical inverse temperature $\\beta_c$ for which\n\n \\begin{equation}\n \\beta>\\beta_c\n \\quad\\Longrightarrow\\quad\n \\lim_{L\\to\\infty}\\tau_L(\\beta,\\varepsilon)=\\infty.\n\\tag{2}\n\\end{equation} \nEquation (2) requires the lifetime in Eq. (1) to diverge at fixed nonzero temperature as $L\\to\\infty$; growth that terminates at a temperature-dependent system size is only partial self-correction.", "url": "https://qiqc-op.com/problem/op_e4a8ae208470f288/", "json": "https://qiqc-op.com/api/problems/op_e4a8ae208470f288.json", "tex": "https://qiqc-op.com/problem/op_e4a8ae208470f288/op_e4a8ae208470f288.tex", "created": "2026-09-01", "updated": "2026-09-11", "createdAt": "2026-09-01T05:13:32.000Z", "updatedAt": "2026-09-11T01:54:51.000Z", "sha256": "bd3e6d2dc7f9b82d1f84e0c29301b31e00b6f3cce45b0b8dbc5229650318d94e" }, { "id": "op_1fc674a1578b1e05", "ulid": "01M26JZEKP1GZB7PW3TZS3BQBD", "aliases": [ "op_1fc674a1578b1e05", "01M26JZEKP1GZB7PW3TZS3BQBD", "op-1fc674a1578b1e05" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 1, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-10T21:18:30.518Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "derived", "posed": null, "areaIds": [ "quantum-communication" ], "topicIds": [ "classical-capacity", "bosonic-channels", "additivity-and-regularization" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [] }, "title": "Low-energy Holevo additivity for a squeezed thermal attenuator", "status": "Unsolved", "fields": [ "Quantum Communication" ], "topics": [ "Classical capacity", "Bosonic channels", "Additivity and regularization" ], "tags": [ "Quantum Communication", "Classical capacity", "Bosonic channels", "Additivity and regularization" ], "statement": "Is energy-constrained Holevo information additive for a squeezed thermal attenuator below its water-filling threshold? Fix $0<\\eta<1$, $b\\geq0$, and $r\\in\\mathbb R\\setminus\\{0\\}$. The channel $\\Phi_{\\eta,b,r}$ mixes an input mode with an independent centered squeezed thermal mode on a beam splitter of transmissivity $\\eta$. Each use has a fresh environment. With $[q,p]=i$, $a=(q+ip)/\\sqrt2$, and $R=(q,p)^T$, use $V_{jk}=\\langle\\{R_j-\\langle R_j\\rangle,R_k-\\langle R_k\\rangle\\}\\rangle/2$. The environment covariance is $(b+1/2)\\operatorname{diag}(e^{2r},e^{-2r})$. Choose an energy $00$, does there exist a real Weyl-ordered polynomial $P(q_1,p_1,\\ldots,q_n,p_n)$ with a self-adjoint realization such that\n\n \\begin{equation}\n \\|\\mathcal U-\\mathcal V_P\\|_{\\diamond,E}<\\varepsilon,\n \\qquad\n \\mathcal V_P(\\rho)=e^{-iP}\\rho e^{iP}?\n\\tag{2}\n\\end{equation} \nThe polynomial and its degree in Eq. (2) may depend on $U,E,\\varepsilon$.", "url": "https://qiqc-op.com/problem/op_5083d02a18761d8a/", "json": "https://qiqc-op.com/api/problems/op_5083d02a18761d8a.json", "tex": "https://qiqc-op.com/problem/op_5083d02a18761d8a/op_5083d02a18761d8a.tex", "created": "2026-09-10", "updated": "2026-09-10", "createdAt": "2026-09-10T21:54:15.000Z", "updatedAt": "2026-09-10T21:54:15.000Z", "sha256": "8c8e726a1739088bfd088b118c6351d0e63bc2040a6e27f320bc999930fec280" }, { "id": "op_55be40726cdf7304", "ulid": "01M26KND1E2P37YSPQCT6FPCCD", "aliases": [ "op_55be40726cdf7304", "01M26KND1E2P37YSPQCT6FPCCD", "op-55be40726cdf7304" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 1, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-10T21:30:29.806Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "derived", "posed": null, "areaIds": [ "quantum-algorithm" ], "topicIds": [ "boson-sampling", "computational-complexity-and-computability" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [ "01M26KNCZSX9BMQTNM7RX75HZ5" ] }, "title": "Anticoncentration of independent complex Gaussian hafnians", "status": "Unsolved", "fields": [ "Quantum algorithm" ], "topics": [ "Boson sampling", "Computational complexity and computability" ], "tags": [ "Quantum algorithm", "Boson sampling", "Computational complexity and computability" ], "statement": "Do independent complex Gaussian hafnians satisfy a polynomial lower-tail bound at their root-mean-square scale? For each integer $n\\geq1$, let $X=X^T\\in\\mathbb C^{2n\\times2n}$ have zero diagonal. The entries above the diagonal are independent with density $\\pi^{-1}e^{-|z|^2}$. Define\n\n \\begin{equation}\n \\operatorname{Haf}X=\\sum_{M\\in\\mathcal M_{2n}}\\prod_{\\{i,j\\}\\in M}X_{ij},\n \\qquad h_n=(2n-1)!!=\\frac{(2n)!}{2^n n!},\n\\tag{1}\n\\end{equation} \nwhere $\\mathcal M_{2n}$ is the set of perfect matchings of $\\{1,\\ldots,2n\\}$. Does there exist a polynomial $p$, positive on $[1,\\infty)^2$, such that\n\n \\begin{equation}\n \\Pr_X\\!\\left[|\\operatorname{Haf}X|<\n \\frac{\\sqrt{h_n}}{p(n,1/\\delta)}\\right]<\\delta\n \\qquad(n\\geq1,\\ 0<\\delta<1)?\n\\tag{2}\n\\end{equation} \nThe definitions in Eq. (1) fix the ensemble and normalization. One polynomial must satisfy Eq. (2) for every $n$ and $\\delta$.", "url": "https://qiqc-op.com/problem/op_55be40726cdf7304/", "json": "https://qiqc-op.com/api/problems/op_55be40726cdf7304.json", "tex": "https://qiqc-op.com/problem/op_55be40726cdf7304/op_55be40726cdf7304.tex", "created": "2026-09-10", "updated": "2026-09-10", "createdAt": "2026-09-10T21:54:15.000Z", "updatedAt": "2026-09-10T21:54:15.000Z", "sha256": "3a0d354c153123b6e4b1b6096e776a580b794420aa616482a53d0bf8565f24d5" }, { "id": "op_64046727b81b4024", "ulid": "01M26KND4S48V6YX37ZM0DZ3D9", "aliases": [ "op_64046727b81b4024", "01M26KND4S48V6YX37ZM0DZ3D9", "op-64046727b81b4024" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 1, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-10T21:30:29.913Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "derived", "posed": null, "areaIds": [ "quantum-algorithm" ], "topicIds": [ "quantum-circuit-complexity", "hamiltonian-simulation" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [ "01M26KND34MGPWHNQXK7W9DXVK" ] }, "title": "Universality of every fixed non-Gaussian polynomial generator", "status": "Unsolved", "fields": [ "Quantum algorithm" ], "topics": [ "Quantum circuit complexity", "Hamiltonian simulation" ], "tags": [ "Quantum algorithm", "Quantum circuit complexity", "Hamiltonian simulation" ], "statement": "Does every fixed non-Gaussian polynomial Hamiltonian become universal when all Gaussian controls are available? Fix $n\\geq1$. On $L^2(\\mathbb R^n)$, let $q_j,p_j$ satisfy $[q_j,p_k]=i\\delta_{jk}$. Let $H_*$ be any real Weyl-ordered polynomial in these operators of degree greater than two. Assume that $H_*$ is essentially self-adjoint on the Schwartz space $\\mathcal S(\\mathbb R^n)$ of smooth, rapidly decreasing functions. Use its unique self-adjoint closure to define $e^{-itH_*}$. Allowed gates are $e^{-itH_*}$ for arbitrary real $t$, together with all Gaussian unitaries. Gaussian unitaries are generated by real polynomials of degree at most two in the canonical operators.\n\nLet $U$ be any unitary satisfying $U\\mathcal S(\\mathbb R^n)\\subseteq\\mathcal S(\\mathbb R^n)$. Fix finite $E\\geq0$ and $\\varepsilon>0$. Does some finite product $V$ of allowed gates satisfy\n\n \\begin{equation}\n \\sup_{\\rho_{AR}:\\operatorname{Tr}(\\rho_A N)\\leq E}\n \\left\\|[(\\mathcal U-\\mathcal V)\\otimes\\operatorname{id}_R](\\rho_{AR})\\right\\|_1\n <\\varepsilon,\n \\qquad N=\\frac12\\sum_{j=1}^n(q_j^2+p_j^2-1)?\n\\tag{1}\n\\end{equation} \nIn Eq. (1), $\\mathcal U(\\rho)=U\\rho U^\\dagger$, $\\mathcal V(\\rho)=V\\rho V^\\dagger$, and $R$ is an arbitrary reference system. The fixed $H_*$ must work for every target $U$, energy $E$, and accuracy $\\varepsilon$.", "url": "https://qiqc-op.com/problem/op_64046727b81b4024/", "json": "https://qiqc-op.com/api/problems/op_64046727b81b4024.json", "tex": "https://qiqc-op.com/problem/op_64046727b81b4024/op_64046727b81b4024.tex", "created": "2026-09-10", "updated": "2026-09-10", "createdAt": "2026-09-10T21:54:15.000Z", "updatedAt": "2026-09-10T21:54:15.000Z", "sha256": "87e0b87798437e20b26c822b30895af94a03e78e0026cfde49d165e72a0f268d" }, { "id": "op_65b01b2a5ef77a6e", "ulid": "01M26KND6EV9BBF5SH8FNWCMD0", "aliases": [ "op_65b01b2a5ef77a6e", "01M26KND6EV9BBF5SH8FNWCMD0", "op-65b01b2a5ef77a6e" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 1, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-10T21:30:29.966Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "derived", "posed": null, "areaIds": [ "quantum-algorithm" ], "topicIds": [ "hamiltonian-simulation" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [] }, "title": "Compact-time validity of the rotating wave approximation", "status": "Solved", "fields": [ "Quantum algorithm" ], "topics": [ "Hamiltonian simulation" ], "tags": [ "Quantum algorithm", "Hamiltonian simulation" ], "statement": "Does the Jaynes–Cummings evolution approximate the Rabi evolution strongly and uniformly on each compact time interval in the high-frequency limit? Work on $\\mathbb C^2\\otimes L^2(\\mathbb R)$. Let $a,a^\\dagger$ be oscillator operators with $[a,a^\\dagger]=1$ and $N=a^\\dagger a$. Write $\\sigma_x,\\sigma_y,\\sigma_z$ for the Pauli matrices and $\\sigma_\\pm=(\\sigma_x\\pm i\\sigma_y)/2$. Fix $\\lambda>0$ and $\\Delta\\in\\mathbb R$. For $\\omega>\\max\\{0,-\\Delta\\}$, define\n\n \\begin{equation}\n \\begin{aligned}\n H_\\omega&=\\frac{\\omega+\\Delta}{2}\\sigma_z+\\omega N+\n \\lambda\\sigma_x(a+a^\\dagger),\\\\\n J_\\omega&=\\frac{\\omega+\\Delta}{2}\\sigma_z+\\omega N+\n \\lambda(\\sigma_+a+\\sigma_-a^\\dagger).\n \\end{aligned}\n\\tag{1}\n\\end{equation} \nTensor factors of the identity are implicit. Use the self-adjoint closures of Eq. (1) from $\\mathbb C^2\\otimes\\mathcal S(\\mathbb R)$. Here $\\mathcal S(\\mathbb R)$ is the Schwartz space. Is it true that every normalized $\\psi\\in\\mathbb C^2\\otimes L^2(\\mathbb R)$ and every finite $T>0$ satisfy\n\n \\begin{equation}\n \\lim_{\\omega\\to\\infty}\\sup_{|t|\\leq T}\n \\|(e^{-itH_\\omega}-e^{-itJ_\\omega})\\psi\\|=0?\n\\tag{2}\n\\end{equation} \nThe vector, time horizon, coupling, and detuning in Eq. (2) are fixed as $\\omega$ grows.", "url": "https://qiqc-op.com/problem/op_65b01b2a5ef77a6e/", "json": "https://qiqc-op.com/api/problems/op_65b01b2a5ef77a6e.json", "tex": "https://qiqc-op.com/problem/op_65b01b2a5ef77a6e/op_65b01b2a5ef77a6e.tex", "created": "2026-09-10", "updated": "2026-09-10", "createdAt": "2026-09-10T21:54:15.000Z", "updatedAt": "2026-09-10T21:54:15.000Z", "sha256": "0452daf07cba7ea1ed9767ee9fe6ecef3c798fc72d1917e063d1c21d6729cc21" }, { "id": "op_68d6ff4ef7be1577", "ulid": "01M26KH5RHYJFRZWBP18Q7QFXJ", "aliases": [ "op_68d6ff4ef7be1577", "01M26KH5RHYJFRZWBP18Q7QFXJ", "op-68d6ff4ef7be1577" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 1, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-10T21:28:11.281Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "derived", "posed": null, "areaIds": [ "quantum-resource-theory" ], "topicIds": [ "gaussian-quantum-information", "entanglement-distillation" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [] }, "title": "Two-way distillable entanglement of mixed two-mode Gaussian states", "status": "Unsolved", "fields": [ "Quantum Resource Theory" ], "topics": [ "Gaussian quantum information", "Entanglement distillation" ], "tags": [ "Quantum Resource Theory", "Gaussian quantum information", "Entanglement distillation" ], "statement": "What is the exact two-way distillable entanglement of a finite-energy mixed two-mode Gaussian state?\n\nLet $\\rho_{AB}$ be a finite-energy mixed two-mode Gaussian density operator. Let $\\Lambda_k$ range over protocols using local operations and unlimited two-way classical communication. The complete protocols are trace preserving, and non-Gaussian operations are allowed. Finite energy means finite total mean photon number. Alice and Bob hold one mode each. Define the Bell-pair density operator and trace norm by\n\n \\begin{equation}\n\\Phi_2:=\\tfrac12(|00\\rangle+|11\\rangle)(\\langle00|+\\langle11|),\n\\qquad\n\\lVert X\\rVert_1:=\\operatorname{Tr}\\sqrt{X^\\dagger X}.\n\\tag{1}\n\\end{equation} \nUsing the Bell pair in Eq. (1), call $r\\geq0$ achievable if an allowed sequence $(\\Lambda_k)_{k\\geq1}$ satisfies\n\n \\begin{equation}\n\\lim_{k\\to\\infty}\\left\\lVert\\Lambda_k(\\rho_{AB}^{\\otimes k})-\\Phi_2^{\\otimes\\lfloor rk\\rfloor}\\right\\rVert_1=0.\n\\tag{2}\n\\end{equation} \nDetermine $D_{\\leftrightarrow}(\\rho_{AB}):=\\sup\\{r\\geq0:r\\text{ is achievable}\\}$, with achievability defined by Eq. (2).", "url": "https://qiqc-op.com/problem/op_68d6ff4ef7be1577/", "json": "https://qiqc-op.com/api/problems/op_68d6ff4ef7be1577.json", "tex": "https://qiqc-op.com/problem/op_68d6ff4ef7be1577/op_68d6ff4ef7be1577.tex", "created": "2026-09-10", "updated": "2026-09-10", "createdAt": "2026-09-10T21:54:15.000Z", "updatedAt": "2026-09-10T21:54:15.000Z", "sha256": "6b3a24489d777023e2260578656ce43e8b7238e94bc46ae29c7d7bd0ffe8a62a" }, { "id": "op_6f6de1a285416bb2", "ulid": "01M26KH5ZAB056C2SMQZFX96S3", "aliases": [ "op_6f6de1a285416bb2", "01M26KH5ZAB056C2SMQZFX96S3", "op-6f6de1a285416bb2" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 1, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-10T21:28:11.498Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "derived", "posed": null, "areaIds": [ "quantum-resource-theory" ], "topicIds": [ "gaussian-quantum-information", "entanglement-distillation", "one-shot-and-finite-blocklength-bounds" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [] }, "title": "Optimal success probability for Gaussian processing of non-Gaussian entanglement", "status": "Unsolved", "fields": [ "Quantum Resource Theory" ], "topics": [ "Gaussian quantum information", "Entanglement distillation", "One-shot and finite-blocklength bounds" ], "tags": [ "Quantum Resource Theory", "Gaussian quantum information", "Entanglement distillation", "One-shot and finite-blocklength bounds" ], "statement": "What is the maximal success probability for preparing an approximate two-mode squeezed vacuum from finitely many copies of a non-Gaussian state?\n\nLet $\\rho$ be a non-Gaussian density operator of one bosonic mode at Alice and one at Bob. Fix an integer $m\\geq1$, squeezing $r>0$, and error tolerance $0<\\varepsilon<1$. The target is the two-mode squeezed vacuum\n\n \\begin{equation}\n\\psi_r:=|\\psi_r\\rangle\\langle\\psi_r|,\\qquad\n|\\psi_r\\rangle:=\\sqrt{1-\\tanh^2r}\\sum_{j=0}^{\\infty}(\\tanh r)^j|j,j\\rangle,\n\\tag{1}\n\\end{equation} \nEquation (1) uses the product photon-number basis.\n\nPreparation from $m$ copies of $\\rho$ uses local Gaussian operations and classical communication. Allowed success maps $\\Lambda$ are trace-nonincreasing completely positive maps. Each implementation is a finite sequence using local Gaussian ancillary states, local Gaussian unitaries, Gaussian measurements or vacuum-projection success branches, discarding, and classical feedforward. Gaussian measurement records may be accepted on a specified measurable set; their probabilities are integrated over that set. A nonvacuum outcome of a vacuum projection terminates that run in failure; its output cannot be reused. The input copies are the only non-Gaussian resource.\n\nFor each allowed map, write $p_\\Lambda:=\\operatorname{Tr}\\Lambda(\\rho^{\\otimes m})$. When $p_\\Lambda>0$, let $\\tau_\\Lambda:=\\Lambda(\\rho^{\\otimes m})/p_\\Lambda$. Define the maximal heralding probability by\n\n \\begin{equation}\nP_G^{(m)}(\\rho,r,\\varepsilon):=\n\\sup_{\\Lambda}\\left\\{p_\\Lambda:p_\\Lambda>0,\\quad\n\\tfrac12\\|\\tau_\\Lambda-\\psi_r\\|_1\\leq\\varepsilon\\right\\}.\n\\tag{2}\n\\end{equation} \nDetermine Eq. (2), with $\\sup\\varnothing:=0$. Here $\\|X\\|_1:=\\operatorname{Tr}\\sqrt{X^\\dagger X}$.", "url": "https://qiqc-op.com/problem/op_6f6de1a285416bb2/", "json": "https://qiqc-op.com/api/problems/op_6f6de1a285416bb2.json", "tex": "https://qiqc-op.com/problem/op_6f6de1a285416bb2/op_6f6de1a285416bb2.tex", "created": "2026-09-10", "updated": "2026-09-10", "createdAt": "2026-09-10T21:54:15.000Z", "updatedAt": "2026-09-10T21:54:15.000Z", "sha256": "197936ab2f3134dc5d71c49fa18c011523a5f83889c9b05be08c71051831a5e6" }, { "id": "op_709c2048154de463", "ulid": "01M26JZEPTR8YAH8EKFW9D0Q24", "aliases": [ "op_709c2048154de463", "01M26JZEPTR8YAH8EKFW9D0Q24", "op-709c2048154de463" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 1, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-10T21:18:30.618Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "source-stated", "posed": null, "areaIds": [ "quantum-communication" ], "topicIds": [ "quantum-capacity", "bosonic-channels" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [ "01M26JZEN93BAESCCMMTRG84M0", "01M26JZET5QZ2FB0WSWESW77V6" ] }, "title": "Two-way quantum capacity of a thermal amplifier", "status": "Unsolved", "fields": [ "Quantum Communication" ], "topics": [ "Quantum capacity", "Bosonic channels" ], "tags": [ "Quantum Communication", "Quantum capacity", "Bosonic channels" ], "statement": "What is the two-way quantum capacity of a thermal amplifier for every gain and thermal noise level? Fix a power gain $\\kappa>1$ and $b\\geq0$. The single-mode thermal amplifier $\\mathcal A_{\\kappa,b}$ is the Gaussian channel realized by two-mode squeezing with output annihilation operator\n\n \\begin{equation}\n a_{\\mathrm{out}}=\\sqrt\\kappa\\,a+\\sqrt{\\kappa-1}\\,e^\\dagger.\n\\tag{1}\n\\end{equation} \nThe independent input and environment modes obey $[a,a^\\dagger]=[e,e^\\dagger]=1$. The environment is a fresh independent thermal mode at each use, with mean photon number $b$ and density operator\n\n \\begin{equation}\n \\tau_b=\\frac1{b+1}\\sum_{n=0}^{\\infty}\\left(\\frac b{b+1}\\right)^n|n\\rangle\\langle n|\\quad(b>0),\\qquad \\tau_0=|0\\rangle\\langle0|.\n\\tag{2}\n\\end{equation} \nHere $|n\\rangle$ is the $n$-photon Fock state. The unused output mode is discarded. Equations (1) and (2) specify the channel on arbitrary input states. Protocols may use arbitrary adaptive local quantum operations and unlimited two-way public classical communication. The parties initially share no entanglement or secret key. There is no input-energy bound; the capacity is the supremum over finite mean input-energy budgets. The two-way quantum capacity $Q_2$ is the supremum of asymptotic qubits transmitted per use with vanishing error. Equivalently, it is the maximal rate of maximally entangled pairs distributed with vanishing trace-distance error.", "url": "https://qiqc-op.com/problem/op_709c2048154de463/", "json": "https://qiqc-op.com/api/problems/op_709c2048154de463.json", "tex": "https://qiqc-op.com/problem/op_709c2048154de463/op_709c2048154de463.tex", "created": "2026-09-10", "updated": "2026-09-10", "createdAt": "2026-09-10T21:54:15.000Z", "updatedAt": "2026-09-10T21:54:15.000Z", "sha256": "ef1428be6e81919d9f28987215e437d1d1ee676fc557d7c3c149efbbcdba4894" }, { "id": "op_71026fbfd41a90c6", "ulid": "01M26K8QB7C9R8CSKZQYP5WRT5", "aliases": [ "op_71026fbfd41a90c6", "01M26K8QB7C9R8CSKZQYP5WRT5", "op-71026fbfd41a90c6" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 1, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-10T21:23:34.375Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "editor-formulated", "posed": null, "areaIds": [ "quantum-communication" ], "topicIds": [ "gaussian-quantum-information", "quantum-channel-structure" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [] }, "title": "Gaussian-input preservation under a Gaussian reference extension", "status": "Solved", "fields": [ "Quantum Communication" ], "topics": [ "Gaussian quantum information", "Quantum channel structure" ], "tags": [ "Quantum Communication", "Gaussian quantum information", "Quantum channel structure" ], "statement": "Does preservation of Gaussian inputs by a trace-decreasing operation imply preservation when a Gaussian reference is attached? Let $\\Phi$ be a completely positive trace-nonincreasing linear map on trace-class operators of one bosonic mode. Assume $\\Phi(\\rho)/\\operatorname{Tr}\\Phi(\\rho)$ is Gaussian for every Gaussian density operator $\\rho$ with positive success probability. Let $\\operatorname{id}_B$ be the identity channel of a second bosonic mode. Must the state in Eq. (1) be Gaussian for every two-mode Gaussian state $\\omega_{AB}$ with positive denominator?\n\n \\begin{equation}\n\\frac{(\\Phi\\otimes\\operatorname{id}_B)(\\omega_{AB})}{\\operatorname{Tr}[(\\Phi\\otimes\\operatorname{id}_B)(\\omega_{AB})]}.\n\\tag{1}\n\\end{equation}", "url": "https://qiqc-op.com/problem/op_71026fbfd41a90c6/", "json": "https://qiqc-op.com/api/problems/op_71026fbfd41a90c6.json", "tex": "https://qiqc-op.com/problem/op_71026fbfd41a90c6/op_71026fbfd41a90c6.tex", "created": "2026-09-10", "updated": "2026-09-10", "createdAt": "2026-09-10T21:54:15.000Z", "updatedAt": "2026-09-10T21:54:15.000Z", "sha256": "155978aa2e85580e51da8fbfc5ce4e3ba3ba477d2ff864685677d779274b3777" }, { "id": "op_71f84253f425fc7e", "ulid": "01M26K8Q9WAW3Q9PKGYVXHHX0C", "aliases": [ "op_71f84253f425fc7e", "01M26K8Q9WAW3Q9PKGYVXHHX0C", "op-71f84253f425fc7e" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 1, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-10T21:23:34.332Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "derived", "posed": null, "areaIds": [ "quantum-resource-theory" ], "topicIds": [ "gaussian-quantum-information" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [] }, "title": "Strict stationary squeezing limit for one-mode thermal diffusion", "status": "Solved", "fields": [ "Quantum Resource Theory" ], "topics": [ "Gaussian quantum information" ], "tags": [ "Quantum Resource Theory", "Gaussian quantum information" ], "statement": "Does every stable one-mode thermal Gaussian diffusion obey the sharp stationary squeezing bound $\\lambda_{\\min}(V_\\infty)>\\nu/2$? Use $R=(q,p)^{\\mathsf T}$, $[q,p]=i$, and covariance $V_{jk}=\\langle\\{R_j-\\langle R_j\\rangle,R_k-\\langle R_k\\rangle\\}\\rangle$. Let $G$ be any real symmetric $2\\times2$ matrix, $\\kappa>0$, and $\\nu=2\\bar n+1$ with $\\bar n\\geq0$. Define the drift and diffusion in Eq. (1).\n\n \\begin{equation}\nA=\\Omega G-\\frac\\kappa2I_2,\\qquad D=\\kappa\\nu I_2,\\qquad \\Omega=\\begin{pmatrix}0&1\\\\-1&0\\end{pmatrix}.\n\\tag{1}\n\\end{equation} \nAssume $A$ is Hurwitz stable: both eigenvalues have negative real part. The stationary covariance $V_\\infty$ solves Eq. (2).\n\n \\begin{equation}\nAV_\\infty+V_\\infty A^{\\mathsf T}+D=0.\n\\tag{2}\n\\end{equation} \nSharpness means that the infimum of $\\lambda_{\\min}(V_\\infty)$ over stable choices of $G$ is $\\nu/2$.", "url": "https://qiqc-op.com/problem/op_71f84253f425fc7e/", "json": "https://qiqc-op.com/api/problems/op_71f84253f425fc7e.json", "tex": "https://qiqc-op.com/problem/op_71f84253f425fc7e/op_71f84253f425fc7e.tex", "created": "2026-09-10", "updated": "2026-09-10", "createdAt": "2026-09-10T21:54:15.000Z", "updatedAt": "2026-09-10T21:54:15.000Z", "sha256": "3d07c88a8dd0022de112b86236bdfc5f7c171a42fc220f41d855575405ba931c" }, { "id": "op_8bb2af22f3465748", "ulid": "01M26K8Q5HX6DE9YV54YEM0S9Z", "aliases": [ "op_8bb2af22f3465748", "01M26K8Q5HX6DE9YV54YEM0S9Z", "op-8bb2af22f3465748" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 1, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-10T21:23:34.193Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "editor-formulated", "posed": null, "areaIds": [ "quantum-metrology" ], "topicIds": [ "gaussian-quantum-information" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [] }, "title": "Single-seed post-processing of Gaussian POVM densities", "status": "Solved", "fields": [ "Quantum metrology" ], "topics": [ "Gaussian quantum information" ], "tags": [ "Quantum metrology", "Gaussian quantum information" ], "statement": "Is every POVM with Gaussian density a classical post-processing of one displaced Gaussian seed? Fix $n\\geq1$ bosonic modes with $[a_j,a_k^\\dagger]=\\delta_{jk}$. Let $(X,\\Sigma)$ be a measurable outcome space and $\\mu$ a positive measure. Let $x\\mapsto\\tau_x$ be a measurable family of Gaussian density operators. Assume $E(B):=\\int_B\\tau_x\\,\\mu(dx)$ defines a POVM and $E(X)=I$, with integrals in the weak operator sense. Define the displacements by Eq. (1).\n\n \\begin{equation}\nD(\\alpha):=\\exp\\!\\left(\\sum_{j=1}^n\\alpha_j a_j^\\dagger-\\overline\\alpha_j a_j\\right),\\qquad \\alpha\\in\\mathbb C^n.\n\\tag{1}\n\\end{equation} \nThe question asks for a Gaussian density operator $\\tau$ and a Markov probability kernel $K$ satisfying Eq. (2). For each $\\alpha$, $B\\mapsto K(B\\mid\\alpha)$ is a probability measure on $(X,\\Sigma)$. For each $B$, $\\alpha\\mapsto K(B\\mid\\alpha)$ is measurable.\n\n \\begin{equation}\nE(B)=\\int_{\\mathbb C^n}K(B\\mid\\alpha)D(\\alpha)\\tau D(\\alpha)^\\dagger\\,\\frac{d^{2n}\\alpha}{\\pi^n}\\qquad(B\\in\\Sigma).\n\\tag{2}\n\\end{equation}", "url": "https://qiqc-op.com/problem/op_8bb2af22f3465748/", "json": "https://qiqc-op.com/api/problems/op_8bb2af22f3465748.json", "tex": "https://qiqc-op.com/problem/op_8bb2af22f3465748/op_8bb2af22f3465748.tex", "created": "2026-09-10", "updated": "2026-09-10", "createdAt": "2026-09-10T21:54:15.000Z", "updatedAt": "2026-09-10T21:54:15.000Z", "sha256": "7b8bebbe1b515efca6cbf4f0142a005976b56c17b17119143daab43bc0af42e3" }, { "id": "op_8f1853475db7ea27", "ulid": "01M26K8Q8DCN2WF4YHQ0RJX373", "aliases": [ "op_8f1853475db7ea27", "01M26K8Q8DCN2WF4YHQ0RJX373", "op-8f1853475db7ea27" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 1, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-10T21:23:34.285Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "derived", "posed": null, "areaIds": [ "quantum-metrology" ], "topicIds": [ "quantum-estimation", "gaussian-quantum-information" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [] }, "title": "Gaussian-measurement equality with the Holevo bound", "status": "Unsolved", "fields": [ "Quantum metrology" ], "topics": [ "Quantum estimation", "Gaussian quantum information" ], "tags": [ "Quantum metrology", "Quantum estimation", "Gaussian quantum information" ], "statement": "When does the optimal single-copy Gaussian-measurement cost equal the Holevo bound? Give necessary and sufficient conditions in terms of the local first moments, covariance, their first derivatives, and the weight matrix defined below. Let $\\rho_\\theta$ be a smooth faithful $n$-mode Gaussian model, with integers $n,p\\geq1$ and $\\theta\\in\\Theta\\subset\\mathbb R^p$ for an open set $\\Theta$. Write $R=(q_1,p_1,\\ldots,q_n,p_n)^{\\mathsf T}$ for the canonical quadrature vector. Its mean vector and covariance are defined in Eq. (1). The commutation convention is Eq. (2).\n\n \\begin{equation}\n\\begin{aligned}\n (d_\\theta)_j&:=\\operatorname{Tr}(\\rho_\\theta R_j),\\\\\n (V_\\theta)_{jk}&:=\\frac12\\operatorname{Tr}(\\rho_\\theta\\{R_j-(d_\\theta)_j,R_k-(d_\\theta)_k\\}).\n\\end{aligned}\n\\tag{1}\n\\end{equation} \n \\begin{equation}\n[R_j,R_k]=i\\Omega_{jk},\\qquad\n\\Omega:=\\bigoplus_{j=1}^{n}\\begin{pmatrix}0&1\\\\-1&0\\end{pmatrix}.\n\\tag{2}\n\\end{equation} \nAssume the symmetric logarithmic derivative quantum Fisher information matrix is nonsingular. Let $W$ be a real positive-definite $p\\times p$ weight matrix.\n\nDefine the optimal single-copy Gaussian-measurement cost by Eq. (3).\n\n \\begin{equation}\nC_G(\\theta,W):=\\inf_{M\\ \\mathrm{Gaussian}}\\operatorname{tr}(WF_M(\\theta)^{-1}),\n\\tag{3}\n\\end{equation} \nwhere $F_M$ is the classical Fisher information matrix of measurement $M$, and singular $F_M$ has infinite cost. Gaussian measurements mean Gaussian-ancilla preparations followed by Gaussian unitaries and homodyne detection with arbitrary classical processing.\n\nDefine the Holevo bound $C_H(\\theta,W)$ as the infimum of Eq. (4).\n\n \\begin{equation}\n\\operatorname{tr}(W\\operatorname{Re}Z)+\\|\\sqrt W\\operatorname{Im}Z\\sqrt W\\|_1\n\\tag{4}\n\\end{equation} \nThe infimum is over Hermitian observables $X_1,\\ldots,X_p$ with finite second moments, subject to Eq. (5).\n\n \\begin{equation}\n\\begin{aligned}\n Z_{jk}&:=\\operatorname{Tr}(\\rho_\\theta X_jX_k),\\\\\n \\operatorname{Tr}(\\rho_\\theta X_j)&=0,\\qquad\n \\operatorname{Tr}((\\partial_k\\rho_\\theta)X_j)=\\delta_{jk}.\n\\end{aligned}\n\\tag{5}\n\\end{equation} \nThe measurement is optimized locally at the specified $\\theta$; its setting is held fixed when taking derivatives for $F_M$. The symmetric logarithmic derivatives $L_j$ satisfy $2\\partial_j\\rho_\\theta=\\rho_\\theta L_j+L_j\\rho_\\theta$. Their Fisher matrix has entries $\\operatorname{Re}\\operatorname{Tr}(\\rho_\\theta L_jL_k)$. The requested criterion must characterize $C_G(\\theta,W)=C_H(\\theta,W)$, rather than merely restate these two optimizations. Equality of the infima is the target. A limiting sequence of Gaussian measurements counts, even if no individual measurement attains the value.", "url": "https://qiqc-op.com/problem/op_8f1853475db7ea27/", "json": "https://qiqc-op.com/api/problems/op_8f1853475db7ea27.json", "tex": "https://qiqc-op.com/problem/op_8f1853475db7ea27/op_8f1853475db7ea27.tex", "created": "2026-09-10", "updated": "2026-09-10", "createdAt": "2026-09-10T21:54:15.000Z", "updatedAt": "2026-09-10T21:54:15.000Z", "sha256": "c5259baadcb6dcd66c6ecb02480c8b581a791c55f57a246c46474f59a0a3acae" }, { "id": "op_903c729ed6eb9ce6", "ulid": "01M26KNCZSX9BMQTNM7RX75HZ5", "aliases": [ "op_903c729ed6eb9ce6", "01M26KNCZSX9BMQTNM7RX75HZ5", "op-903c729ed6eb9ce6" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 1, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-10T21:30:29.753Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "derived", "posed": null, "areaIds": [ "quantum-algorithm" ], "topicIds": [ "boson-sampling", "computational-complexity-and-computability" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [ "01M26KND1E2P37YSPQCT6FPCCD" ] }, "title": "Average-case additive hardness of squared complex Gaussian hafnians", "status": "Unsolved", "fields": [ "Quantum algorithm" ], "topics": [ "Boson sampling", "Computational complexity and computability" ], "tags": [ "Quantum algorithm", "Boson sampling", "Computational complexity and computability" ], "statement": "Is additive estimation of squared complex Gaussian hafnians $\\#\\mathrm P$-hard under randomized polynomial-time oracle reductions? For $n\\geq1$, let $X=X^T\\in\\mathbb C^{2n\\times2n}$ have zero diagonal. Its entries above the diagonal are independent with density $\\pi^{-1}e^{-|z|^2}$. Let $\\mathcal M_{2n}$ be the perfect matchings of $\\{1,\\ldots,2n\\}$ and define\n\n \\begin{equation}\n \\operatorname{Haf}X=\\sum_{M\\in\\mathcal M_{2n}}\\prod_{\\{i,j\\}\\in M}X_{ij},\n \\qquad h_n=(2n-1)!!=\\frac{(2n)!}{2^n n!}.\n\\tag{1}\n\\end{equation} \nGiven $0<\\varepsilon,\\delta<1$, an estimator must output $\\widehat p\\in\\mathbb R$ with\n\n \\begin{equation}\n \\Pr\\!\\left[\\left|\\widehat p-|\\operatorname{Haf}X|^2\\right|\n \\leq\\varepsilon h_n\\right]\\geq1-\\delta.\n\\tag{2}\n\\end{equation} \nProbability in Eq. (2) includes both $X$ and the estimator’s randomness. The binary input is a truncated, rounded copy $\\widetilde X$. Use $\\operatorname{poly}(n,\\log(1/\\varepsilon),\\log(1/\\delta))$ bits per entry. Choose the precision so that the squared hafnians of $X$ and $\\widetilde X$ differ by at most $\\varepsilon h_n/2$, except with probability $\\delta/2$. Reduction cost is polynomial in the binary input length, $1/\\varepsilon$, and $1/\\delta$. Here $\\#\\mathrm P$ is the class of functions counting accepting paths of nondeterministic polynomial-time machines. The target normalization $h_n$ is defined in Eq. (1).", "url": "https://qiqc-op.com/problem/op_903c729ed6eb9ce6/", "json": "https://qiqc-op.com/api/problems/op_903c729ed6eb9ce6.json", "tex": "https://qiqc-op.com/problem/op_903c729ed6eb9ce6/op_903c729ed6eb9ce6.tex", "created": "2026-09-10", "updated": "2026-09-10", "createdAt": "2026-09-10T21:54:15.000Z", "updatedAt": "2026-09-10T21:54:15.000Z", "sha256": "1b4693e7b8536feb107b06292cbd003049447572570335b21671c491f94c60ab" }, { "id": "op_9966229e2ac69eee", "ulid": "01M26JZET5QZ2FB0WSWESW77V6", "aliases": [ "op_9966229e2ac69eee", "01M26JZET5QZ2FB0WSWESW77V6", "op-9966229e2ac69eee" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 1, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-10T21:18:30.725Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "source-stated", "posed": null, "areaIds": [ "quantum-communication" ], "topicIds": [ "quantum-capacity", "bosonic-channels" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [ "01M26JZEPTR8YAH8EKFW9D0Q24", "01M26JZEN93BAESCCMMTRG84M0" ] }, "title": "Unassisted quantum capacity of a thermal amplifier", "status": "Unsolved", "fields": [ "Quantum Communication" ], "topics": [ "Quantum capacity", "Bosonic channels" ], "tags": [ "Quantum Communication", "Quantum capacity", "Bosonic channels" ], "statement": "What is the unassisted quantum capacity of a thermal amplifier for every gain and thermal noise level? Fix a power gain $\\kappa>1$ and $b\\geq0$. The single-mode thermal amplifier $\\mathcal A_{\\kappa,b}$ is the Gaussian channel realized by two-mode squeezing with output annihilation operator\n\n \\begin{equation}\n a_{\\mathrm{out}}=\\sqrt\\kappa\\,a+\\sqrt{\\kappa-1}\\,e^\\dagger.\n\\tag{1}\n\\end{equation} \nThe independent input and environment modes obey $[a,a^\\dagger]=[e,e^\\dagger]=1$. The environment is a fresh independent thermal mode at each use, with mean photon number $b$ and density operator\n\n \\begin{equation}\n \\tau_b=\\frac1{b+1}\\sum_{n=0}^{\\infty}\\left(\\frac b{b+1}\\right)^n|n\\rangle\\langle n|\\quad(b>0),\\qquad \\tau_0=|0\\rangle\\langle0|.\n\\tag{2}\n\\end{equation} \nHere $|n\\rangle$ is the $n$-photon Fock state. The unused output mode is discarded. Equations (1) and (2) specify the channel on arbitrary input states. The capacity $Q(\\mathcal A_{\\kappa,b})$ is the supremum of asymptotic qubits transmitted per channel use by block codes with vanishing error. Arbitrary encodings across uses and joint decoding are allowed. No classical assistance or preshared entanglement is available. There is no fixed input-energy bound: take the supremum of the capacities over all finite mean photon budgets.", "url": "https://qiqc-op.com/problem/op_9966229e2ac69eee/", "json": "https://qiqc-op.com/api/problems/op_9966229e2ac69eee.json", "tex": "https://qiqc-op.com/problem/op_9966229e2ac69eee/op_9966229e2ac69eee.tex", "created": "2026-09-10", "updated": "2026-09-10", "createdAt": "2026-09-10T21:54:15.000Z", "updatedAt": "2026-09-10T21:54:15.000Z", "sha256": "874e383bf2af38068207968d00b6c0980baaecbf780b78e4a974b12fe0cf61ff" }, { "id": "op_b52e28b95f7677a7", "ulid": "01M26K8Q6Z6DTG3558MEC99KST", "aliases": [ "op_b52e28b95f7677a7", "01M26K8Q6Z6DTG3558MEC99KST", "op-b52e28b95f7677a7" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 1, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-10T21:23:34.239Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "editor-formulated", "posed": null, "areaIds": [ "quantum-resource-theory" ], "topicIds": [ "gaussian-quantum-information" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [] }, "title": "Optimal output monitoring for parametric-oscillator squeezing", "status": "Solved", "fields": [ "Quantum Resource Theory" ], "topics": [ "Gaussian quantum information" ], "tags": [ "Quantum Resource Theory", "Gaussian quantum information" ], "statement": "Can any causal output measurement improve the mean conditional position squeezing achieved by ideal homodyne detection? Consider one mode with $[q,p]=i$, $a=(q+ip)/\\sqrt2$, and $0\\leq\\chi<1/2$. The oscillator interacts with a vacuum Markov bath at unit damping rate. Its dynamics are Eq. (1).\n\n \\begin{equation}\nH_\\chi=-\\frac\\chi2(qp+pq),\\qquad\n\\dot\\rho=-i[H_\\chi,\\rho]+\\mathcal D[a]\\rho,\\qquad\n\\mathcal D[a]\\rho=a\\rho a^\\dagger-\\frac12\\{a^\\dagger a,\\rho\\}.\n\\tag{1}\n\\end{equation} \nThe initial oscillator state is the unconditional stationary state. The measurement $\\mathsf M$ may be adaptive or non-Gaussian and may access the output field up to the current time. It does not apply control operations to the oscillator. Let $\\rho_c(t)$ be the oscillator state conditioned on the measurement record. The mean is over all records, without postselection. Define the optimum by Eq. (2).\n\n \\begin{equation}\ns_{\\mathrm{all}}(\\chi)=\\inf_{\\mathsf M}\\liminf_{t\\to\\infty}\\mathbb E_{\\mathsf M}\\!\\left[2\\operatorname{Var}_{\\rho_c(t)}q\\right].\n\\tag{2}\n\\end{equation} \nDoes $s_{\\mathrm{all}}(\\chi)=1-2\\chi$ hold throughout the stated range?", "url": "https://qiqc-op.com/problem/op_b52e28b95f7677a7/", "json": "https://qiqc-op.com/api/problems/op_b52e28b95f7677a7.json", "tex": "https://qiqc-op.com/problem/op_b52e28b95f7677a7/op_b52e28b95f7677a7.tex", "created": "2026-09-10", "updated": "2026-09-10", "createdAt": "2026-09-10T21:54:15.000Z", "updatedAt": "2026-09-10T21:54:15.000Z", "sha256": "0c471d54fec50abaaae15fbca1b6e128ff021414b4c2bb3493f53085ec5783dc" }, { "id": "op_ba39e7a80122b256", "ulid": "01M26JZEHFZ0RFR2137NXD72BE", "aliases": [ "op_ba39e7a80122b256", "01M26JZEHFZ0RFR2137NXD72BE", "op-ba39e7a80122b256" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 1, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-10T21:18:30.447Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "source-stated", "posed": null, "areaIds": [ "quantum-communication" ], "topicIds": [ "bosonic-channels", "matrix-and-entropy-inequalities" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [ "01M1Q787QRE9WF0NXMX32BQDCQ" ] }, "title": "Entropy photon-number inequality", "status": "Unsolved", "fields": [ "Quantum Communication" ], "topics": [ "Bosonic channels", "Matrix and entropy inequalities" ], "tags": [ "Quantum Communication", "Bosonic channels", "Matrix and entropy inequalities" ], "statement": "Does the entropy photon-number inequality hold for every pair of independent finite-energy bosonic inputs? Let $n\\geq1$ and $0\\leq\\eta\\leq1$. The input is $\\rho_A\\otimes\\rho_B$, where each factor is an $n$-mode state with finite total mean photon number. Correlations among the modes within either factor are allowed. Mix corresponding input modes on beam splitters and retain the outputs\n\n \\begin{equation}\n c_j=\\sqrt\\eta\\,a_j+\\sqrt{1-\\eta}\\,b_j,\\qquad j=1,\\ldots,n.\n\\tag{1}\n\\end{equation} \nLet $\\rho_C$ be the joint state of the modes in Eq. (1). Use natural logarithms, $S(\\rho)=-\\operatorname{Tr}\\rho\\ln\\rho$, and $g(x)=(x+1)\\ln(x+1)-x\\ln x$ for $x\\geq0$, with $0\\ln0=0$. The proposed inequality is\n\n \\begin{equation}\n g^{-1}\\!\\left(\\frac{S(\\rho_C)}n\\right)\n \\geq\\eta g^{-1}\\!\\left(\\frac{S(\\rho_A)}n\\right)\n +(1-\\eta)g^{-1}\\!\\left(\\frac{S(\\rho_B)}n\\right).\n\\tag{2}\n\\end{equation} \nProve Eq. (2) in this full domain or provide a physical input pair that violates it.", "url": "https://qiqc-op.com/problem/op_ba39e7a80122b256/", "json": "https://qiqc-op.com/api/problems/op_ba39e7a80122b256.json", "tex": "https://qiqc-op.com/problem/op_ba39e7a80122b256/op_ba39e7a80122b256.tex", "created": "2026-09-10", "updated": "2026-09-10", "createdAt": "2026-09-10T21:54:15.000Z", "updatedAt": "2026-09-10T21:54:15.000Z", "sha256": "ac3c8e6b4992318163cb44f462db0dbd6667f257f12d9c7c4c0656b8200904bd" }, { "id": "op_c509b64359fbdc2d", "ulid": "01M26KNCWR2T3G8NTDRJ6G3ECR", "aliases": [ "op_c509b64359fbdc2d", "01M26KNCWR2T3G8NTDRJ6G3ECR", "op-c509b64359fbdc2d" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 1, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-10T21:30:29.656Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "source-stated", "posed": null, "areaIds": [ "quantum-algorithm" ], "topicIds": [ "boson-sampling", "computational-complexity-and-computability" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [ "01M20DFNK3W29RR12PZ50N40AH" ] }, "title": "Permanent anticoncentration for complex Gaussian matrices", "status": "Solved", "fields": [ "Quantum algorithm" ], "topics": [ "Boson sampling", "Computational complexity and computability" ], "tags": [ "Quantum algorithm", "Boson sampling", "Computational complexity and computability" ], "statement": "Do complex Gaussian permanents satisfy a polynomial lower-tail bound at their root-mean-square scale? For each integer $n\\geq1$, let $X\\in\\mathbb C^{n\\times n}$ have independent entries with density $\\pi^{-1}e^{-|z|^2}$. Write $S_n$ for the permutations of $\\{1,\\ldots,n\\}$ and define\n\n \\begin{equation}\n \\operatorname{Per}X=\\sum_{\\pi\\in S_n}\\prod_{j=1}^nX_{j,\\pi(j)}.\n\\tag{1}\n\\end{equation} \nDoes there exist a polynomial $p$, positive on $[1,\\infty)^2$, such that the permanent in Eq. (1) satisfies\n\n \\begin{equation}\n \\Pr_X\\!\\left[|\\operatorname{Per}X|<\n \\frac{\\sqrt{n!}}{p(n,1/\\delta)}\\right]<\\delta\n \\qquad(n\\geq1,\\ 0<\\delta<1)?\n\\tag{2}\n\\end{equation} \nThe same polynomial must work for every pair $(n,\\delta)$ in Eq. (2).", "url": "https://qiqc-op.com/problem/op_c509b64359fbdc2d/", "json": "https://qiqc-op.com/api/problems/op_c509b64359fbdc2d.json", "tex": "https://qiqc-op.com/problem/op_c509b64359fbdc2d/op_c509b64359fbdc2d.tex", "created": "2026-09-10", "updated": "2026-09-10", "createdAt": "2026-09-10T21:54:15.000Z", "updatedAt": "2026-09-10T21:54:15.000Z", "sha256": "3911b20578f7260b658ba6e02e892ca265a99d1d35e7e8ac4bf37b41b07acd34" }, { "id": "op_d436ff9fb9cb0ce5", "ulid": "01M26KNCY5ZMJCE77MG11QCQT6", "aliases": [ "op_d436ff9fb9cb0ce5", "01M26KNCY5ZMJCE77MG11QCQT6", "op-d436ff9fb9cb0ce5" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 1, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-10T21:30:29.701Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "derived", "posed": null, "areaIds": [ "quantum-algorithm" ], "topicIds": [ "boson-sampling", "computational-complexity-and-computability" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [] }, "title": "Fully polynomial sampling of boson sampling with constant photon transmission", "status": "Unsolved", "fields": [ "Quantum algorithm" ], "topics": [ "Boson sampling", "Computational complexity and computability" ], "tags": [ "Quantum algorithm", "Boson sampling", "Computational complexity and computability" ], "statement": "For every fixed rational transmission $0<\\eta<1$, is boson sampling with independent photon loss classically samplable in fully polynomial time? Let $m\\geq n\\geq1$ and draw $U\\in U(m)$ from normalized Haar measure. Inject one perfectly indistinguishable photon into each of the first $n$ input modes. Each photon survives independently with probability $\\eta$. The photon-counting distribution is\n\n \\begin{equation}\n p_{\\eta,U}(s)=\n \\sum_{\\substack{T\\subseteq\\{1,\\ldots,n\\}\\\\|T|=|s|}}\n \\eta^{|T|}(1-\\eta)^{n-|T|}\n \\frac{|\\operatorname{Per}(U_{s,T})|^2}{\\prod_{j=1}^{m}s_j!},\n \\quad s\\in\\mathbb N_0^m,\\quad |s|=\\sum_js_j.\n\\tag{1}\n\\end{equation} \nIn Eq. (1), $U_{s,T}$ selects columns in $T$ and repeats row $j$ exactly $s_j$ times. For a $k\\times k$ matrix $A$, $\\operatorname{Per}A=\\sum_{\\pi\\in S_k}\\prod_{j=1}^kA_{j,\\pi(j)}$; the empty permanent equals one. A randomized classical algorithm receives $n,m,\\varepsilon$ and a finite description of $U$, with $0<\\varepsilon<1$. Use $\\operatorname{poly}(n,m,\\log(1/\\varepsilon))$ bits to encode the matrix. Choose the precision so that its contribution to total variation is at most $\\varepsilon/2$. For every $n,m,\\varepsilon$, require an output law $q_U$ satisfying\n\n \\begin{equation}\n \\mathbb E_{U\\sim\\mathrm{Haar}}\\operatorname{TV}(q_U,p_{\\eta,U})\\leq\\varepsilon,\n \\qquad\n \\operatorname{TV}(q,p)=\\frac12\\sum_{s\\in\\mathbb N_0^m}|q(s)-p(s)|.\n\\tag{2}\n\\end{equation} \nCan Eq. (2) be achieved in time polynomial in $n,m,1/\\varepsilon$ and the binary input length? The polynomial may depend on the fixed $\\eta$, but its exponent cannot depend on $\\varepsilon$.", "url": "https://qiqc-op.com/problem/op_d436ff9fb9cb0ce5/", "json": "https://qiqc-op.com/api/problems/op_d436ff9fb9cb0ce5.json", "tex": "https://qiqc-op.com/problem/op_d436ff9fb9cb0ce5/op_d436ff9fb9cb0ce5.tex", "created": "2026-09-10", "updated": "2026-09-10", "createdAt": "2026-09-10T21:54:15.000Z", "updatedAt": "2026-09-10T21:54:15.000Z", "sha256": "68add40049d13057e26fb3d4bfc35816dcf1e6ca85d2aa74159c806d3263cbef" }, { "id": "op_dc53476a203c535b", "ulid": "01M26KH5M08P2ATTX0CB1N1E09", "aliases": [ "op_dc53476a203c535b", "01M26KH5M08P2ATTX0CB1N1E09", "op-dc53476a203c535b" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 1, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-10T21:28:11.136Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "derived", "posed": null, "areaIds": [ "quantum-resource-theory" ], "topicIds": [ "gaussian-quantum-information", "resource-conversion" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [] }, "title": "Asymptotic nonclassical-state conversion by linear optics", "status": "Unsolved", "fields": [ "Quantum Resource Theory" ], "topics": [ "Gaussian quantum information", "Resource conversion" ], "tags": [ "Quantum Resource Theory", "Gaussian quantum information", "Resource conversion" ], "statement": "What is the optimal asymptotic conversion rate between finite-energy bosonic states under passive linear-optical protocols?\n\nLet $\\rho$ and $\\sigma$ be finite-energy density operators on finitely many bosonic modes, with $\\sigma$ nonclassical. Finite energy means finite total mean photon number. Classical states form the trace-norm closed convex hull of multimode coherent states. The allowed deterministic protocols $\\Lambda_k$ use passive linear-optical unitaries, ancillary coherent-state mixtures, destructive measurements, classical feed-forward, and discarding modes. Measurements may be non-Gaussian. The complete protocol is trace preserving; failure outcomes cannot be omitted.\n\nUse the trace norm $\\lVert X\\rVert_1:=\\operatorname{Tr}\\sqrt{X^\\dagger X}$. A rate $r\\geq0$ is achievable if an allowed sequence $(\\Lambda_k)_{k\\geq1}$ satisfies\n\n \\begin{equation}\n\\lim_{k\\to\\infty}\\left\\lVert\\Lambda_k(\\rho^{\\otimes k})-\\sigma^{\\otimes\\lfloor rk\\rfloor}\\right\\rVert_1=0.\n\\tag{1}\n\\end{equation} \nDetermine $R_{\\mathrm{LO}}(\\rho\\to\\sigma):=\\sup\\{r\\geq0:r\\text{ is achievable}\\}$, with achievability defined by Eq. (1).", "url": "https://qiqc-op.com/problem/op_dc53476a203c535b/", "json": "https://qiqc-op.com/api/problems/op_dc53476a203c535b.json", "tex": "https://qiqc-op.com/problem/op_dc53476a203c535b/op_dc53476a203c535b.tex", "created": "2026-09-10", "updated": "2026-09-10", "createdAt": "2026-09-10T21:54:15.000Z", "updatedAt": "2026-09-10T21:54:15.000Z", "sha256": "a10dfadcaeb053e8ff52919e6ecd6954092b4e9bc1b119647eb2acedbec943d1" }, { "id": "op_f408a3c300b9b214", "ulid": "01M26KH5VV4PMPT9VB7KSSMS65", "aliases": [ "op_f408a3c300b9b214", "01M26KH5VV4PMPT9VB7KSSMS65", "op-f408a3c300b9b214" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 1, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-10T21:28:11.387Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "source-stated", "posed": null, "areaIds": [ "quantum-resource-theory" ], "topicIds": [ "gaussian-quantum-information", "quantum-separability" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [] }, "title": "Exact covariance criterion for bipartite Gaussian separability", "status": "Solved", "fields": [ "Quantum Resource Theory" ], "topics": [ "Gaussian quantum information", "Quantum separability" ], "tags": [ "Quantum Resource Theory", "Gaussian quantum information", "Quantum separability" ], "statement": "What necessary and sufficient condition on a covariance matrix characterizes bipartite Gaussian separability?\n\nLet $m,n\\geq1$ be integers. Let $\\rho_V$ be a zero-mean Gaussian state with $m$ modes held by Alice and $n$ modes held by Bob. Its finite real covariance matrix and canonical commutators are\n\n \\begin{equation}\nV_{jk}:=\\operatorname{Tr}\\rho_V\\{R_j,R_k\\},\\qquad\n[R_j,R_k]=i(\\Omega_{m+n})_{jk},\\qquad\n\\Omega_k:=\\bigoplus_{j=1}^{k}\\begin{pmatrix}0&1\\\\-1&0\\end{pmatrix}.\n\\tag{1}\n\\end{equation} \nSeparability means that $\\rho_V$ belongs to the trace-norm closed convex hull of product density operators. Determine separability from the covariance in Eq. (1).", "url": "https://qiqc-op.com/problem/op_f408a3c300b9b214/", "json": "https://qiqc-op.com/api/problems/op_f408a3c300b9b214.json", "tex": "https://qiqc-op.com/problem/op_f408a3c300b9b214/op_f408a3c300b9b214.tex", "created": "2026-09-10", "updated": "2026-09-10", "createdAt": "2026-09-10T21:54:15.000Z", "updatedAt": "2026-09-10T21:54:15.000Z", "sha256": "4a0b6a84fd580b522b9e0793619e09f0a93bec34936fabd3c70047098749887c" }, { "id": "op_7249a79e1534f6ab", "ulid": "01M20CKB78AVP4GX3MKGMJ1VAE", "aliases": [ "op_7249a79e1534f6ab", "01M20CKB78AVP4GX3MKGMJ1VAE", "op-7249a79e1534f6ab" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 1, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-08T11:31:35.784Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "editor-formulated", "posed": null, "areaIds": [ "quantum-algorithm" ], "topicIds": [ "computational-complexity-and-computability" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [] }, "title": "Polynomial-time quantum algorithm for Graph Isomorphism", "status": "Unsolved", "fields": [ "Quantum algorithm" ], "topics": [ "Computational complexity and computability" ], "tags": [ "Quantum algorithm", "Computational complexity and computability" ], "statement": "Does the Graph Isomorphism problem admit a polynomial-time quantum algorithm?\n\nLet $G=(V_G,E_G)$ and $H=(V_H,E_H)$ be finite simple graphs with $|V_G|=|V_H|=n$. The graphs are isomorphic if there exists a bijection $\\pi:V_G\\to V_H$ satisfying\n\n \\begin{equation}\n \\{u,v\\}\\in E_G\n \\Longleftrightarrow\n \\{\\pi(u),\\pi(v)\\}\\in E_H\n \\qquad\n \\forall u,v\\in V_G .\n\\tag{1}\n\\end{equation} \nThe open question is whether there is a bounded-error quantum algorithm running in time polynomial in $n$ that, given $G$ and $H$, decides whether a bijection satisfying (1) exists.", "url": "https://qiqc-op.com/problem/op_7249a79e1534f6ab/", "json": "https://qiqc-op.com/api/problems/op_7249a79e1534f6ab.json", "tex": "https://qiqc-op.com/problem/op_7249a79e1534f6ab/op_7249a79e1534f6ab.tex", "created": "2026-09-08", "updated": "2026-09-10", "createdAt": "2026-09-08T11:55:53.000Z", "updatedAt": "2026-09-10T10:50:30.000Z", "sha256": "e7a94fec5dc87e84e94ae712d4143e97c3579c826514290c0d64d969fec45fa9" }, { "id": "op_25d9dee5435ea835", "ulid": "01M22P0HY0R1RBEABK7MZW488E", "aliases": [ "op_25d9dee5435ea835", "01M22P0HY0R1RBEABK7MZW488E", "op-25d9dee5435ea835" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 1, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-09T08:54:34.688Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "source-stated", "posed": "2024", "areaIds": [ "quantum-algorithm" ], "topicIds": [ "hamiltonian-simulation", "computational-complexity-and-computability" ], "keywords": [ "low-energy dynamics", "spectral gap amplification", "polynomial approximation", "precision dependence" ], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [] }, "title": "Optimal precision dependence of low-energy Hamiltonian simulation", "status": "Unsolved", "fields": [ "Quantum algorithm" ], "topics": [ "Hamiltonian simulation", "Computational complexity and computability" ], "tags": [ "Quantum algorithm", "Hamiltonian simulation", "Computational complexity and computability" ], "statement": "What is the tight precision dependence of worst-case query complexity for low-energy simulation in the regime (2)? Let $A\\in\\mathbb{C}^{N\\times N}$, $\\lVert A\\rVert\\leq1$, $H=\\lambda A^\\dagger A$, and $P_\\Delta=\\mathbf{1}_{[0,\\Delta]}(H)$, where $\\lambda>0$ and $0<\\Delta\\leq\\lambda$. Assume an exact block encoding $(\\langle0^m\\rvert\\otimes I_N)V_A(\\lvert0^m\\rangle\\otimes I_N)=A$, with controlled and inverse calls. Count these queries; input-state preparation is excluded. For known $t>0$ and $0<\\epsilon<1/2$, a unitary simulator $W$ must satisfy (1) uniformly on the promised subspace:\n\n \\begin{equation}\n \\sup_{\\substack{\\lVert\\psi\\rVert=1\\\\P_\\Delta\\lvert\\psi\\rangle=\\lvert\\psi\\rangle}}\n \\left\\lVert W(\\lvert0^a\\rangle\\lvert\\psi\\rangle)\n -\\lvert0^a\\rangle e^{-itH}\\lvert\\psi\\rangle\\right\\rVert\\leq\\epsilon.\n\\tag{1}\n\\end{equation} \nHere $a$ counts workspace qubits. Consider asymptotic families with $\\epsilon\\to0$ satisfying\n\n \\begin{equation}\n \\epsilon=o(t\\Delta),\\qquad\n t\\Delta=o\\bigl(\\log(1/\\epsilon)\\bigr),\\qquad\n \\log(1/\\epsilon)=o(t\\lambda).\n\\tag{2}\n\\end{equation} \nDetermine whether the known $O(\\sqrt{t\\lambda\\log(1/\\epsilon)})$ upper bound has optimal precision dependence.", "url": "https://qiqc-op.com/problem/op_25d9dee5435ea835/", "json": "https://qiqc-op.com/api/problems/op_25d9dee5435ea835.json", "tex": "https://qiqc-op.com/problem/op_25d9dee5435ea835/op_25d9dee5435ea835.tex", "created": "2026-09-09", "updated": "2026-09-09", "createdAt": "2026-09-09T09:12:15.000Z", "updatedAt": "2026-09-09T09:12:15.000Z", "sha256": "23e5679f7b7760e1ae37e7439a39f8eb05203c74e9b33a01e9e4cd465f745332" }, { "id": "op_b0666933a3d77eba", "ulid": "01M22P0HX43A7EQD9XKQ1G7QQJ", "aliases": [ "op_b0666933a3d77eba", "01M22P0HX43A7EQD9XKQ1G7QQJ", "op-b0666933a3d77eba" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 1, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-09T08:54:34.660Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "source-stated", "posed": "2024", "areaIds": [ "quantum-algorithm" ], "topicIds": [ "quantum-linear-algebra", "computational-complexity-and-computability" ], "keywords": [ "quantum linear systems", "block encoding", "variable-time amplitude amplification", "oracle tradeoffs" ], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [] }, "title": "Simultaneously optimal queries to both quantum linear-system oracles", "status": "Unsolved", "fields": [ "Quantum algorithm" ], "topics": [ "Quantum linear algebra", "Computational complexity and computability" ], "tags": [ "Quantum algorithm", "Quantum linear algebra", "Computational complexity and computability" ], "statement": "Can one quantum linear-system algorithm attain both query bounds in (2) simultaneously? Let $A\\in\\mathbb{C}^{N\\times N}$ be invertible, with exact block-encoding oracle $O_A$ for $A/\\alpha_A$ and state-preparation oracle $O_b\\lvert0\\rangle=\\lvert b\\rangle$. Given bounds $\\alpha_A\\geq\\lVert A\\rVert$, $\\alpha_{A^{-1}}\\geq\\lVert A^{-1}\\rVert$, define (1):\n\n \\begin{equation}\n K=\\alpha_A\\alpha_{A^{-1}},\\qquad\n p=\\frac{\\lVert A^{-1}\\lvert b\\rangle\\rVert^2}{\\alpha_{A^{-1}}^2},\\qquad\n \\lvert x\\rangle=\\frac{A^{-1}\\lvert b\\rangle}{\\lVert A^{-1}\\lvert b\\rangle\\rVert}.\n\\tag{1}\n\\end{equation} \nAssume a constant-factor estimate of $p$ is supplied. For every $0<\\epsilon<1/2$, require a state $\\lvert\\widetilde{x}\\rangle$ with $\\lVert\\lvert\\widetilde{x}\\rangle-\\lvert x\\rangle\\rVert\\leq\\epsilon$, with success probability at least $2/3$, using\n\n \\begin{equation}\n Q_b=O(p^{-1/2}),\\qquad Q_A=O\\bigl(K\\log(1/\\epsilon)\\bigr).\n\\tag{2}\n\\end{equation} \nQueries include inverse and controlled oracle calls; constants must be uniform over admissible inputs.", "url": "https://qiqc-op.com/problem/op_b0666933a3d77eba/", "json": "https://qiqc-op.com/api/problems/op_b0666933a3d77eba.json", "tex": "https://qiqc-op.com/problem/op_b0666933a3d77eba/op_b0666933a3d77eba.tex", "created": "2026-09-09", "updated": "2026-09-09", "createdAt": "2026-09-09T09:12:15.000Z", "updatedAt": "2026-09-09T09:12:15.000Z", "sha256": "02f3bdff90d6c8e5049a81234324b123d399e7e458c470d99c5b12fc5eba2bc4" }, { "id": "op_0c63e4d2400a95b1", "ulid": "01M22DB0YP7E8RZH3CD4XBSAFR", "aliases": [ "op_0c63e4d2400a95b1", "01M22DB0YP7E8RZH3CD4XBSAFR", "op-0c63e4d2400a95b1" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 1, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-09T06:23:00.566Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "source-stated", "posed": null, "areaIds": [ "quantum-metrology", "quantum-resource-theory" ], "topicIds": [ "quantum-state-discrimination", "local-operations-and-classical-communication" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [] }, "title": "LOCC discrimination of three maximally entangled states", "status": "Unsolved", "fields": [ "Quantum metrology", "Quantum Resource Theory" ], "topics": [ "Quantum state discrimination", "Local operations and classical communication" ], "tags": [ "Quantum metrology", "Quantum Resource Theory", "Quantum state discrimination", "Local operations and classical communication" ], "statement": "Can every set of three orthogonal maximally entangled pure states on $\\mathbb C^d\\otimes\\mathbb C^d$, for every integer $d\\geq4$, be distinguished exactly from one copy using unrestricted two-way local operations and classical communication (LOCC)? Alice and Bob know the set $\\mathcal S:=\\{|\\psi_i\\rangle\\}_{i=1}^3$ and hold systems $A$ and $B$, respectively. The density operators $\\rho_i:=|\\psi_i\\rangle\\langle\\psi_i|$ satisfy Eq. (1):\n\n \\begin{equation}\n\\langle\\psi_i|\\psi_j\\rangle=\\delta_{ij},\n\\qquad \\operatorname{Tr}_B\\rho_i=\\frac{I_A}{d},\n\\qquad i,j\\in\\{1,2,3\\}.\n\\tag{1}\n\\end{equation} \nHere $I_A$ is the identity on Alice’s system and $\\delta_{ij}$ is the Kronecker delta. The task is to implement a positive-operator-valued measurement $\\mathbb M:=\\{M_i\\}_{i=1}^3$ by LOCC, with no additional shared entanglement and no requirement to preserve the input state. Writing $I_{AB}$ for the joint identity, perfect discrimination means Eq. (2):\n\n \\begin{equation}\n\\begin{gathered}\nM_i\\geq0,\\qquad \\sum_{i=1}^3M_i=I_{AB},\\\\\n\\operatorname{Tr}(M_i\\rho_j)=\\delta_{ij}\n\\quad\\text{for all }i,j\\in\\{1,2,3\\}.\n\\end{gathered}\n\\tag{2}\n\\end{equation} \nThe measurement must be exactly implementable by LOCC, rather than merely approximable by a sequence of LOCC measurements.", "url": "https://qiqc-op.com/problem/op_0c63e4d2400a95b1/", "json": "https://qiqc-op.com/api/problems/op_0c63e4d2400a95b1.json", "tex": "https://qiqc-op.com/problem/op_0c63e4d2400a95b1/op_0c63e4d2400a95b1.tex", "created": "2026-09-09", "updated": "2026-09-09", "createdAt": "2026-09-09T09:07:10.000Z", "updatedAt": "2026-09-09T09:07:10.000Z", "sha256": "860b54e86f2526f450d367f2720bf8b8675f730bcdbe606032902b930e580ee1" }, { "id": "op_0ca7986d256cf0df", "ulid": "01M22DB10WQKKPR643DRENCH01", "aliases": [ "op_0ca7986d256cf0df", "01M22DB10WQKKPR643DRENCH01", "op-0ca7986d256cf0df" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 1, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-09T06:23:00.636Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "source-stated", "posed": null, "areaIds": [ "quantum-communication", "quantum-resource-theory" ], "topicIds": [ "entanglement-cost", "quantum-communication-complexity", "local-operations-and-classical-communication" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [ "01M1HME780TJ3Z7X332QY1GWFR" ] }, "title": "Exponential entanglement cost with simultaneous classical communication", "status": "Unsolved", "fields": [ "Quantum Communication", "Quantum Resource Theory" ], "topics": [ "Entanglement cost", "Quantum communication complexity", "Local operations and classical communication" ], "tags": [ "Quantum Communication", "Quantum Resource Theory", "Entanglement cost", "Quantum communication complexity", "Local operations and classical communication" ], "statement": "Does there exist a family of bipartite unitaries requiring exponentially many shared Bell pairs for fixed-error implementation when classical communication is limited to one simultaneous exchange? For every integer $n\\geq1$, let $U_n$ act on $\\mathcal H_{A_n}\\otimes\\mathcal H_{B_n}=(\\mathbb C^2)^{\\otimes n}\\otimes(\\mathbb C^2)^{\\otimes n}$, with $n$ qubits held by each party. The allowed protocols $\\Lambda\\in\\mathrm{LOBC}$ use arbitrary local quantum operations and one simultaneous exchange of classical messages: each outgoing message is fixed before the incoming message is read. No quantum communication or shared entanglement beyond the supplied Bell pairs is allowed.\n\nDefine the Bell-pair state and target channel by Eq. (1), where $\\rho$ is an arbitrary input density operator:\n\n \\begin{equation}\n\\begin{gathered}\n|\\Phi_2\\rangle:=\\frac{|00\\rangle+|11\\rangle}{\\sqrt2},\\qquad\n\\Phi_2:=|\\Phi_2\\rangle\\langle\\Phi_2|,\\\\\n\\mathcal U_n(\\rho):=U_n\\rho U_n^\\dagger.\n\\end{gathered}\n\\tag{1}\n\\end{equation} \nFor $0\\leq\\varepsilon<1/10$, the minimum Bell-pair cost is Eq. (2):\n\n \\begin{equation}\nE_{\\parallel}^{\\varepsilon}(U_n):=\\min\\left\\{\nE\\in\\mathbb Z_{\\geq0}:\\ \\begin{gathered}\n\\exists\\Lambda\\in\\mathrm{LOBC},\\\\\n\\frac12\\left\\|\\Lambda\\bigl(\\,\\cdot\\,\\otimes\\Phi_2^{\\otimes E}\\bigr)-\\mathcal U_n\\right\\|_{\\diamond}\\leq\\varepsilon\n\\end{gathered}\\right\\}.\n\\tag{2}\n\\end{equation} \nHere $\\|\\cdot\\|_{\\diamond}$ is the diamond norm, including arbitrary reference systems, and the minimum is $+\\infty$ if the feasible set is empty. Each $\\Lambda$ is a deterministic channel returning the two prescribed $n$-qubit outputs after discarding auxiliary systems. The question asks whether there exist such a family and constants $c>0$ and $\\varepsilon_0\\in(0,1/10)$ satisfying Eq. (3):\n\n \\begin{equation}\nE_{\\parallel}^{\\varepsilon_0}(U_n)\\geq2^{cn}\n\\qquad\\text{for all sufficiently large }n.\n\\tag{3}\n\\end{equation}", "url": "https://qiqc-op.com/problem/op_0ca7986d256cf0df/", "json": "https://qiqc-op.com/api/problems/op_0ca7986d256cf0df.json", "tex": "https://qiqc-op.com/problem/op_0ca7986d256cf0df/op_0ca7986d256cf0df.tex", "created": "2026-09-09", "updated": "2026-09-09", "createdAt": "2026-09-09T09:07:10.000Z", "updatedAt": "2026-09-09T09:07:10.000Z", "sha256": "09489c8fb22eed1afc7fc4fabaed16433e8c63b3bb86fe87b10201339cec3358" }, { "id": "op_21cb3e1c3ed33976", "ulid": "01M22C448YAVGY33AV04GHTFGC", "aliases": [ "op_21cb3e1c3ed33976", "01M22C448YAVGY33AV04GHTFGC", "op-21cb3e1c3ed33976" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 1, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-09T06:01:46.014Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "derived", "posed": null, "areaIds": [ "quantum-resource-theory" ], "topicIds": [ "quantum-state-preparation", "resource-conversion" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [ "01M22C4460W3FYGAFEKQT833BJ" ] }, "title": "Quasi-linear graph-state resources for Pauli pairability", "status": "Unsolved", "fields": [ "Quantum Resource Theory" ], "topics": [ "Quantum state preparation", "Resource conversion" ], "tags": [ "Quantum Resource Theory", "Quantum state preparation", "Resource conversion" ], "statement": "Does there exist an absolute constant $a\\geq0$ such that $N_{\\mathrm P}(k)=O(k[\\log_2(k+1)]^a)$ as the integer $k\\geq2$ tends to infinity? Here $N_{\\mathrm P}(k)$ is the smallest number $n\\geq2k$ of parties holding one qubit each of a graph state as defined in Eq. (1):\n\n \\begin{equation}\n\\begin{aligned}\n|G\\rangle&:=\\prod_{\\{u,v\\}\\in E}CZ_{uv}|+\\rangle^{\\otimes n},\n\\qquad G=(V,E),\\quad |V|=n,\\\\\n|+\\rangle&:=\\frac{|0\\rangle+|1\\rangle}{\\sqrt2},\n\\qquad CZ:=\\operatorname{diag}(1,1,1,-1).\n\\end{aligned}\n\\tag{1}\n\\end{equation} \nEvery requested set of $k$ disjoint pairs of labelled parties $(a_1,b_1),\\ldots,(a_k,b_k)$ must be able to obtain $\\bigotimes_{j=1}^k|\\Phi^+\\rangle_{a_jb_j}$ deterministically, where $|\\Phi^+\\rangle:=(|00\\rangle+|11\\rangle)/\\sqrt2$. The allowed operations are single-qubit Clifford unitaries (unitaries normalizing the Pauli group), destructive single-qubit Pauli measurements on discarded parties, and classical communication.\n\nEquivalently, define $N_{\\mathrm P}(k)$ by Eq. (2), with $G$ ranging over finite simple graphs:\n\n \\begin{equation}\nN_{\\mathrm P}(k):=\\min\\left\\{\n|V(G)|:\\ \\begin{gathered}\n|V(G)|\\geq2k,\\\\\n\\forall S\\subseteq V(G)\\text{ with }|S|=2k,\\\\\n\\forall M\\text{ a perfect matching on }S,\\quad M\\leq_{\\mathrm{vm}}G\n\\end{gathered}\n\\right\\}.\n\\tag{2}\n\\end{equation} \nA perfect matching consists of $k$ disjoint edges covering $S$. The relation $M\\leq_{\\mathrm{vm}}G$ means reachability, with surviving labels fixed, by vertex deletion and local complementation, which toggles edges between distinct neighbors of a vertex.", "url": "https://qiqc-op.com/problem/op_21cb3e1c3ed33976/", "json": "https://qiqc-op.com/api/problems/op_21cb3e1c3ed33976.json", "tex": "https://qiqc-op.com/problem/op_21cb3e1c3ed33976/op_21cb3e1c3ed33976.tex", "created": "2026-09-09", "updated": "2026-09-09", "createdAt": "2026-09-09T09:07:10.000Z", "updatedAt": "2026-09-09T09:07:10.000Z", "sha256": "20813114e84d6137d6cfaff8a7834fa0b0cde38e27b02099b188e1f50fe4f398" }, { "id": "op_2e43f525333b67c0", "ulid": "01M22MTNSG0TQ51EAG3EFY48JP", "aliases": [ "op_2e43f525333b67c0", "01M22MTNSG0TQ51EAG3EFY48JP", "op-2e43f525333b67c0" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 1, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-09T08:33:53.456Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "source-stated", "posed": null, "areaIds": [ "quantum-resource-theory" ], "topicIds": [ "entanglement-cost", "ppt-preserving-operations", "entanglement-measures" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [ "01M1Q787QRPDH1Y9ADAGSB1AGN" ] }, "title": "Second-level collapse of the exact PPT entanglement-cost hierarchy", "status": "Unsolved", "fields": [ "Quantum Resource Theory" ], "topics": [ "Entanglement cost", "PPT-preserving operations", "Entanglement measures" ], "tags": [ "Quantum Resource Theory", "Entanglement cost", "PPT-preserving operations", "Entanglement measures" ], "statement": "Does the semidefinite hierarchy for exact PPT entanglement cost collapse at its second level for every finite-dimensional bipartite density operator $\\rho_{AB}$? Let $\\Gamma=\\operatorname{id}_A\\otimes T_B$ denote partial transpose. For an integer $p\\geq0$, define\n\n \\begin{equation}\n E_{\\chi,p}(\\rho):=\\log_2\\min_{S_0,\\ldots,S_p}\\left\\{\\operatorname{Tr}S_p:\\ -S_i\\leq S_{i-1}^{\\Gamma}\\leq S_i\\ (0\\leq i\\leq p),\\ S_{-1}=\\rho\\right\\},\n\\tag{1}\n\\end{equation} \nwhere the variables in Eq. (1) are Hermitian operators on $A\\otimes B$ and the inequalities are in the positive-semidefinite order. Prove or disprove\n\n \\begin{equation}\n E_{\\chi,3}(\\rho)=E_{\\chi,2}(\\rho)\\qquad\\text{for every }\\rho_{AB}.\n\\tag{2}\n\\end{equation} \nA counterexample to Eq. (2) must establish a strict gap.", "url": "https://qiqc-op.com/problem/op_2e43f525333b67c0/", "json": "https://qiqc-op.com/api/problems/op_2e43f525333b67c0.json", "tex": "https://qiqc-op.com/problem/op_2e43f525333b67c0/op_2e43f525333b67c0.tex", "created": "2026-09-09", "updated": "2026-09-09", "createdAt": "2026-09-09T09:07:10.000Z", "updatedAt": "2026-09-09T09:07:10.000Z", "sha256": "7dd250c9ac199c355453f0dd5482caeb70e40f24e7f6ca1c530124950cfc20c6" }, { "id": "op_396109643100aaca", "ulid": "01M22MXHD69PEA36XK1QY3ARAY", "aliases": [ "op_396109643100aaca", "01M22MXHD69PEA36XK1QY3ARAY", "op-396109643100aaca" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 1, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-09T08:35:27.270Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "source-stated", "posed": null, "areaIds": [ "quantum-communication" ], "topicIds": [ "quantum-source-coding" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [] }, "title": "Distributed compression of bipartite pure-state ensembles", "status": "Unsolved", "fields": [ "Quantum Communication" ], "topics": [ "Quantum source coding" ], "tags": [ "Quantum Communication", "Quantum source coding" ], "statement": "Determine the achievable rate region for independent local compression of an arbitrary finite ensemble of bipartite pure states. Let $\\mathcal E=\\{p_i,|\\phi_i\\rangle^{AB}\\}_{i\\in\\mathcal I}$, where $A$ and $B$ are finite-dimensional and $p_i>0$ with $\\sum_i p_i=1$. Write $p_{i^n}=\\prod_{t=1}^n p_{i_t}$, $|\\phi_{i^n}\\rangle:=\\bigotimes_{t=1}^n|\\phi_{i_t}\\rangle$, and $\\phi_{i^n}:=|\\phi_{i^n}\\rangle\\langle\\phi_{i^n}|$. Alice and Bob receive the respective shares of $\\phi_{i^n}$, without the classical label $i^n$. They apply independent completely positive trace-preserving encoders $\\mathcal E_{A,n}:\\mathcal L(A^{\\otimes n})\\to\\mathcal L(M_{A,n})$ and $\\mathcal E_{B,n}:\\mathcal L(B^{\\otimes n})\\to\\mathcal L(M_{B,n})$, and send both quantum messages to one receiver. The receiver applies a completely positive trace-preserving decoder $\\mathcal D_n$ with output $\\widehat A^{\\otimes n}\\widehat B^{\\otimes n}$, where $\\widehat A\\cong A$ and $\\widehat B\\cong B$. The parties have no initially shared entanglement and exchange no other messages. Define the average squared fidelity by\n\n \\begin{equation}\n F_n:=\\sum_{i^n}p_{i^n}\\langle\\phi_{i^n}|\\mathcal D_n\\circ(\\mathcal E_{A,n}\\otimes\\mathcal E_{B,n})(\\phi_{i^n})|\\phi_{i^n}\\rangle.\n\\tag{1}\n\\end{equation} \nA nonnegative pair $(R_A,R_B)$ is achievable if there is a sequence of such codes with $F_n\\to1$ in Eq. (1) and\n\n \\begin{equation}\n \\limsup_{n\\to\\infty}\\frac1n\\log_2\\dim M_{A,n}\\le R_A,\\qquad\n \\limsup_{n\\to\\infty}\\frac1n\\log_2\\dim M_{B,n}\\le R_B.\n\\tag{2}\n\\end{equation} \nGive necessary and sufficient conditions on $(R_A,R_B)$ for membership in the closure $\\mathcal R(\\mathcal E)$ of the pairs satisfying Eq. (2).", "url": "https://qiqc-op.com/problem/op_396109643100aaca/", "json": "https://qiqc-op.com/api/problems/op_396109643100aaca.json", "tex": "https://qiqc-op.com/problem/op_396109643100aaca/op_396109643100aaca.tex", "created": "2026-09-09", "updated": "2026-09-09", "createdAt": "2026-09-09T09:07:10.000Z", "updatedAt": "2026-09-09T09:07:10.000Z", "sha256": "79fc599ac76d759b7ab9f949a2072b1a74caa723a81812eba0c5137a1ea8a8bc" }, { "id": "op_58610efec5dbe564", "ulid": "01M22N8KF12VPYYFTZNP5YC0S4", "aliases": [ "op_58610efec5dbe564", "01M22N8KF12VPYYFTZNP5YC0S4", "op-58610efec5dbe564" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 1, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-09T08:41:29.825Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "source-stated", "posed": null, "areaIds": [ "quantum-cryptography" ], "topicIds": [ "secret-key-distillation", "quantum-relative-entropy" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [] }, "title": "Exact privacy-amplification exponent under relative-entropy security", "status": "Unsolved", "fields": [ "Quantum Cryptography" ], "topics": [ "Secret-key distillation", "Quantum relative entropy" ], "tags": [ "Quantum Cryptography", "Secret-key distillation", "Quantum relative entropy" ], "statement": "What is the optimal exponential rate of decay of relative-entropy insecurity achievable by two-universal hashing against quantum side information at every positive extraction rate below the conditional entropy? Let $\\rho_{AE}=\\sum_{a\\in\\mathcal A}p_a\\lvert a\\rangle\\langle a\\rvert\\otimes\\rho_E^a$ be a fixed finite-dimensional classical–quantum state, with $\\rho_E=\\sum_a p_a\\rho_E^a$. All logarithms are base two. Write $S(\\sigma)=-\\operatorname{Tr}\\sigma\\log_2\\sigma$ and $H(A|E)_\\rho=S(\\rho_{AE})-S(\\rho_E)$.\n\nFor $n$ independent copies, choose a public random hash $F_n:\\mathcal A^n\\to\\{1,\\ldots,M_n\\}$, independently of the source. Two-universality means\n\n \\begin{equation}\n \\Pr\\{F_n(a^n)=F_n(b^n)\\}\\leq M_n^{-1}\\quad\\text{for every }a^n\\neq b^n.\n\\tag{1}\n\\end{equation} \nFor a realized function $f$, let $\\rho^f_{Z_nE^n}$ be the state obtained by applying $f$ to the classical register of $\\rho_{AE}^{\\otimes n}$, and put $\\tau_{M_n}=I_{Z_n}/M_n$. With $D(\\omega\\|\\sigma)=\\operatorname{Tr}\\omega(\\log_2\\omega-\\log_2\\sigma)$, define\n\n \\begin{equation}\n e_I(\\rho_{AE},R):=\\sup_{\\{(M_n,F_n)\\}}\\liminf_{n\\to\\infty}-\\frac1n\\log_2\\mathbb E_{F_n}D\\!\\left(\\rho^{F_n}_{Z_nE^n}\\middle\\|\\tau_{M_n}\\otimes\\rho_E^{\\otimes n}\\right),\n\\tag{2}\n\\end{equation} \nwhere the supremum ranges over sequences satisfying Eq. (1) and $n^{-1}\\log_2 M_n\\to R$, and $-\\log_2 0=+\\infty$. The hash family may depend on the fixed source state. Determine Eq. (2) for arbitrary $\\rho_{AE}$ and $00$ be density operators on a finite-dimensional Hilbert space $\\mathcal H$, with $[\\rho,\\sigma]\\neq0$. The hypotheses $\\rho^{\\otimes n}$ and $\\sigma^{\\otimes n}$ have equal prior probability. Each of $n$ parties holds exactly one tensor factor. For a class $\\mathcal C_n$ of two-outcome POVMs, define the optimal error by Eq. (1):\n\n \\begin{equation}\np_e^{\\mathcal C}(n):=\\inf_{\\{T,I-T\\}\\in\\mathcal C_n}\\frac12\\left[\\operatorname{Tr}\\rho^{\\otimes n}(I-T)+\\operatorname{Tr}\\sigma^{\\otimes n}T\\right].\n\\tag{1}\n\\end{equation} \nHere $\\mathrm{SEP}_n$ consists of tests for which both effects are fully separable: each effect has the form $\\sum_j A_{1,j}\\otimes\\cdots\\otimes A_{n,j}$ with $A_{k,j}\\geq0$. The class $\\mathrm{LOCC}_n$ consists of tests exactly implementable by a finite number of rounds of local operations and classical messages, with arbitrary two-way communication and no shared entangled resource. The number of rounds can depend on $n$. No party may jointly operate on two input copies.\n\nTo avoid assuming convergence, define lower and upper exponents as in Eq. (2):\n\n \\begin{equation}\n\\underline\\xi_{\\mathcal C}(\\rho,\\sigma):=\\liminf_{n\\to\\infty}-\\frac1n\\log p_e^{\\mathcal C}(n),\n\\qquad\n\\overline\\xi_{\\mathcal C}(\\rho,\\sigma):=\\limsup_{n\\to\\infty}-\\frac1n\\log p_e^{\\mathcal C}(n).\n\\tag{2}\n\\end{equation} \nAll logarithms are natural. For a finite-outcome single-copy POVM $M=\\{M_\\omega\\}_\\omega$, put $P^M_\\rho(\\omega):=\\operatorname{Tr}(\\rho M_\\omega)$ and define the fixed-measurement Chernoff exponent in Eq. (3):\n\n \\begin{equation}\nC_M(\\rho,\\sigma):=\\sup_M\\max_{0\\leq s\\leq1}\\left[-\\log\\sum_\\omega P^M_\\rho(\\omega)^{1-s}P^M_\\sigma(\\omega)^s\\right].\n\\tag{3}\n\\end{equation} \nThe question is whether Eq. (4) holds for every such pair, and, if it fails, what the correct exponents are:\n\n \\begin{equation}\n\\underline\\xi_{\\mathrm{LOCC}}=\\overline\\xi_{\\mathrm{LOCC}}\n=\\underline\\xi_{\\mathrm{SEP}}=\\overline\\xi_{\\mathrm{SEP}}\n=C_M(\\rho,\\sigma).\n\\tag{4}\n\\end{equation}", "url": "https://qiqc-op.com/problem/op_a370855db65d4d24/", "json": "https://qiqc-op.com/api/problems/op_a370855db65d4d24.json", "tex": "https://qiqc-op.com/problem/op_a370855db65d4d24/op_a370855db65d4d24.tex", "created": "2026-09-09", "updated": "2026-09-09", "createdAt": "2026-09-09T09:07:10.000Z", "updatedAt": "2026-09-09T09:07:10.000Z", "sha256": "55a4aed8c98583e4ce13a66945d3e7111a933245dc7660a90450cd61eb05cb3a" }, { "id": "op_a40ad449c54093d7", "ulid": "01M22MSW6TF2CK8PB6W3W7WZCC", "aliases": [ "op_a40ad449c54093d7", "01M22MSW6TF2CK8PB6W3W7WZCC", "op-a40ad449c54093d7" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 1, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-09T08:33:27.258Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "source-stated", "posed": "2020-03-31", "areaIds": [ "quantum-communication" ], "topicIds": [ "quantum-channel-structure" ], "keywords": [ "random-unitary decomposition", "mixed-unitary rank", "symmetric Werner-Holevo channel", "orthogonal symmetric unitary basis" ], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [] }, "title": "Optimal random-unitary decomposition of symmetric Werner–Holevo channels", "status": "Unsolved", "fields": [ "Quantum Communication" ], "topics": [ "Quantum channel structure" ], "tags": [ "Quantum Communication", "Quantum channel structure" ], "statement": "For every odd integer $d\\geq5$, do there exist $r=d(d+1)/2$ unitary operators $U_1,\\ldots,U_r$ on $\\mathbb C^d$ such that\n\n \\begin{equation}\n \\Phi_d(X):=\\frac{\\operatorname{Tr}(X)I_d+X^{\\mathsf T}}{d+1}\n =\\frac1r\\sum_{j=1}^{r}U_j XU_j^\\dagger\n \\qquad\\text{for every }X\\in\\mathcal L(\\mathbb C^d)?\n\\tag{1}\n\\end{equation} \nThe transpose in Eq. (1) is taken in a fixed orthonormal basis.", "url": "https://qiqc-op.com/problem/op_a40ad449c54093d7/", "json": "https://qiqc-op.com/api/problems/op_a40ad449c54093d7.json", "tex": "https://qiqc-op.com/problem/op_a40ad449c54093d7/op_a40ad449c54093d7.tex", "created": "2026-09-09", "updated": "2026-09-09", "createdAt": "2026-09-09T09:07:10.000Z", "updatedAt": "2026-09-09T09:07:10.000Z", "sha256": "61fa90cbcaf68e6ad10f81d4b0e48a6b2745e8e19ba531bdd05a8559ac49433e" }, { "id": "op_17bba2d8f41697f6", "ulid": "01M21QD4KR3RBCKD1WEK2XSD4K", "aliases": [ "op_17bba2d8f41697f6", "01M21QD4KR3RBCKD1WEK2XSD4K", "op-17bba2d8f41697f6" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 1, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-08T23:59:41.176Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "source-stated", "posed": null, "areaIds": [ "quantum-algorithm" ], "topicIds": [ "computational-complexity-and-computability" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [] }, "title": "Quantum query complexity of Welded Tree path-finding", "status": "Unsolved", "fields": [ "Quantum algorithm" ], "topics": [ "Computational complexity and computability" ], "tags": [ "Quantum algorithm", "Computational complexity and computability" ], "statement": "Does finding an explicit ENTRANCE-to-EXIT path in the standard Welded Tree oracle problem require exponentially many quantum queries?\n\nFor an integer $n\\geq 2$, construct a graph $G_n$ from two complete binary trees of depth $n$. Designate the root of the left tree as ENTRANCE $s$ and the root of the right tree as EXIT $t$. Join the $2^n$ leaves of the left tree and the $2^n$ leaves of the right tree into a uniformly random cycle that alternates between leaves of the two trees. Assign the vertices distinct random binary labels of length $2n$.\n\nThe algorithm is given the label of $s$ and coherent quantum-query access to an adjacency-list oracle. The oracle is a static function of vertex labels: for every valid vertex label, including labels guessed without prior traversal, it returns the labels of the vertex’s neighbors, and it returns distinguished invalid outputs only for labels that are not vertices and for missing neighbor slots. Validity of a queried label is independent of the algorithm’s query history, and coherent queries may be arbitrary superpositions of label strings.\n\nThe path-finding task is to output an explicit sequence of vertex labels $v_0,v_1,\\ldots,v_\\ell$ satisfying\n\n \\begin{equation}\n v_0=s,\n \\qquad\n v_\\ell=t,\n \\qquad\n \\{v_{j-1},v_j\\}\\in E(G_n)\n \\quad\n \\text{for every }1\\leq j\\leq\\ell .\n\\tag{1}\n\\end{equation} \nLet $Q_{\\mathrm{WTPath}}(n)$ denote the minimum number of oracle queries used by a quantum algorithm that outputs a valid path satisfying (1) with bounded error on the standard random Welded Tree oracle distribution. The open question is whether there exists a constant $c>0$ such that, for all sufficiently large $n$,\n\n \\begin{equation}\n Q_{\\mathrm{WTPath}}(n)\n \\geq\n 2^{cn}.\n\\tag{2}\n\\end{equation} \nEquivalently, does the exponential quantum-query lower bound in (2) hold for unrestricted quantum algorithms?", "url": "https://qiqc-op.com/problem/op_17bba2d8f41697f6/", "json": "https://qiqc-op.com/api/problems/op_17bba2d8f41697f6.json", "tex": "https://qiqc-op.com/problem/op_17bba2d8f41697f6/op_17bba2d8f41697f6.tex", "created": "2026-09-09", "updated": "2026-09-09", "createdAt": "2026-09-09T00:04:31.000Z", "updatedAt": "2026-09-09T03:31:48.000Z", "sha256": "52308086d0d6d4e4578d0a6739836fe805eb005906cfbeb7061095a49d2f1f15" }, { "id": "op_c4726b570e765f51", "ulid": "01M220TAAFBFS38AFQF94N28WR", "aliases": [ "op_c4726b570e765f51", "01M220TAAFBFS38AFQF94N28WR", "op-c4726b570e765f51" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 1, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-09T02:44:10.191Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "editor-formulated", "posed": null, "areaIds": [ "quantum-cryptography", "quantum-communication" ], "topicIds": [ "secret-key-distillation", "local-operations-and-classical-communication" ], "keywords": [ "amplitude damping", "two-way secret-key capacity", "authenticated public communication" ], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [ "01M1HME780DDSDKPH6BERTWRWB" ] }, "title": "Two-way secret-key capacity of the qubit amplitude-damping channel", "status": "Unsolved", "fields": [ "Quantum Cryptography", "Quantum Communication" ], "topics": [ "Secret-key distillation", "Local operations and classical communication" ], "tags": [ "Quantum Cryptography", "Quantum Communication", "Secret-key distillation", "Local operations and classical communication" ], "statement": "What is the exact secret-key capacity $K(\\mathcal A_p)$ of the qubit amplitude-damping channel for every damping probability $p\\in[0,1]$? Define the channel by\n\n \\begin{equation}\n \\begin{aligned}\n \\mathcal A_p(\\rho)&=A_0\\rho A_0^\\dagger+A_1\\rho A_1^\\dagger,\\\\\n A_0&=\\lvert0\\rangle\\!\\langle0\\rvert+\\sqrt{1-p}\\,\\lvert1\\rangle\\!\\langle1\\rvert,\n \\qquad A_1=\\sqrt p\\,\\lvert0\\rangle\\!\\langle1\\rvert.\n \\end{aligned}\n\\tag{1}\n\\end{equation} \nIn Eq. (1), $\\rho$ is a qubit density operator and $p$ is the excited-state decay probability. Alice may send systems to Bob through arbitrarily many independent uses of this channel. They may interleave these uses with arbitrary adaptive local quantum operations and unlimited authenticated two-way public classical communication. No preshared entanglement or secret key is available, apart from a vanishing-rate authentication seed. Eve holds the channel environments and the entire public transcript. The capacity is the supremum of asymptotic rates $\\liminf_{n\\to\\infty}\\log_2 M_n/n$ attainable by protocols producing keys in an alphabet of size $M_n$, with disagreement probability tending to zero and trace-distance secrecy from a uniform key independent of Eve tending to zero. Determine this capacity in secret bits per channel use, with matching achievability and converse bounds.", "url": "https://qiqc-op.com/problem/op_c4726b570e765f51/", "json": "https://qiqc-op.com/api/problems/op_c4726b570e765f51.json", "tex": "https://qiqc-op.com/problem/op_c4726b570e765f51/op_c4726b570e765f51.tex", "created": "2026-09-09", "updated": "2026-09-09", "createdAt": "2026-09-09T03:08:34.000Z", "updatedAt": "2026-09-09T03:08:34.000Z", "sha256": "a6c7f363e32d164f0a4f48188dc3a8cd9aee0256dde7054ce94ba89ebc0a9c80" }, { "id": "op_edcb345719181830", "ulid": "01M220TA5N18AWYMVTHV44QY39", "aliases": [ "op_edcb345719181830", "01M220TA5N18AWYMVTHV44QY39", "op-edcb345719181830" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 1, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-09T02:44:10.037Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "editor-formulated", "posed": null, "areaIds": [ "quantum-error-correction" ], "topicIds": [ "quantum-coding-theory" ], "keywords": [ "asymptotic relative distance", "Rains bound", "nonadditive quantum codes" ], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [] }, "title": "Qubit code distance bound", "status": "Unsolved", "fields": [ "Quantum Error Correction" ], "topics": [ "Quantum coding theory" ], "tags": [ "Quantum Error Correction", "Quantum coding theory" ], "statement": "Does there exist a sequence of qubit quantum codes $Q_j\\subseteq(\\mathbb C^2)^{\\otimes n_j}$, with $n_j\\to\\infty$, $\\dim Q_j\\geq2$, and minimum distances $d_j$, satisfying\n\n \\begin{equation}\n \\limsup_{j\\to\\infty}\\frac{d_j}{n_j}=\\frac{3-\\sqrt3}{4}?\n\\tag{1}\n\\end{equation} \nHere $d_j$ is the least weight of a Pauli operator $E$ for which $P_jEP_j$ is not a scalar multiple of the projector $P_j$ onto $Q_j$; weight counts nonidentity tensor factors. Equation (1) allows nonadditive and degenerate codes, and the coding rate $\\log_2(\\dim Q_j)/n_j$ may tend to zero.", "url": "https://qiqc-op.com/problem/op_edcb345719181830/", "json": "https://qiqc-op.com/api/problems/op_edcb345719181830.json", "tex": "https://qiqc-op.com/problem/op_edcb345719181830/op_edcb345719181830.tex", "created": "2026-09-09", "updated": "2026-09-09", "createdAt": "2026-09-09T03:08:34.000Z", "updatedAt": "2026-09-09T03:08:34.000Z", "sha256": "4497cc6f33e0c355057f9be96c64459261e7a3f903377709579fe7ae1798e087" }, { "id": "op_36ac6718d2c37628", "ulid": "01M20QZS0HSJKXWFBD1YS9VVTS", "aliases": [ "op_36ac6718d2c37628", "01M20QZS0HSJKXWFBD1YS9VVTS", "op-36ac6718d2c37628" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 1, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-08T14:50:37.457Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "editor-formulated", "posed": null, "areaIds": [ "quantum-algorithm" ], "topicIds": [ "computational-complexity-and-computability" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [ "01M20H9K0GJGBPXAKZBJ2YEBSH", "01M20J3M572VGZM83E2GJ0AG2D" ] }, "title": "Polynomial-time quantum algorithm for approximate Shortest Vector Problem", "status": "Unsolved", "fields": [ "Quantum algorithm" ], "topics": [ "Computational complexity and computability" ], "tags": [ "Quantum algorithm", "Computational complexity and computability" ], "statement": "Does the polynomial-factor approximate Shortest Vector Problem admit a polynomial-time quantum algorithm?\n\nLet $B\\in\\mathbb Q^{n\\times n}$ be a nonsingular lattice basis whose entries have polynomially bounded bit length, and let\n\n \\begin{equation}\n \\mathcal L(B)=\\{Bz:z\\in\\mathbb Z^n\\}\\subset\\mathbb R^n .\n\\tag{1}\n\\end{equation} \nDefine the length of a shortest nonzero lattice vector by\n\n \\begin{equation}\n \\lambda_1(\\mathcal L)=\\min_{v\\in\\mathcal L\\setminus\\{0\\}}|v|_2 .\n\\tag{2}\n\\end{equation} \nFor an approximation factor $\\gamma=\\gamma(n)\\geq 1$, the $\\gamma$-approximate Shortest Vector Problem asks for a nonzero vector $v\\in\\mathcal L(B)$ satisfying\n\n \\begin{equation}\n |v|_2\\leq\\gamma(n)\\lambda_1(\\mathcal L(B)).\n\\tag{3}\n\\end{equation} \nThe open question is whether, for polynomial approximation factors such as $\\gamma(n)=n^c$ for a fixed constant $c>0$, there exists a bounded-error quantum algorithm that outputs a vector satisfying (3) in time polynomial in the bit length of $B$.", "url": "https://qiqc-op.com/problem/op_36ac6718d2c37628/", "json": "https://qiqc-op.com/api/problems/op_36ac6718d2c37628.json", "tex": "https://qiqc-op.com/problem/op_36ac6718d2c37628/op_36ac6718d2c37628.tex", "created": "2026-09-08", "updated": "2026-09-09", "createdAt": "2026-09-08T14:54:06.000Z", "updatedAt": "2026-09-09T03:08:34.000Z", "sha256": "620d35fc3f535eb67d41d8d559c0dcaee83a31a8abef3d80b8f2dc0668bb2f50" }, { "id": "op_90a05e57e44c086e", "ulid": "01M20J3M572VGZM83E2GJ0AG2D", "aliases": [ "op_90a05e57e44c086e", "01M20J3M572VGZM83E2GJ0AG2D", "op-90a05e57e44c086e" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 1, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-08T13:07:52.103Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "editor-formulated", "posed": null, "areaIds": [ "quantum-algorithm" ], "topicIds": [ "computational-complexity-and-computability" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [ "01M20H9K0GJGBPXAKZBJ2YEBSH", "01M20QZS0HSJKXWFBD1YS9VVTS" ] }, "title": "Polynomial-time quantum algorithm for the Dihedral Hidden Subgroup Problem", "status": "Unsolved", "fields": [ "Quantum algorithm" ], "topics": [ "Computational complexity and computability" ], "tags": [ "Quantum algorithm", "Computational complexity and computability" ], "statement": "Does the Dihedral Hidden Subgroup Problem admit a quantum algorithm whose running time is polynomial in the input length?\n\nFor a positive integer $N$, let the dihedral group be\n\n \\begin{equation}\n D_N=\\langle r,s\\mid r^N=1,\\ s^2=1,\\ srs=r^{-1}\\rangle ,\n\\tag{1}\n\\end{equation} \nso that $D_N$ has order $2N$. Consider an oracle $f:D_N\\to X$ promised to hide a subgroup generated by an unknown reflection. Equivalently, for an unknown $a\\in\\mathbb Z_N$, let\n\n \\begin{equation}\n H_a=\\{1,sr^a\\},\n \\qquad\n f(x)=f(y)\\iff xH_a=yH_a .\n\\tag{2}\n\\end{equation} \nGiven quantum oracle access to $f$, the task is to recover $a$, and hence $H_a$ in (2). The open question is whether this can be done with bounded error using a number of elementary quantum operations polynomial in $\\log N$, where $D_N$ is defined by (1).", "url": "https://qiqc-op.com/problem/op_90a05e57e44c086e/", "json": "https://qiqc-op.com/api/problems/op_90a05e57e44c086e.json", "tex": "https://qiqc-op.com/problem/op_90a05e57e44c086e/op_90a05e57e44c086e.tex", "created": "2026-09-08", "updated": "2026-09-09", "createdAt": "2026-09-08T13:12:36.000Z", "updatedAt": "2026-09-09T03:08:34.000Z", "sha256": "31c96de012a7111efcf84c262a06cc525f855ed1b2f7aa05f92232ebaf09a1e4" }, { "id": "op_291a943fdec5bd1d", "ulid": "01M20868F4GEPX6FBJZF8T0GRJ", "aliases": [ "op_291a943fdec5bd1d", "01M20868F4GEPX6FBJZF8T0GRJ", "op-291a943fdec5bd1d" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 1, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-08T10:14:32.676Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "derived", "posed": null, "areaIds": [ "quantum-resource-theory", "quantum-communication" ], "topicIds": [ "entanglement-measures", "quantum-capacity", "private-capacity" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [ "01M1HME7809M71BG24CSMYKA8A", "01M1HME780NTMHKFB95TSXKKFW", "01M20868D5Z2B0W2KKC7QGRGRR" ] }, "title": "Squashed entanglement of the qubit depolarizing channel", "status": "Unsolved", "fields": [ "Quantum Resource Theory", "Quantum Communication" ], "topics": [ "Entanglement measures", "Quantum capacity", "Private capacity" ], "tags": [ "Quantum Resource Theory", "Quantum Communication", "Entanglement measures", "Quantum capacity", "Private capacity" ], "statement": "What is the squashed entanglement $E_{\\mathrm{sq}}(\\Lambda_p)$ of the qubit depolarizing channel, defined for a mixing parameter $p\\in[0,1]$ by\n\n \\begin{equation}\n \\Lambda_p(\\rho):=(1-p)\\,\\rho+p\\,\\mathrm{Tr}[\\rho]\\,\\frac{I}{2}\n =\\bigl(1-\\tfrac{3p}{4}\\bigr)\\rho\n +\\tfrac{p}{4}\\bigl(X\\rho X+Y\\rho Y+Z\\rho Z\\bigr),\n\\tag{1}\n\\end{equation} \nwhere $\\rho$ ranges over qubit states, $I$ is the qubit identity, and $X,Y,Z$ are the Pauli matrices, so that the total Pauli error probability, read off from the second expression in Eq. (1), is $\\tfrac{3p}{4}$? Let $U:A'\\to BE$ be an isometric extension of $\\Lambda_p$, let $\\varphi_{RA'}$ range over pure bipartite input states, and let $S:E\\to E'$ range over completely positive trace-preserving maps of arbitrary finite output dimension. For \\(\\omega_{RBE'}:=(\\mathrm{id}_{RB}\\otimes S)\\bigl[(\\mathrm{id}_R\\otimes\nU)\\,\\varphi_{RA'}\\,(\\mathrm{id}_R\\otimes U^\\dagger)\\bigr]\\) the squashed entanglement of the channel is\n\n \\begin{equation}\n E_{\\mathrm{sq}}(\\Lambda_p)\n :=\\max_{\\varphi_{RA'}}\\ \\frac{1}{2}\\inf_{S}I(R;B|E')_\\omega,\n\\tag{2}\n\\end{equation} \nwhere $I(R;B|E')_\\omega$ is the conditional mutual information and all logarithms are base 2 [TGW14]. The quantity in Eq. (2) is additive over tensor products of channels and upper-bounds the two-way-assisted quantum and private capacities, $Q_2(\\Lambda_p)\\le E_{\\mathrm{sq}}(\\Lambda_p)$ and $P_2(\\Lambda_p)\\le E_{\\mathrm{sq}}(\\Lambda_p)$ [TGW14]; this is its operational role, since an exact value would be a single-letter upper bound on secret and quantum transmission over depolarizing noise. Determine $E_{\\mathrm{sq}}(\\Lambda_p)$ as an explicit function of $p$ on $[0,1]$.", "url": "https://qiqc-op.com/problem/op_291a943fdec5bd1d/", "json": "https://qiqc-op.com/api/problems/op_291a943fdec5bd1d.json", "tex": "https://qiqc-op.com/problem/op_291a943fdec5bd1d/op_291a943fdec5bd1d.tex", "created": "2026-09-08", "updated": "2026-09-09", "createdAt": "2026-09-08T10:25:04.000Z", "updatedAt": "2026-09-09T03:08:34.000Z", "sha256": "c0112ac8aab398c9f264c96e752e8d9705419a366a3dd597adbd0cf882395ffb" }, { "id": "op_d2813fe3fcdf09ad", "ulid": "01M20868D5Z2B0W2KKC7QGRGRR", "aliases": [ "op_d2813fe3fcdf09ad", "01M20868D5Z2B0W2KKC7QGRGRR", "op-d2813fe3fcdf09ad" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 1, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-08T10:14:32.613Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "source-stated", "posed": null, "areaIds": [ "quantum-communication", "quantum-cryptography" ], "topicIds": [ "private-capacity", "additivity-and-regularization" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [ "01M1HME7809M71BG24CSMYKA8A", "01M1Q787QR8FTR00QF4PMHHWPE", "01M20868F4GEPX6FBJZF8T0GRJ" ] }, "title": "Private capacity of the qubit depolarizing channel", "status": "Unsolved", "fields": [ "Quantum Communication", "Quantum Cryptography" ], "topics": [ "Private capacity", "Additivity and regularization" ], "tags": [ "Quantum Communication", "Quantum Cryptography", "Private capacity", "Additivity and regularization" ], "statement": "What is the private classical capacity $P(\\Lambda_p)$ of the qubit depolarizing channel, defined for a mixing parameter $p\\in[0,1]$ by\n\n \\begin{equation}\n \\Lambda_p(\\rho):=(1-p)\\,\\rho+p\\,\\mathrm{Tr}[\\rho]\\,\\frac{I}{2}\n =\\bigl(1-\\tfrac{3p}{4}\\bigr)\\rho\n +\\tfrac{p}{4}\\bigl(X\\rho X+Y\\rho Y+Z\\rho Z\\bigr),\n\\tag{1}\n\\end{equation} \nwhere $\\rho$ ranges over qubit states, $I$ is the qubit identity, and $X,Y,Z$ are the Pauli matrices? The second expression in Eq. (1) exhibits the Pauli error probabilities \\(\\bigl(p_I,p_X,p_Y,p_Z\\bigr)=\\bigl(1-\\tfrac{3p}{4},\\tfrac{p}{4},\\tfrac{p}{4},\n\\tfrac{p}{4}\\bigr)\\), with total Pauli error probability $\\tfrac{3p}{4}$. Fix a Stinespring isometry $V:A\\to BE$ of $\\Lambda_p$ and a finite ensemble $\\{q_x,\\rho_x\\}_x$ of qubit states, and set $\\omega^{XBE}:=\\sum_x q_x\\,|x\\rangle\\!\\langle x|^X\\otimes V\\rho_xV^\\dagger$. The one-shot private information and the private classical capacity of the channel are\n\n \\begin{equation}\n P^{(1)}(\\Lambda_p):=\\max_{\\{q_x,\\rho_x\\}}\n \\bigl[I(X;B)_\\omega-I(X;E)_\\omega\\bigr],\n \\qquad\n P(\\Lambda_p):=\\sup_{n\\ge1}\\frac{1}{n}\\,\n P^{(1)}\\bigl(\\Lambda_p^{\\otimes n}\\bigr),\n\\tag{2}\n\\end{equation} \nwhere $I(\\cdot\\,;\\cdot)_\\omega$ is the quantum mutual information and all logarithms are base 2 [Dev05], [CWY04]. Determine the value of $P(\\Lambda_p)$ in Eq. (2) as an explicit function of $p$ on $[0,1]$. For context, every channel satisfies $P\\ge\\mathcal Q$, where $\\mathcal Q$ denotes the quantum capacity, the supremum of achievable coherent-information rates: for any input state $\\rho=\\sum_x q_x\\,\\psi_x$ decomposed into an ensemble of pure states, each $V\\psi_xV^\\dagger$ is pure on $BE$, so its $B$ and $E$ marginals have equal entropy, the conditional-entropy terms cancel in $I(X;B)_\\omega-I(X;E)_\\omega$, and the private information of the ensemble equals the coherent information $S(\\Lambda_p(\\rho))-S(\\Lambda_p^{c}(\\rho))$ at $\\rho$, where $S(\\cdot)$ is the von Neumann entropy and $\\Lambda_p^{c}(\\sigma):=\\mathrm{Tr}_B[V\\sigma V^\\dagger]$ is the complementary channel; the identity holds verbatim for every tensor power, so the private-information maximum dominates the coherent-information maximum for the channel and all its powers, giving $P\\ge\\mathcal Q$ [Dev05]. Whether $P(\\Lambda_p)$ and $\\mathcal Q(\\Lambda_p)$ differ anywhere on $0\\chi(\\mathcal N_1)+\\chi(\\mathcal N_2),\n\\tag{3}\n\\end{equation} \nA solution must specify the finite states or channel operators completely, give their dimensions, and certify the corresponding inequality in (3) for the quantities defined in (1) and (2). For example, a certificate may combine rigorous one-copy bounds with an explicit two-copy input or ensemble. A simple closed-form construction must include a proof; a computational construction must include the actual instance and reproducible error bounds. An asymptotic algorithm or an existence proof alone does not supply such an instance. No universal upper bound on the witness dimensions is imposed.\n\nThe two formulations share the universal additivity equivalences discussed in Progress. Any use of a reduction to produce a witness must also specify the resulting data and certificate; the equivalence alone does not guarantee practical size. Counterexamples for minimum output Rényi entropy at orders $p\\ne1$ do not answer this question.", "url": "https://qiqc-op.com/problem/op_25e23d6e92ebce9d/", "json": "https://qiqc-op.com/api/problems/op_25e23d6e92ebce9d.json", "tex": "https://qiqc-op.com/problem/op_25e23d6e92ebce9d/op_25e23d6e92ebce9d.tex", "created": "2026-09-08", "updated": "2026-09-09", "createdAt": "2026-09-08T08:42:45.000Z", "updatedAt": "2026-09-09T03:08:34.000Z", "sha256": "a0c3b1fce466f77de56866553565021483c5d9cd0b5642493ceaeb305741ddc8" }, { "id": "op_54aa8f0f61ecc5b1", "ulid": "01M1Q787QRH2ASM04Q5SDG2H88", "aliases": [ "op_54aa8f0f61ecc5b1", "01M1Q787QRH2ASM04Q5SDG2H88", "op-54aa8f0f61ecc5b1" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 1, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-04T22:04:59Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "source-stated", "posed": null, "areaIds": [ "quantum-communication", "quantum-algorithm" ], "topicIds": [ "quantum-capacity", "computational-complexity-and-computability" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [] }, "title": "Computability of ordinary quantum capacity", "status": "Unsolved", "fields": [ "Quantum Communication", "Quantum algorithm" ], "topics": [ "Quantum capacity", "Computational complexity and computability" ], "tags": [ "Quantum Communication", "Quantum algorithm", "Quantum capacity", "Computational complexity and computability" ], "statement": "Is the ordinary unassisted quantum capacity a computable function of a finite description of a finite-dimensional quantum channel? Let $\\mathcal N:\\mathcal L(A)\\to\\mathcal L(B)$ be specified either by an exact finite list of Kraus matrices with computable algebraic entries (including rational entries) or by a finite quantum circuit over a fixed algebraic gate set, allowing state preparation and partial trace. For a complementary channel $\\mathcal N^c$, define coherent information and quantum capacity by\n\n \\begin{equation}\n I_c(\\rho,\\mathcal N)\n :=S(\\mathcal N(\\rho))-S(\\mathcal N^c(\\rho)),\n \\qquad\n Q(\\mathcal N)\n :=\\sup_{n\\geq1}\\frac1n\n \\max_{\\rho_{A^{\\otimes n}}}\n I_c(\\rho_{A^{\\otimes n}},\\mathcal N^{\\otimes n}),\n\\tag{1}\n\\end{equation} \nwhere $S(\\tau):=-\\operatorname{Tr}(\\tau\\log_2\\tau)$. Writing $\\langle\\mathcal N\\rangle$ for either finite encoding above, the question is whether there exists a Turing machine $T$ satisfying\n\n \\begin{equation}\n \\forall\\,\\langle\\mathcal N\\rangle\\ \\forall k\\in\\mathbb N:\n \\quad\n T(\\langle\\mathcal N\\rangle,k)=q_{\\mathcal N,k}\\in\\mathbb Q,\n \\qquad\n |q_{\\mathcal N,k}-Q(\\mathcal N)|\\leq2^{-k}.\n\\tag{2}\n\\end{equation} \nDetermine whether the algorithm in Eq. (2) exists for the capacity in Eq. (1), without imposing a running-time bound.", "url": "https://qiqc-op.com/problem/op_54aa8f0f61ecc5b1/", "json": "https://qiqc-op.com/api/problems/op_54aa8f0f61ecc5b1.json", "tex": "https://qiqc-op.com/problem/op_54aa8f0f61ecc5b1/op_54aa8f0f61ecc5b1.tex", "created": "2026-09-03", "updated": "2026-09-09", "createdAt": "2026-09-03T02:22:57.000Z", "updatedAt": "2026-09-09T03:08:34.000Z", "sha256": "f794ba55ab5e7030bd84b57f896e2c7eebbf031bb6fd4a62c28be9c9c04707f5" }, { "id": "op_a4600b38b94042a8", "ulid": "01M1Q787QRN9XH5T5717HHCXHG", "aliases": [ "op_a4600b38b94042a8", "01M1Q787QRN9XH5T5717HHCXHG", "op-a4600b38b94042a8" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 1, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-04T22:04:59Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "source-stated", "posed": null, "areaIds": [ "quantum-metrology" ], "topicIds": [ "channel-discrimination", "quantum-hypothesis-testing", "superchannels-and-quantum-combs" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [] }, "title": "Parallel versus nested-adaptive superchannel discrimination", "status": "Unsolved", "fields": [ "Quantum metrology" ], "topics": [ "Channel discrimination", "Quantum hypothesis testing", "Superchannels and quantum combs" ], "tags": [ "Quantum metrology", "Channel discrimination", "Quantum hypothesis testing", "Superchannels and quantum combs" ], "statement": "Can nested-adaptive strategies improve the Stein exponent for discriminating two finite-dimensional quantum superchannels? A superchannel maps channels $\\mathcal N:A\\to B$ to channels $C\\to D$ and admits a realization\n\n \\begin{equation}\n \\Theta(\\mathcal N)\n =\\mathcal D\\circ(\\mathcal N\\otimes\\operatorname{id}_S)\\circ\\mathcal E,\n \\qquad\n \\mathcal E:C\\to A S,\n \\quad \\mathcal D:B S\\to D.\n\\tag{1}\n\\end{equation} \nThe memory system $S$ in Eq. (1) is internal to the superchannel. In a fully parallel $n$-use strategy, one applies $\\Theta_i^{\\otimes n}$ to a joint $n$-partite inserted channel and then tests the resulting output on a joint input state. A nested-adaptive strategy may instead recursively insert the channel produced by one tested use into the channel slot of another, with arbitrary compatible CPTP maps between uses and a final binary measurement. For $\\mathsf S\\in\\{\\mathrm{par},\\mathrm{nest}\\}$, let\n\n \\begin{equation}\n \\begin{aligned}\n \\beta_{\\varepsilon,n}^{\\mathsf S}(\\Theta_1\\|\\Theta_2)\n &:=\\inf\\{\\beta_n(P):P\\in\\mathsf S_n,\\ \\alpha_n(P)\\leq\\varepsilon\\},\\\\\n \\zeta_{\\mathsf S}(\\Theta_1\\|\\Theta_2)\n &:=\\lim_{\\varepsilon\\downarrow0}\\liminf_{n\\to\\infty}\n -\\frac1n\\log_2\n \\beta_{\\varepsilon,n}^{\\mathsf S}(\\Theta_1\\|\\Theta_2),\n \\end{aligned}\n\\tag{2}\n\\end{equation} \nwhere $\\alpha_n$ and $\\beta_n$ are the type-I and type-II errors. Is\n\n \\begin{equation}\n \\zeta_{\\mathrm{nest}}(\\Theta_1\\|\\Theta_2)\n =\\zeta_{\\mathrm{par}}(\\Theta_1\\|\\Theta_2)\n\\tag{3}\n\\end{equation} \nfor every pair $\\Theta_1,\\Theta_2$?", "url": "https://qiqc-op.com/problem/op_a4600b38b94042a8/", "json": "https://qiqc-op.com/api/problems/op_a4600b38b94042a8.json", "tex": "https://qiqc-op.com/problem/op_a4600b38b94042a8/op_a4600b38b94042a8.tex", "created": "2026-09-03", "updated": "2026-09-09", "createdAt": "2026-09-03T02:22:57.000Z", "updatedAt": "2026-09-09T03:08:34.000Z", "sha256": "3d62b56d8b4ed0d5a77135cbceb23e81bfd09359b2aab5d6bbc67cf6d5a61d9d" }, { "id": "op_cbc0bf88b109b122", "ulid": "01M1Q787QR701HKYB3YDFJ15TK", "aliases": [ "op_cbc0bf88b109b122", "01M1Q787QR701HKYB3YDFJ15TK", "op-cbc0bf88b109b122" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 1, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-04T22:04:59Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "source-stated", "posed": null, "areaIds": [ "quantum-communication" ], "topicIds": [ "quantum-recovery", "quantum-relative-entropy", "matrix-and-entropy-inequalities" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [] }, "title": "Ordinary-Petz fidelity remainder for relative-entropy data processing", "status": "Solved", "fields": [ "Quantum Communication" ], "topics": [ "Quantum recovery", "Quantum relative entropy", "Matrix and entropy inequalities" ], "tags": [ "Quantum Communication", "Quantum recovery", "Quantum relative entropy", "Matrix and entropy inequalities" ], "statement": "Does the ordinary Petz recovery map give a universal fidelity remainder for monotonicity of quantum relative entropy? Let $\\mathcal N:\\mathcal L(A)\\to\\mathcal L(B)$ be a finite-dimensional quantum channel, and let $\\rho,\\sigma\\in\\mathcal D(A)$ satisfy $\\operatorname{supp}\\rho\\subseteq\\operatorname{supp}\\sigma$. With inverses taken on the relevant supports, define the Petz map by\n\n \\begin{equation}\n \\mathcal P_{\\sigma,\\mathcal N}(X)\n :=\\sigma^{1/2}\\mathcal N^{\\dagger}\\!\\left(\n \\mathcal N(\\sigma)^{-1/2}X\n \\mathcal N(\\sigma)^{-1/2}\n \\right)\\sigma^{1/2},\n\\tag{1}\n\\end{equation} \nwhere $\\mathcal N^{\\dagger}$ is the Hilbert–Schmidt adjoint. Write \\(D(\\tau\\Vert\\omega):=\\operatorname{Tr}[\\tau(\\log_2\\tau-\n\\log_2\\omega)]\\) and use squared fidelity $F(\\tau,\\omega):=\\lVert\\sqrt\\tau\\sqrt\\omega\\rVert_1^2$. The proposed remainder bound for the map in Eq. (1) is\n\n \\begin{equation}\n D(\\rho\\Vert\\sigma)\n -D\\!\\left(\\mathcal N(\\rho)\\middle\\Vert\\mathcal N(\\sigma)\\right)\n \\stackrel{?}{\\geq}\n -\\log_2 F\\!\\left(\n \\rho,\n \\mathcal P_{\\sigma,\\mathcal N}(\\mathcal N(\\rho))\n \\right).\n\\tag{2}\n\\end{equation} \nDetermine whether Eq. (2) holds for every such triple $(\\rho,\\sigma,\\mathcal N)$.", "url": "https://qiqc-op.com/problem/op_cbc0bf88b109b122/", "json": "https://qiqc-op.com/api/problems/op_cbc0bf88b109b122.json", "tex": "https://qiqc-op.com/problem/op_cbc0bf88b109b122/op_cbc0bf88b109b122.tex", "created": "2026-09-03", "updated": "2026-09-09", "createdAt": "2026-09-03T02:22:57.000Z", "updatedAt": "2026-09-09T03:08:34.000Z", "sha256": "1c644888a01a50fb8a2d98bd19aac445025d948c8a4c2108c909e23fe64c2e95" }, { "id": "op_a3a8680c50800797", "ulid": "01M1Q787QRBXA9T9KBKMSKZBMY", "aliases": [ "op_a3a8680c50800797", "01M1Q787QRBXA9T9KBKMSKZBMY", "op-a3a8680c50800797" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 1, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-04T22:04:59Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "source-stated", "posed": null, "areaIds": [ "quantum-resource-theory" ], "topicIds": [ "entanglement-measures", "bell-diagonal-states" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [] }, "title": "Entanglement of formation of generalized Bell-diagonal states", "status": "Unsolved", "fields": [ "Quantum Resource Theory" ], "topics": [ "Entanglement measures", "Bell-diagonal states" ], "tags": [ "Quantum Resource Theory", "Entanglement measures", "Bell-diagonal states" ], "statement": "For every local dimension $d\\geq3$, determine the entanglement of formation of an arbitrary Weyl–Bell-diagonal state. Let $\\omega_d:=\\exp(2\\pi i/d)$ and define the generalized Pauli operators and their associated Bell basis by\n\n \\begin{equation}\n X\\lvert j\\rangle:=\\lvert j+1\\!\\!\\pmod d\\rangle,\n \\qquad\n Z\\lvert j\\rangle:=\\omega_d^j\\lvert j\\rangle,\n \\qquad\n \\lvert\\Phi_{a,b}\\rangle\n :=(I\\otimes X^aZ^b)\\lvert\\Phi_d\\rangle,\n \\qquad\n \\lvert\\Phi_d\\rangle:=\\frac1{\\sqrt d}\\sum_{j=0}^{d-1}\\lvert j,j\\rangle,\n\\tag{1}\n\\end{equation} \nwhere $a,b\\in\\mathbb Z_d$. In this problem, “Pauli-diagonal” means diagonal in the generalized Bell basis in Eq. (1). Thus the state is\n\n \\begin{equation}\n \\rho_{\\mathbf p}\n :=\\sum_{a,b\\in\\mathbb Z_d}p_{a,b}\n \\lvert\\Phi_{a,b}\\rangle\\!\\langle\\Phi_{a,b}\\rvert,\n \\qquad\n p_{a,b}\\geq0,\n \\qquad\n \\sum_{a,b\\in\\mathbb Z_d}p_{a,b}=1.\n\\tag{2}\n\\end{equation} \nFor every probability array $\\mathbf p$ in Eq. (2), determine an evaluable exact formula and an optimal pure-state ensemble for\n\n \\begin{equation}\n E_F(\\rho_{\\mathbf p})\n :=\\inf_{\\rho_{\\mathbf p}=\\sum_i q_i\n \\lvert\\psi_i\\rangle\\!\\langle\\psi_i\\rvert}\n \\sum_i q_i\\,\n S\\!\\left(\\operatorname{Tr}_B\n \\lvert\\psi_i\\rangle\\!\\langle\\psi_i\\rvert\\right),\n \\qquad\n S(\\sigma):=-\\operatorname{Tr}(\\sigma\\log_2\\sigma).\n\\tag{3}\n\\end{equation} \nThe infimum in Eq. (3) is over finite pure-state ensembles with $q_i\\geq0$ and $\\sum_iq_i=1$.", "url": "https://qiqc-op.com/problem/op_a3a8680c50800797/", "json": "https://qiqc-op.com/api/problems/op_a3a8680c50800797.json", "tex": "https://qiqc-op.com/problem/op_a3a8680c50800797/op_a3a8680c50800797.tex", "created": "2026-09-02", "updated": "2026-09-09", "createdAt": "2026-09-02T21:32:33.000Z", "updatedAt": "2026-09-09T03:08:34.000Z", "sha256": "2239f5ab3214f92801d9da8d0c8f83b4be1d96f1db075b4fe8ad3c7f489a9660" }, { "id": "op_b315f0d0b6ddbdee", "ulid": "01M1HME7803QE7KXJDNM1ACKBP", "aliases": [ "op_b315f0d0b6ddbdee", "01M1HME7803QE7KXJDNM1ACKBP", "op-b315f0d0b6ddbdee", "v2-minimal-dimensions-for-strict-transpose-degradability", "open-problem-v2-problem-54" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 2, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-04T22:04:59Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "source-stated", "posed": null, "areaIds": [ "quantum-communication" ], "topicIds": [ "channel-degradability", "quantum-channel-structure" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [] }, "title": "Minimal dimensions for strict transpose degradability", "status": "Unsolved", "fields": [ "Quantum Communication" ], "topics": [ "Channel degradability", "Quantum channel structure" ], "tags": [ "Quantum Communication", "Channel degradability", "Quantum channel structure" ], "statement": "What are the componentwise-minimal dimension triples $(d_A,d_B,d_E)$ that admit a transpose-degradable but nondegradable channel? Let $d_X:=\\dim X$ and let $V:A\\to B\\otimes E$ be a support-minimal isometry, meaning that the channels\n\n \\begin{equation}\n \\Phi_V(X):=\\operatorname{Tr}_E(VXV^\\dagger),\n \\qquad\n \\Phi_V^c(X):=\\operatorname{Tr}_B(VXV^\\dagger)\n\\tag{1}\n\\end{equation} \nsatisfy $\\operatorname{supp}(\\Phi_V(I_A))=B$ and $\\operatorname{supp}(\\Phi_V^c(I_A))=E$. Equation (1) is strictly transpose degradable when, for a fixed-basis transpose $\\mathsf T_E$, its factorization properties are\n\n \\begin{equation}\n \\begin{aligned}\n &\\exists\\ \\mathcal D:\\mathcal L(B)\\to\\mathcal L(E)\\ \\text{CPTP},\n &&\\mathsf T_E\\circ\\Phi_V^c=\\mathcal D\\circ\\Phi_V,\\\\\n &\\nexists\\ \\widetilde{\\mathcal D}:\\mathcal L(B)\\to\\mathcal L(E)\\\n \\text{CPTP},\n &&\\Phi_V^c=\\widetilde{\\mathcal D}\\circ\\Phi_V.\n \\end{aligned}\n\\tag{2}\n\\end{equation} \nA feasible triple is componentwise minimal if no distinct feasible $(d'_A,d'_B,d'_E)$ satisfies $d'_X\\leq d_X$ for every $X\\in\\{A,B,E\\}$. Determine all minimal triples satisfying Eq. (2).", "url": "https://qiqc-op.com/problem/op_b315f0d0b6ddbdee/", "json": "https://qiqc-op.com/api/problems/op_b315f0d0b6ddbdee.json", "tex": "https://qiqc-op.com/problem/op_b315f0d0b6ddbdee/op_b315f0d0b6ddbdee.tex", "created": "2026-09-02", "updated": "2026-09-09", "createdAt": "2026-09-02T00:20:51.000Z", "updatedAt": "2026-09-09T03:08:34.000Z", "sha256": "39c30e50160652c7cac1c6096cfce995201b39b6fb7b4bee076c3b140d62eae2" }, { "id": "op_2beed65d248be57a", "ulid": "01M1HME7803ZFMDQWV9SVP8FH8", "aliases": [ "op_2beed65d248be57a", "01M1HME7803ZFMDQWV9SVP8FH8", "op-2beed65d248be57a", "v2-positivity-threshold-for-thermal-attenuator-quantum-capacity", "open-problem-v2-problem-50" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 2, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-04T22:04:59Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "source-stated", "posed": null, "areaIds": [ "quantum-communication" ], "topicIds": [ "quantum-capacity", "bosonic-channels" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [] }, "title": "Positivity threshold for thermal-attenuator quantum capacity", "status": "Unsolved", "fields": [ "Quantum Communication" ], "topics": [ "Quantum capacity", "Bosonic channels" ], "tags": [ "Quantum Communication", "Quantum capacity", "Bosonic channels" ], "statement": "For $0<\\eta<1$ and $0<\\nu<\\infty$, let the single-mode bosonic thermal attenuator be\n\n \\begin{equation}\n \\Phi_{\\eta,\\nu}(\\rho_A)\n :=\\operatorname{Tr}_{E'}\\!\\left[\n U_\\eta(\\rho_A\\otimes\\tau_{\\nu,E})U_\\eta^\\dagger\n \\right],\n \\qquad\n \\tau_{\\nu,E}:=\\sum_{k=0}^{\\infty}\n \\frac{\\nu^k}{(\\nu+1)^{k+1}}\n |k\\rangle_E\\!\\langle k|_E,\n\\tag{1}\n\\end{equation} \nwhere $U_\\eta:AE\\to BE'$ is a beam-splitter unitary and $\\tau_{\\nu,E}$ acts on the environment mode $E$. For an environment mode of angular frequency $\\omega_E$ at temperature $T$, $\\nu=(e^{\\hbar\\omega_E/(k_{\\rm B}T)}-1)^{-1}$. Thus $00$; $\\nu=0$ corresponds to $T=0$, while $\\nu\\to\\infty$ as $T\\to\\infty$. Define its unassisted quantum capacity and its critical transmissivity by\n\n \\begin{equation}\n \\mathcal Q(\\Phi_{\\eta,\\nu})\n :=\\lim_{n\\to\\infty}\\frac1n\\sup_{\\rho_{A^n}}\n I_{\\rm c}(\\rho_{A^n},\\Phi_{\\eta,\\nu}^{\\otimes n}),\n \\qquad\n \\eta_{\\rm c}(\\nu)\n :=\\inf\\{\\eta\\in(0,1):\\mathcal Q(\\Phi_{\\eta,\\nu})>0\\},\n\\tag{2}\n\\end{equation} \nwhere \\(I_{\\rm c}(\\rho,\\mathcal N)\n:=S(\\mathcal N(\\rho))-S(\\mathcal N^{\\rm c}(\\rho))\\). Determine the exact threshold curve in Eq. (2) for the channel in Eq. (1).", "url": "https://qiqc-op.com/problem/op_2beed65d248be57a/", "json": "https://qiqc-op.com/api/problems/op_2beed65d248be57a.json", "tex": "https://qiqc-op.com/problem/op_2beed65d248be57a/op_2beed65d248be57a.tex", "created": "2026-09-01", "updated": "2026-09-09", "createdAt": "2026-09-01T05:13:32.000Z", "updatedAt": "2026-09-09T03:08:34.000Z", "sha256": "621797b4e01ec51537d75aaee0ffbf3c5ca74253f4ca16581553ac7a49188af6" }, { "id": "op_439ae5e7e9b3b043", "ulid": "01M1HME780TFDMAP1RAR9PDCSJ", "aliases": [ "op_439ae5e7e9b3b043", "01M1HME780TFDMAP1RAR9PDCSJ", "op-439ae5e7e9b3b043", "v2-unclassified-existence-parameters-for-homogeneous-ame-states", "open-problem-v2-problem-45" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 2, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-04T22:04:59Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "source-stated", "posed": null, "areaIds": [ "quantum-resource-theory", "quantum-error-correction" ], "topicIds": [ "absolutely-maximally-entangled-states", "quantum-coding-theory" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [] }, "title": "Unclassified existence parameters for homogeneous AME states", "status": "Unsolved", "fields": [ "Quantum Resource Theory", "Quantum Error Correction" ], "topics": [ "Absolutely maximally entangled states", "Quantum coding theory" ], "tags": [ "Quantum Resource Theory", "Quantum Error Correction", "Absolutely maximally entangled states", "Quantum coding theory" ], "statement": "For which of the parameter pairs specified below does an absolutely maximally entangled state exist? A normalized vector $|\\psi\\rangle\\in(\\mathbb C^d)^{\\otimes n}$ is an $\\operatorname{AME}(n,d)$ state when\n\n \\begin{equation}\n \\operatorname{Tr}_{S^c}\\!\\left(|\\psi\\rangle\\!\\langle\\psi|\\right)\n =\\frac{I_{d^{|S|}}}{d^{|S|}}\n \\quad\\text{for every }S\\subseteq\\{1,\\ldots,n\\}\n \\text{ with }|S|\\leq\\left\\lfloor\\frac n2\\right\\rfloor.\n\\tag{1}\n\\end{equation} \nDetermine whether a state satisfying Eq. (1) exists for every pair in\n\n \\begin{equation}\n \\mathcal U:=\\bigl\\{(8,6),(8,10),(9,6),(9,10),(10,6),(10,10),\n (11,3),(11,6),(11,10),(12,6),(12,10)\\bigr\\}.\n\\tag{2}\n\\end{equation} \nThus the task is to classify every pair in Eq. (2) by existence or nonexistence, without restricting to stabilizer or minimal-support states.", "url": "https://qiqc-op.com/problem/op_439ae5e7e9b3b043/", "json": "https://qiqc-op.com/api/problems/op_439ae5e7e9b3b043.json", "tex": "https://qiqc-op.com/problem/op_439ae5e7e9b3b043/op_439ae5e7e9b3b043.tex", "created": "2026-09-01", "updated": "2026-09-09", "createdAt": "2026-09-01T05:13:32.000Z", "updatedAt": "2026-09-09T03:08:34.000Z", "sha256": "1aac6c94a243ced4708f7eb0bca91edd017dbde315025da6aa1e9c76ec798c8e" }, { "id": "op_6ba929179cc40c0a", "ulid": "01M1HME780146X04XW01Y1DZHB", "aliases": [ "op_6ba929179cc40c0a", "01M1HME780146X04XW01Y1DZHB", "op-6ba929179cc40c0a", "v2-weyl-heisenberg-covariant-sics-in-every-dimension", "open-problem-v2-problem-18" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 2, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-04T22:04:59Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "source-stated", "posed": null, "areaIds": [ "quantum-metrology" ], "topicIds": [ "symmetric-informationally-complete-measurements" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [] }, "title": "Weyl–Heisenberg-covariant SICs in every dimension", "status": "Unsolved", "fields": [ "Quantum metrology" ], "topics": [ "Symmetric informationally complete measurements" ], "tags": [ "Quantum metrology", "Symmetric informationally complete measurements" ], "statement": "Does every finite dimension admit a symmetric informationally complete measurement that is a single Weyl–Heisenberg orbit? For an integer $d\\geq2$, let $\\omega_d=e^{2\\pi i/d}$ and define shift, phase, and displacement operators on the computational basis by\n\n \\begin{equation}\n X_d\\lvert j\\rangle=\\lvert j+1\\!\\!\\pmod d\\rangle,\n \\qquad\n Z_d\\lvert j\\rangle=\\omega_d^j\\lvert j\\rangle,\n \\qquad\n D_{p,q}=X_d^pZ_d^q,\n \\quad (p,q)\\in\\mathbb Z_d^2.\n\\tag{1}\n\\end{equation} \nEquation (1) fixes a phase convention that does not affect the orbit of rank-one projectors. The question is whether, for every $d\\geq2$, there is a unit vector $\\lvert\\phi\\rangle\\in\\mathbb C^d$ satisfying\n\n \\begin{equation}\n \\bigl|\\langle\\phi\\rvert D_{p,q}\\lvert\\phi\\rangle\\bigr|^2\n =\\frac{1}{d+1}\n \\qquad\n \\text{for every }(p,q)\\in\\mathbb Z_d^2\\setminus\\{(0,0)\\}.\n\\tag{2}\n\\end{equation} \nIf Eq. (2) holds, the $d^2$ projectors in the Weyl–Heisenberg orbit of $\\lvert\\phi\\rangle$ form a SIC.", "url": "https://qiqc-op.com/problem/op_6ba929179cc40c0a/", "json": "https://qiqc-op.com/api/problems/op_6ba929179cc40c0a.json", "tex": "https://qiqc-op.com/problem/op_6ba929179cc40c0a/op_6ba929179cc40c0a.tex", "created": "2026-09-01", "updated": "2026-09-09", "createdAt": "2026-09-01T05:13:32.000Z", "updatedAt": "2026-09-09T03:08:34.000Z", "sha256": "d1b8171da77e6daf34134761dbaab2d1c8ec731348c8f8b21b5d3e8f7111eaf5" }, { "id": "op_7a9051ff6d0a1739", "ulid": "01M1HME780J76RC69YY1FTM06V", "aliases": [ "op_7a9051ff6d0a1739", "01M1HME780J76RC69YY1FTM06V", "op-7a9051ff6d0a1739", "v2-entanglement-cost-of-an-amplitude-damping-channel-choi-state", "open-problem-v2-problem-8" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 2, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-04T22:04:59Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "source-stated", "posed": null, "areaIds": [ "quantum-resource-theory" ], "topicIds": [ "entanglement-cost", "local-operations-and-classical-communication" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [] }, "title": "Entanglement cost of an amplitude-damping-channel Choi state", "status": "Unsolved", "fields": [ "Quantum Resource Theory" ], "topics": [ "Entanglement cost", "Local operations and classical communication" ], "tags": [ "Quantum Resource Theory", "Entanglement cost", "Local operations and classical communication" ], "statement": "What is the entanglement cost of the Choi state of the qubit amplitude-damping channel\n\n \\begin{equation}\n \\mathcal A_p(\\rho)=A_0\\rho A_0^\\dagger+A_1\\rho A_1^\\dagger,\n \\qquad 0\\le p\\le1?\n\\tag{1}\n\\end{equation} \nThe Kraus operators in Eq. (1) are\n\n \\begin{equation}\n \\begin{aligned}\n A_0&=\\lvert0\\rangle\\!\\langle0\\rvert\n +\\sqrt{1-p}\\,\\lvert1\\rangle\\!\\langle1\\rvert\n =\\begin{pmatrix}1&0\\\\0&\\sqrt{1-p}\\end{pmatrix},\\\\\n A_1&=\\sqrt p\\,\\lvert0\\rangle\\!\\langle1\\rvert\n =\\begin{pmatrix}0&\\sqrt p\\\\0&0\\end{pmatrix}.\n \\end{aligned}\n\\tag{2}\n\\end{equation} \nIn Eq. (2), $p$ is the decay probability of the excited state. Let $\\lvert\\Phi^+\\rangle_{RA}=(\\lvert00\\rangle+\\lvert11\\rangle)/\\sqrt2$. The normalized Choi state of the channel in Eq. (1) is\n\n \\begin{equation}\n \\begin{aligned}\n \\omega_p^{RB}\n &:=(\\operatorname{id}_R\\otimes\\mathcal A_p)\n (\\lvert\\Phi^+\\rangle\\!\\langle\\Phi^+\\rvert_{RA})\\\\\n &=\\frac12\\Bigl[\n \\lvert00\\rangle\\!\\langle00\\rvert\n +\\sqrt{1-p}\\bigl(\\lvert00\\rangle\\!\\langle11\\rvert\n +\\lvert11\\rangle\\!\\langle00\\rvert\\bigr)\n +(1-p)\\lvert11\\rangle\\!\\langle11\\rvert\n +p\\lvert10\\rangle\\!\\langle10\\rvert\n \\Bigr].\n \\end{aligned}\n\\tag{3}\n\\end{equation} \nHere the first and second entries in each ket in Eq. (3) label $R$ and $B$, respectively. Thus the question is to determine $E_C(\\omega_p)$, the asymptotic number of ebits per copy required to prepare many copies of $\\omega_p$ by local operations and classical communication.", "url": "https://qiqc-op.com/problem/op_7a9051ff6d0a1739/", "json": "https://qiqc-op.com/api/problems/op_7a9051ff6d0a1739.json", "tex": "https://qiqc-op.com/problem/op_7a9051ff6d0a1739/op_7a9051ff6d0a1739.tex", "created": "2026-09-01", "updated": "2026-09-09", "createdAt": "2026-09-01T05:13:32.000Z", "updatedAt": "2026-09-09T03:08:34.000Z", "sha256": "3e16bed50a0839dbcafdff69f15979500c5a80a469a5d6dbeb6e14b7654a019a" }, { "id": "op_a34f0e2d6489068f", "ulid": "01M1HME7804M1QPND87GJGK3MH", "aliases": [ "op_a34f0e2d6489068f", "01M1HME7804M1QPND87GJGK3MH", "op-a34f0e2d6489068f", "v2-statistical-strength-of-cglmp-measurements", "open-problem-v2-problem-25" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 2, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-04T22:04:59Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "source-stated", "posed": null, "areaIds": [ "quantum-resource-theory", "quantum-metrology" ], "topicIds": [ "bell-nonlocality", "quantum-hypothesis-testing" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [] }, "title": "Statistical strength of CGLMP measurements", "status": "Unsolved", "fields": [ "Quantum Resource Theory", "Quantum metrology" ], "topics": [ "Bell nonlocality", "Quantum hypothesis testing" ], "tags": [ "Quantum Resource Theory", "Quantum metrology", "Bell nonlocality", "Quantum hypothesis testing" ], "statement": "For every $d\\geq3$, do the standard CGLMP Fourier–phase measurements maximize the relative-entropy statistical strength against local realism among all projective $d$-outcome measurements on the fixed state $\\lvert\\Phi_d\\rangle=d^{-1/2}\\sum_{j=0}^{d-1}\\lvert j,j\\rangle$, when the setting distribution is also optimized? For a behavior $p(a,b\\mid x,y)$, a distribution $\\mu(x,y)$ on the four setting pairs, and the local polytope $\\mathcal L$, define\n\n \\begin{equation}\n S(p;\\mu)\n :=\\inf_{\\ell\\in\\mathcal L}\n \\sum_{a,b,x,y}\\mu(x,y)p(a,b\\mid x,y)\n \\log_2\\!\\frac{p(a,b\\mid x,y)}{\\ell(a,b\\mid x,y)},\n\\tag{1}\n\\end{equation} \nSet $S^\\star(p):=\\sup_{\\mu\\in\\Delta(\\{0,1\\}^2)}S(p;\\mu)$; by Eq. (1), this is the optimized asymptotic evidence rate against the best local model. The candidate bases are\n\n \\begin{equation}\n \\begin{aligned}\n \\lvert a;x\\rangle\n &=\\frac{1}{\\sqrt d}\\sum_{j=0}^{d-1}\n \\exp\\!\\left(\\frac{2\\pi i}{d}j(a+\\alpha_x)\\right)\\lvert j\\rangle,\\\\\n \\lvert b;y\\rangle\n &=\\frac{1}{\\sqrt d}\\sum_{j=0}^{d-1}\n \\exp\\!\\left(\\frac{2\\pi i}{d}j(-b+\\beta_y)\\right)\\lvert j\\rangle,\n \\end{aligned}\n\\tag{2}\n\\end{equation} \nwhere $\\alpha_0=0$, $\\alpha_1=-1/2$, $\\beta_0=1/4$, and $\\beta_1=3/4$. Let $p_{\\mathrm{CGLMP}}$ denote the behavior produced on $\\lvert\\Phi_d\\rangle$ by the bases in Eq. (2), and write $p_M$ for the behavior produced by any other measurement choice $M$ on $\\lvert\\Phi_d\\rangle$. The conjectured optimality of Eq. (2) is\n\n \\begin{equation}\n S^\\star(p_{\\mathrm{CGLMP}})\n =\\sup_M S^\\star(p_M),\n\\tag{3}\n\\end{equation} \nwhere the supremum is over two projective $d$-outcome measurements per party. Equation (3) is the question to be resolved.", "url": "https://qiqc-op.com/problem/op_a34f0e2d6489068f/", "json": "https://qiqc-op.com/api/problems/op_a34f0e2d6489068f.json", "tex": "https://qiqc-op.com/problem/op_a34f0e2d6489068f/op_a34f0e2d6489068f.tex", "created": "2026-09-01", "updated": "2026-09-09", "createdAt": "2026-09-01T05:13:32.000Z", "updatedAt": "2026-09-09T03:08:34.000Z", "sha256": "0c9da5fb7a0acf6ce00762685fe9df17af491aaf03bc041b924ff93f6270fbd4" }, { "id": "op_aa21ac5ebca8b888", "ulid": "01M1HME780JH0D9Y0RQ750ZZPY", "aliases": [ "op_aa21ac5ebca8b888", "01M1HME780JH0D9Y0RQ750ZZPY", "op-aa21ac5ebca8b888", "v2-entanglement-cost-of-a-qubit-bell-diagonal-state", "open-problem-v2-problem-7" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 2, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-04T22:04:59Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "source-stated", "posed": null, "areaIds": [ "quantum-resource-theory" ], "topicIds": [ "entanglement-cost", "bell-diagonal-states", "local-operations-and-classical-communication" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [] }, "title": "Entanglement cost of a qubit Bell-diagonal state", "status": "Unsolved", "fields": [ "Quantum Resource Theory" ], "topics": [ "Entanglement cost", "Bell-diagonal states", "Local operations and classical communication" ], "tags": [ "Quantum Resource Theory", "Entanglement cost", "Bell-diagonal states", "Local operations and classical communication" ], "statement": "What is the entanglement cost of a qubit Bell-diagonal state for an arbitrary probability vector $\\mathbf p=(p_I,p_X,p_Y,p_Z)$? Let $\\lvert\\Phi^+\\rangle=(\\lvert00\\rangle+\\lvert11\\rangle)/\\sqrt2$ and $\\lvert\\Phi_P\\rangle=(I\\otimes P)\\lvert\\Phi^+\\rangle$ for $P\\in\\{I,X,Y,Z\\}$. The state is\n\n \\begin{equation}\n \\rho_{\\mathbf p}\n :=\\sum_{P\\in\\{I,X,Y,Z\\}}p_P\n \\lvert\\Phi_P\\rangle\\!\\langle\\Phi_P\\rvert,\n \\qquad p_P\\geq0,\n \\qquad \\sum_{P\\in\\{I,X,Y,Z\\}}p_P=1.\n\\tag{1}\n\\end{equation} \nFor the state in Eq. (1), determine $E_C(\\rho_{\\mathbf p})$ for every $\\mathbf p$ in the probability simplex, where $E_C$ is the infimum asymptotic rate of ebits consumed by LOCC protocols that prepare $\\rho_{\\mathbf p}^{\\otimes n}$ with trace-norm error tending to zero.", "url": "https://qiqc-op.com/problem/op_aa21ac5ebca8b888/", "json": "https://qiqc-op.com/api/problems/op_aa21ac5ebca8b888.json", "tex": "https://qiqc-op.com/problem/op_aa21ac5ebca8b888/op_aa21ac5ebca8b888.tex", "created": "2026-09-01", "updated": "2026-09-09", "createdAt": "2026-09-01T05:13:32.000Z", "updatedAt": "2026-09-09T03:08:34.000Z", "sha256": "a6c3fd5908e6a1f5c597d497787300176d4c030a89f3d91eaf65cd513b03b092" }, { "id": "op_b65cf15705065d81", "ulid": "01M1HME78030A51WENEAWKSA90", "aliases": [ "op_b65cf15705065d81", "01M1HME78030A51WENEAWKSA90", "op-b65cf15705065d81", "v2-energy-constrained-quantum-capacity-of-a-thermal-attenuator", "open-problem-v2-problem-51" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 2, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-04T22:04:59Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "source-stated", "posed": null, "areaIds": [ "quantum-communication" ], "topicIds": [ "quantum-capacity", "bosonic-channels" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [] }, "title": "Energy-constrained quantum capacity of a thermal attenuator", "status": "Unsolved", "fields": [ "Quantum Communication" ], "topics": [ "Quantum capacity", "Bosonic channels" ], "tags": [ "Quantum Communication", "Quantum capacity", "Bosonic channels" ], "statement": "For $0<\\eta<1$ and $0<\\nu<\\infty$, let the single-mode bosonic thermal attenuator be\n\n \\begin{equation}\n \\Phi_{\\eta,\\nu}(\\rho_A)\n :=\\operatorname{Tr}_{E'}\\!\\left[\n U_\\eta(\\rho_A\\otimes\\tau_{\\nu,E})U_\\eta^\\dagger\n \\right],\n \\qquad\n \\tau_{\\nu,E}:=\\sum_{k=0}^{\\infty}\n \\frac{\\nu^k}{(\\nu+1)^{k+1}}\n |k\\rangle_E\\!\\langle k|_E,\n\\tag{1}\n\\end{equation} \nwhere $U_\\eta:AE\\to BE'$ is a beam-splitter unitary and $\\tau_{\\nu,E}$ acts on the environment mode $E$. For an environment mode of angular frequency $\\omega_E$ at temperature $T$, $\\nu=(e^{\\hbar\\omega_E/(k_{\\rm B}T)}-1)^{-1}$. Thus $00$; $\\nu=0$ corresponds to $T=0$, while $\\nu\\to\\infty$ as $T\\to\\infty$. For a finite mean input photon number $00$ and $00$ and $0\\zeta_{\\mathrm{nest}}(\\Theta_1\\|\\Theta_2),\n\\tag{3}\n\\end{equation} \nwhere $\\mathrm{fg}$ denotes fully general strategies? If not, prove equality in Eq. (3) with $>$ replaced by $=$ for all superchannel pairs.", "url": "https://qiqc-op.com/problem/op_02bb8f8228649ac3/", "json": "https://qiqc-op.com/api/problems/op_02bb8f8228649ac3.json", "tex": "https://qiqc-op.com/problem/op_02bb8f8228649ac3/op_02bb8f8228649ac3.tex", "created": "2026-09-03", "updated": "2026-09-04", "createdAt": "2026-09-03T02:22:57.000Z", "updatedAt": "2026-09-04T23:38:14.000Z", "sha256": "c22990104178f1d81384b56196e6d0139c3eb176e106271a92dac11acf10a7f8" }, { "id": "op_2c9c3abb0983004c", "ulid": "01M1Q787QR4C97BS9QTECX5TNG", "aliases": [ "op_2c9c3abb0983004c", "01M1Q787QR4C97BS9QTECX5TNG", "op-2c9c3abb0983004c" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 1, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-04T22:04:59Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "source-stated", "posed": null, "areaIds": [ "quantum-algorithm", "quantum-communication" ], "topicIds": [ "decoding-algorithms", "classical-capacity", "quantum-state-discrimination" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [] }, "title": "Uniformly efficient HSW pretty-good decoding", "status": "Unsolved", "fields": [ "Quantum algorithm", "Quantum Communication" ], "topics": [ "Decoding algorithms", "Classical capacity", "Quantum state discrimination" ], "tags": [ "Quantum algorithm", "Quantum Communication", "Decoding algorithms", "Classical capacity", "Quantum state discrimination" ], "statement": "Can Holevo–Schumacher–Westmoreland codebooks operating at every rate below their ensemble Holevo information be chosen so that their square-root, or pretty-good, measurements have uniform quantum implementations whose cost is polynomial in the blocklength and the logarithm of the codebook size? Fix an efficiently implementable finite-dimensional memoryless channel $\\mathcal N:A\\to B$ and an efficiently preparable ensemble $\\{p(x),\\phi_x^A\\}_{x\\in\\mathcal X}$. For\n\n \\begin{equation}\n \\omega_{XB}:=\\sum_{x\\in\\mathcal X}p(x)|x\\rangle\\!\\langle x|_X\n \\otimes\\mathcal N(\\phi_x),\n \\qquad\n \\chi(\\mathcal N,p):=I(X;B)_\\omega,\n\\tag{1}\n\\end{equation} \nfix a rate $00$ sufficiently small as a function of the gap $\\chi(\\mathcal N,p)-R$. A length-$n$ codebook consists of $M_n$ words $x^n(m)$, with $n^{-1}\\log_2M_n\\to R$, and corresponding output states\n\n \\begin{equation}\n \\rho_m^{B^n}:=\\bigotimes_{i=1}^{n}\\mathcal N(\\phi_{x_i(m)}),\n \\qquad m\\in\\{1,\\ldots,M_n\\}.\n\\tag{2}\n\\end{equation} \nLet $\\Pi_{\\bar\\rho,\\delta}^{(n)}$ be the typical projector of $\\bar\\rho^B:=\\sum_x p(x)\\mathcal N(\\phi_x)$, and let $\\Pi_{m,\\delta}^{(n)}$ be the conditionally typical projector for the codeword $x^n(m)$ and the product state in Eq. (2). The detection operators used in the standard HSW decoder are\n\n \\begin{equation}\n \\Gamma_{m,n}\n :=\\Pi_{\\bar\\rho,\\delta}^{(n)}\n \\Pi_{m,\\delta}^{(n)}\n \\Pi_{\\bar\\rho,\\delta}^{(n)},\n \\qquad\n G_n:=\\sum_{m=1}^{M_n}\\Gamma_{m,n}.\n\\tag{3}\n\\end{equation} \nUsing the Moore–Penrose inverse on $\\operatorname{supp}G_n$, their square-root measurement, including its failure outcome, is\n\n \\begin{equation}\n \\Lambda_{m,n}:=G_n^{-1/2}\\Gamma_{m,n}G_n^{-1/2},\n \\qquad\n \\Lambda_{0,n}:=I-\\sum_{m=1}^{M_n}\\Lambda_{m,n}.\n\\tag{4}\n\\end{equation} \nThe codebook must be succinct rather than an explicit list of exponentially many words. Require uniform polynomial-size coherent circuits that (i) compute $x_i(m)$, (ii) prepare a purification $|\\psi_m\\rangle_{B^nR_n}$ of $\\rho_m^{B^n}$ coherently in $m$, and (iii) give a controlled block encoding of the operators in Eq. (3). Denote these circuits by $C_n$, $O_n$, and $U_{\\Gamma,n}$, so that\n\n \\begin{equation}\n \\begin{aligned}\n C_n|m,i,0\\rangle&=|m,i,x_i(m)\\rangle,\\\\\n O_n|m,0\\rangle&=|m\\rangle|\\psi_m\\rangle,\n \\qquad \\operatorname{Tr}_{R_n}|\\psi_m\\rangle\\!\\langle\\psi_m|=\\rho_m,\\\\\n (\\langle0|_Z\\otimes I)U_{\\Gamma,n}(|0\\rangle_Z\\otimes I)\n &=\\sum_{m=1}^{M_n}|m\\rangle\\!\\langle m|\\otimes\\Gamma_{m,n}.\n \\end{aligned}\n\\tag{5}\n\\end{equation} \nThe promise in Eq. (5) includes efficient inverse circuits and a classical algorithm that outputs their gates in time polynomial in $n$ and $\\log M_n$.\n\nThe target is a uniform decoder circuit producing a POVM $\\{\\widetilde\\Lambda_{j,n}\\}_{j=0}^{M_n}$ whose output distribution, averaged over transmitted messages, approximates that of Eq. (4). For every $0<\\varepsilon<1/2$, require\n\n \\begin{equation}\n \\Delta_n\n :=\\frac{1}{2M_n}\\sum_{m=1}^{M_n}\\sum_{j=0}^{M_n}\n \\left|\n \\operatorname{Tr}\\!\\left[\n (\\widetilde\\Lambda_{j,n}-\\Lambda_{j,n})\\rho_m^{B^n}\n \\right]\n \\right|\n \\leq\\varepsilon,\n\\tag{6}\n\\end{equation} \nwith gate and oracle complexity $\\operatorname{poly}(n,\\log M_n,\\log(1/\\varepsilon))$, rather than polynomial in $M_n$ or in the inverse of an exponentially small singular value. Taking, for example, an inverse-polynomial sequence $\\varepsilon=\\varepsilon_n\\to0$, the implemented decoder must retain vanishing average message error. The problem is to construct such codebooks and decoders for every instance in Eq. (1), or to prove that this uniform target is impossible under the access model in Eq. (5).", "url": "https://qiqc-op.com/problem/op_2c9c3abb0983004c/", "json": "https://qiqc-op.com/api/problems/op_2c9c3abb0983004c.json", "tex": "https://qiqc-op.com/problem/op_2c9c3abb0983004c/op_2c9c3abb0983004c.tex", "created": "2026-09-03", "updated": "2026-09-04", "createdAt": "2026-09-03T02:22:57.000Z", "updatedAt": "2026-09-04T23:38:14.000Z", "sha256": "eac466a7693c6d2c7192ebc43c275dc6c30a8719fe3ec3609324e0f874d75c28" }, { "id": "op_3cd14aef409b226b", "ulid": "01M1Q787QRE9WF0NXMX32BQDCQ", "aliases": [ "op_3cd14aef409b226b", "01M1Q787QRE9WF0NXMX32BQDCQ", "op-3cd14aef409b226b" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 1, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-04T22:04:59Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "source-stated", "posed": null, "areaIds": [ "quantum-communication" ], "topicIds": [ "bosonic-channels", "matrix-and-entropy-inequalities" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [] }, "title": "Multimode constrained output entropy of a pure-loss channel", "status": "Unsolved", "fields": [ "Quantum Communication" ], "topics": [ "Bosonic channels", "Matrix and entropy inequalities" ], "tags": [ "Quantum Communication", "Bosonic channels", "Matrix and entropy inequalities" ], "statement": "Is Guha’s multimode Strong Conjecture 2 true for every correlated input state? Fix $n\\geq1$, $K\\geq0$, and $0<\\eta<1$. Let the modes $A_1,\\ldots,A_n$ be in the vacuum and let $\\rho_{B^n}$ be an arbitrary joint $n$-mode state. Apply identical beam splitters whose output annihilation operators satisfy\n\n \\begin{equation}\n \\hat c_i=\\sqrt{\\eta}\\,\\hat a_i\n +\\sqrt{1-\\eta}\\,\\hat b_i,\n \\qquad i\\in\\{1,\\ldots,n\\},\n\\tag{1}\n\\end{equation} \nand denote the resulting joint state of $C_1,\\ldots,C_n$ by $\\rho_{C^n}$. In the convention of Eq. (1), $\\eta$ multiplies the vacuum $A$ port, so the attenuator from the information-bearing $B$ port to $C$ has transmissivity $1-\\eta$. Define the thermal entropy function by\n\n \\begin{equation}\n g(x):=(x+1)\\log_2(x+1)-x\\log_2x,\n \\qquad x\\geq0,\n\\tag{2}\n\\end{equation} \nwith $0\\log_2 0:=0$. Using the function in Eq. (2), impose the sole input-entropy constraint $S(\\rho_{B^n})=ng(K)$. Does the state produced in Eq. (1) always obey\n\n \\begin{equation}\n S(\\rho_{C^n})\\geq ng((1-\\eta)K)?\n\\tag{3}\n\\end{equation} \nEquality in Eq. (3) is attained when $\\rho_{B^n}$ is a tensor product of $n$ thermal states of mean photon number $K$; the conjecture asserts that arbitrary correlations cannot lower the output entropy further.", "url": "https://qiqc-op.com/problem/op_3cd14aef409b226b/", "json": "https://qiqc-op.com/api/problems/op_3cd14aef409b226b.json", "tex": "https://qiqc-op.com/problem/op_3cd14aef409b226b/op_3cd14aef409b226b.tex", "created": "2026-09-03", "updated": "2026-09-04", "createdAt": "2026-09-03T02:22:57.000Z", "updatedAt": "2026-09-04T23:38:14.000Z", "sha256": "e472b68bfe2150139aa8a330d98ff92480b41c7b312f285438cb0baf30ab7653" }, { "id": "op_3ea0de34a1fe6e0b", "ulid": "01M1Q787QR8FTR00QF4PMHHWPE", "aliases": [ "op_3ea0de34a1fe6e0b", "01M1Q787QR8FTR00QF4PMHHWPE", "op-3ea0de34a1fe6e0b" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 1, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-04T22:04:59Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "source-stated", "posed": null, "areaIds": [ "quantum-communication", "quantum-cryptography" ], "topicIds": [ "quantum-capacity", "private-capacity", "additivity-and-regularization" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [] }, "title": "Universal finite truncation of quantum and private capacities", "status": "Solved", "fields": [ "Quantum Communication", "Quantum Cryptography" ], "topics": [ "Quantum capacity", "Private capacity", "Additivity and regularization" ], "tags": [ "Quantum Communication", "Quantum Cryptography", "Quantum capacity", "Private capacity", "Additivity and regularization" ], "statement": "Do there exist channel-independent finite integers $m_Q$ and $m_P$ that determine, respectively, the quantum capacity and the private classical capacity of every finite-dimensional quantum channel? Let $V_{A\\to BE}$ be a Stinespring isometry for a channel and its complement, and define coherent information by\n\n \\begin{equation}\n \\mathcal N_{A\\to B}(\\rho):=\\operatorname{Tr}_{E}[V\\rho V^\\dagger],\n \\qquad\n \\mathcal N^c_{A\\to E}(\\rho):=\\operatorname{Tr}_{B}[V\\rho V^\\dagger],\n \\qquad\n I_{\\rm c}(\\rho,\\mathcal N):=\n S(\\mathcal N(\\rho))-S(\\mathcal N^c(\\rho)).\n\\tag{1}\n\\end{equation} \nFor each $m\\geq1$, use Eq. (1) to set\n\n \\begin{equation}\n Q^{(m)}(\\mathcal N)\n :=\\frac1m\\max_{\\rho_{A^m}}\n I_{\\rm c}(\\rho_{A^m},\\mathcal N^{\\otimes m}),\n \\qquad\n Q(\\mathcal N):=\\sup_{m\\geq1}Q^{(m)}(\\mathcal N).\n\\tag{2}\n\\end{equation} \nFor an ensemble $\\{p_x,\\rho_x^{A^m}\\}$, let its joint channel output be\n\n \\begin{equation}\n \\omega^{XB^mE^m}\n :=\\sum_x p_x|x\\rangle\\!\\langle x|^X\\otimes\n V^{\\otimes m}\\rho_x^{A^m}(V^\\dagger)^{\\otimes m}.\n\\tag{3}\n\\end{equation} \nIn terms of the state in Eq. (3), define\n\n \\begin{equation}\n P^{(m)}(\\mathcal N)\n :=\\frac1m\\max_{\\{p_x,\\rho_x^{A^m}\\}}\n \\bigl[I(X;B^m)_\\omega-I(X;E^m)_\\omega\\bigr],\n \\qquad\n P(\\mathcal N):=\\sup_{m\\geq1}P^{(m)}(\\mathcal N).\n\\tag{4}\n\\end{equation} \nThe question is whether there exist finite integers $m_Q$ and $m_P$, independent of the channel dimensions and of $\\mathcal N$, such that\n\n \\begin{equation}\n Q(\\mathcal N)=Q^{(m_Q)}(\\mathcal N)\n \\quad\\text{and}\\quad\n P(\\mathcal N)=P^{(m_P)}(\\mathcal N)\n \\quad\\text{for every finite-dimensional }\\mathcal N.\n\\tag{5}\n\\end{equation}", "url": "https://qiqc-op.com/problem/op_3ea0de34a1fe6e0b/", "json": "https://qiqc-op.com/api/problems/op_3ea0de34a1fe6e0b.json", "tex": "https://qiqc-op.com/problem/op_3ea0de34a1fe6e0b/op_3ea0de34a1fe6e0b.tex", "created": "2026-09-03", "updated": "2026-09-04", "createdAt": "2026-09-03T02:22:57.000Z", "updatedAt": "2026-09-04T23:38:14.000Z", "sha256": "fad7c8eafc8f8890f188968a9211f44b1a8e847ca65a68c1c42a1c5799126b89" }, { "id": "op_a59d7cc1c843edb5", "ulid": "01M1Q787QRJWENG45M2E70WZXW", "aliases": [ "op_a59d7cc1c843edb5", "01M1Q787QRJWENG45M2E70WZXW", "op-a59d7cc1c843edb5" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 1, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-04T22:04:59Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "source-stated", "posed": null, "areaIds": [ "quantum-communication" ], "topicIds": [ "quantum-capacity", "channel-degradability" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [] }, "title": "Exactly solvable nondegradable quantum channels", "status": "Solved", "fields": [ "Quantum Communication" ], "topics": [ "Quantum capacity", "Channel degradability" ], "tags": [ "Quantum Communication", "Quantum capacity", "Channel degradability" ], "statement": "Does there exist a finite-dimensional quantum channel that is neither degradable nor antidegradable and whose unassisted quantum capacity is known exactly? For a channel $\\mathcal N_{A\\to B}$ with complementary channel $\\mathcal N^c_{A\\to E}$, degradability and antidegradability mean, respectively, that there is a channel $\\mathcal D$ or $\\mathcal A$ satisfying\n\n \\begin{equation}\n \\mathcal N^c=\\mathcal D_{B\\to E}\\circ\\mathcal N,\n \\qquad\\text{or}\\qquad\n \\mathcal N=\\mathcal A_{E\\to B}\\circ\\mathcal N^c.\n\\tag{1}\n\\end{equation} \nThe channel sought must satisfy neither identity in Eq. (1). Its quantum capacity is defined by the coherent-information regularization\n\n \\begin{equation}\n Q(\\mathcal N)\n :=\\sup_{n\\geq1}\\frac1n\\max_{\\rho_{A^n}}\n \\left[\n S(\\mathcal N^{\\otimes n}(\\rho_{A^n}))\n -S((\\mathcal N^c)^{\\otimes n}(\\rho_{A^n}))\n \\right].\n\\tag{2}\n\\end{equation} \nThe question asks for an explicit channel outside both classes in Eq. (1) together with an exact evaluation of Eq. (2).", "url": "https://qiqc-op.com/problem/op_a59d7cc1c843edb5/", "json": "https://qiqc-op.com/api/problems/op_a59d7cc1c843edb5.json", "tex": "https://qiqc-op.com/problem/op_a59d7cc1c843edb5/op_a59d7cc1c843edb5.tex", "created": "2026-09-03", "updated": "2026-09-04", "createdAt": "2026-09-03T02:22:57.000Z", "updatedAt": "2026-09-04T23:38:14.000Z", "sha256": "4398cabe343f7e86d434895c97f0099a09bf5d6d51663dced39cb3821b6c93d9" }, { "id": "op_ad12295d8fdfb02a", "ulid": "01M1Q787QRDHHK1971H9D8YPN9", "aliases": [ "op_ad12295d8fdfb02a", "01M1Q787QRDHHK1971H9D8YPN9", "op-ad12295d8fdfb02a" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 1, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-04T22:04:59Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "source-stated", "posed": null, "areaIds": [ "quantum-communication", "quantum-resource-theory" ], "topicIds": [ "classical-capacity", "entanglement-assisted-communication", "additivity-and-regularization" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [] }, "title": "Structural criterion for classical–entanglement trade-off advantage", "status": "Unsolved", "fields": [ "Quantum Communication", "Quantum Resource Theory" ], "topics": [ "Classical capacity", "Entanglement-assisted communication", "Additivity and regularization" ], "tags": [ "Quantum Communication", "Quantum Resource Theory", "Classical capacity", "Entanglement-assisted communication", "Additivity and regularization" ], "statement": "Characterize the finite-dimensional quantum channels for which joint classical–entanglement coding strictly outperforms time sharing between unassisted and unlimited-entanglement classical communication. For a channel $\\mathcal N:A'\\to B$, let $C_{\\rm CE}(\\mathcal N,e)$ be the supremum of asymptotically achievable classical rates when at most $e$ ebits per channel use are consumed. Operationally,\n\n \\begin{equation}\n C_{\\rm CE}(\\mathcal N,e)\n :=\\sup\\left\\{\n R:\\begin{array}{l}\n \\text{there are length-$n$ classical-message codes with}\\\\\n n^{-1}\\log_2M_n\\to R,\\quad\n \\limsup_{n\\to\\infty}n^{-1}\\log_2K_n\\leq e,\\quad\n P_{\\rm err}^{(n)}\\to0\n \\end{array}\n \\right\\},\n\\tag{1}\n\\end{equation} \nwhere $K_n$ in Eq. (1) is the Schmidt rank of the preshared maximally entangled resource. Define the unassisted and unlimited-entanglement endpoints by\n\n \\begin{equation}\n C_0(\\mathcal N):=C_{\\rm CE}(\\mathcal N,0),\n \\qquad\n C_{\\rm EA}(\\mathcal N):=\\sup_{e\\geq0}C_{\\rm CE}(\\mathcal N,e),\n \\qquad\n e_{\\rm EA}(\\mathcal N)\n :=\\inf\\{e:C_{\\rm CE}(\\mathcal N,e)=C_{\\rm EA}(\\mathcal N)\\}.\n\\tag{2}\n\\end{equation} \nFor $e_{\\rm EA}(\\mathcal N)>0$, endpoint time sharing gives\n\n \\begin{equation}\n C_{\\rm TS}(\\mathcal N,e)\n :=\\begin{cases}\n \\left(1-\\dfrac{e}{e_{\\rm EA}(\\mathcal N)}\\right)C_0(\\mathcal N)\n +\\dfrac{e}{e_{\\rm EA}(\\mathcal N)}C_{\\rm EA}(\\mathcal N),\n &0\\leq e\\leq e_{\\rm EA}(\\mathcal N),\\\\[2mm]\n C_{\\rm EA}(\\mathcal N),&e\\geq e_{\\rm EA}(\\mathcal N).\n \\end{cases}\n\\tag{3}\n\\end{equation} \nIf the infimum in Eq. (2) is not attained, interpret Eq. (3) through its operational closure; if $e_{\\rm EA}(\\mathcal N)=0$, set $C_{\\rm TS}(\\mathcal N,e)=C_{\\rm EA}(\\mathcal N)$. Give intrinsic necessary and sufficient conditions for strict suboptimality of this benchmark:\n\n \\begin{equation}\n \\exists e>0:\\qquad\n C_{\\rm CE}(\\mathcal N,e)>C_{\\rm TS}(\\mathcal N,e).\n\\tag{4}\n\\end{equation}", "url": "https://qiqc-op.com/problem/op_ad12295d8fdfb02a/", "json": "https://qiqc-op.com/api/problems/op_ad12295d8fdfb02a.json", "tex": "https://qiqc-op.com/problem/op_ad12295d8fdfb02a/op_ad12295d8fdfb02a.tex", "created": "2026-09-03", "updated": "2026-09-04", "createdAt": "2026-09-03T02:22:57.000Z", "updatedAt": "2026-09-04T23:38:14.000Z", "sha256": "c48afafbecb46a1f5d1a5d8acee5e239476beca5bd2baea68d088962448deb78" }, { "id": "op_c9b532d3a7389c77", "ulid": "01M1Q787QRSHGZH7NDSMFG88GH", "aliases": [ "op_c9b532d3a7389c77", "01M1Q787QRSHGZH7NDSMFG88GH", "op-c9b532d3a7389c77" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 1, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-04T22:04:59Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "source-stated", "posed": null, "areaIds": [ "quantum-communication" ], "topicIds": [ "quantum-source-coding", "strong-converses" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [] }, "title": "Strong converse for general mixed-state quantum compression", "status": "Unsolved", "fields": [ "Quantum Communication" ], "topics": [ "Quantum source coding", "Strong converses" ], "tags": [ "Quantum Communication", "Quantum source coding", "Strong converses" ], "statement": "Does general finite-dimensional i.i.d. mixed-state quantum compression obey an unrestricted strong converse? Let $\\rho^{AR}$ be a finite-dimensional source state, where only $A$ is available to the encoder and $R$ is an inaccessible reference. Fix a Koashi–Imoto isometry $U:A\\to CNQ$ for which the source has the form\n\n \\begin{equation}\n \\omega^{CNQR}\n :=(U\\otimes I_R)\\rho^{AR}(U^\\dagger\\otimes I_R)\n =\\sum_j p_j|j\\rangle\\!\\langle j|^C\n \\otimes\\omega_j^N\\otimes\\rho_j^{QR},\n\\tag{1}\n\\end{equation} \nwhere $C$ is classical, $N$ is redundant relative to $R$ conditioned on $C$, and $Q$ carries the remaining source–reference correlations. At blocklength $n$, allow arbitrary encoding and decoding channels $\\mathcal E_n:A^{\\otimes n}\\to M_n$ and $\\mathcal D_n:M_n\\to\\widehat A^{\\otimes n}$, with $\\widehat A\\cong A$. Their reference-preserving squared fidelity is\n\n \\begin{equation}\n F_n:=F\\!\\left(\n (\\rho^{AR})^{\\otimes n},\n \\left[(\\mathcal D_n\\circ\\mathcal E_n)\n \\otimes\\operatorname{id}_{R^{\\otimes n}}\\right]\n ((\\rho^{AR})^{\\otimes n})\n \\right),\n \\qquad\n F(\\tau,\\zeta):=\\lVert\\sqrt\\tau\\sqrt\\zeta\\rVert_1^2.\n\\tag{2}\n\\end{equation} \nThe optimal first-order qubit rate is $S(CQ)_\\omega$. Determine whether every sequence of unrestricted channels defining Eq. (2) satisfies the strong-converse implication\n\n \\begin{equation}\n \\limsup_{n\\to\\infty}\\frac1n\\log_2|M_n|0$, define the channel by\n\n \\begin{equation}\n \\mathcal N_\\sigma(\\rho)\n :=\\frac{1}{\\pi\\sigma^2}\\int_{\\mathbb C}\n e^{-\\lvert\\alpha\\rvert^2/\\sigma^2}\n D(\\alpha)\\rho D(\\alpha)^\\dagger\\,d^2\\alpha,\n \\qquad\n D(\\alpha):=e^{\\alpha a^\\dagger-\\alpha^*a},\n\\tag{1}\n\\end{equation} \nwhere $a$ is the mode annihilation operator. Equivalently, writing $\\alpha=(x+iy)/\\sqrt2$, the channel in Eq. (1) adds independent classical Gaussian shifts $x,y\\sim\\mathcal N(0,\\sigma^2)$ to the two canonical quadratures. With $\\hat n_j:=a_j^\\dagger a_j$, define its energy-unconstrained capacity as\n\n \\begin{equation}\n \\begin{aligned}\n \\mathcal Q(\\mathcal N_\\sigma)\n &:={\\sup}_{00$.", "url": "https://qiqc-op.com/problem/op_1e7f431ed9013d1e/", "json": "https://qiqc-op.com/api/problems/op_1e7f431ed9013d1e.json", "tex": "https://qiqc-op.com/problem/op_1e7f431ed9013d1e/op_1e7f431ed9013d1e.tex", "created": "2026-09-02", "updated": "2026-09-04", "createdAt": "2026-09-02T23:47:21.000Z", "updatedAt": "2026-09-04T23:38:14.000Z", "sha256": "325229a90df0f6248aa6ea011eaac8e4cb5827930823d14b6abe06f223f3ce9e" }, { "id": "op_76e284219621a785", "ulid": "01M1Q787QR0M0RT7TK205931W8", "aliases": [ "op_76e284219621a785", "01M1Q787QR0M0RT7TK205931W8", "op-76e284219621a785" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 1, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-04T22:04:59Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "source-stated", "posed": null, "areaIds": [ "quantum-metrology" ], "topicIds": [ "quantum-state-discrimination" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [] }, "title": "Convergence of the JRF iteration for mixed-state discrimination", "status": "Unsolved", "fields": [ "Quantum metrology" ], "topics": [ "Quantum state discrimination" ], "tags": [ "Quantum metrology", "Quantum state discrimination" ], "statement": "Does the Ježek–Řeháček–Fiurášek (JRF) iteration, initialized by the uniform POVM, converge to a globally optimal minimum-error measurement for every finite ensemble containing mixed quantum states? Let $m\\geq2$, let $p_i>0$ be prior probabilities, and let $\\rho_i$ be density operators on a finite-dimensional Hilbert space. Define\n\n \\begin{equation}\n \\xi_i:=p_i\\rho_i,\n \\qquad\n \\operatorname{Tr}\\rho_i=1,\n \\qquad\n \\sum_{i=1}^{m}p_i=1,\n \\qquad\n \\mathcal H_0:=\\operatorname{supp}\\!\\left(\\sum_{i=1}^{m}\\xi_i\\right),\n\\tag{1}\n\\end{equation} \nwhere at least one $\\rho_i$ has rank greater than one. On the signal space $\\mathcal H_0$ in Eq. (1), the optimal guessing probability is\n\n \\begin{equation}\n P_{\\mathrm{opt}}\n :=\\max_{\\substack{\\Pi_i\\succeq0\\\\\n \\sum_{i=1}^{m}\\Pi_i=I_{\\mathcal H_0}}}\n \\sum_{i=1}^{m}\\operatorname{Tr}(\\xi_i\\Pi_i).\n\\tag{2}\n\\end{equation} \nStarting from $\\Pi_i^{(0)}:=I_{\\mathcal H_0}/m$, define the JRF iterates by\n\n \\begin{equation}\n \\Lambda_k\n :=\\left(\\sum_{j=1}^{m}\n \\xi_j\\Pi_j^{(k)}\\xi_j\\right)^{1/2},\n \\qquad\n \\Pi_i^{(k+1)}\n :=\\Lambda_k^{-1}\\xi_i\\Pi_i^{(k)}\\xi_i\\Lambda_k^{-1}\n \\quad (i=1,\\ldots,m).\n\\tag{3}\n\\end{equation} \nFor this initialization, $\\Lambda_k$ is inverted on $\\mathcal H_0$ and the operators in Eq. (3) form a POVM at every step. Determine whether there always exists an optimal POVM $\\{\\Pi_i^\\star\\}_{i=1}^{m}$ attaining Eq. (2) such that\n\n \\begin{equation}\n \\lim_{k\\to\\infty}\n \\sum_{i=1}^{m}\\lVert\\Pi_i^{(k)}-\\Pi_i^\\star\\rVert_2=0,\n \\qquad\n \\lim_{k\\to\\infty}\n \\sum_{i=1}^{m}\\operatorname{Tr}(\\xi_i\\Pi_i^{(k)})\n =P_{\\mathrm{opt}},\n\\tag{4}\n\\end{equation} \nwhere $\\lVert\\cdot\\rVert_2$ is the Hilbert–Schmidt norm. If Eq. (4) fails, construct an explicit mixed-state counterexample and characterize its limiting behavior.", "url": "https://qiqc-op.com/problem/op_76e284219621a785/", "json": "https://qiqc-op.com/api/problems/op_76e284219621a785.json", "tex": "https://qiqc-op.com/problem/op_76e284219621a785/op_76e284219621a785.tex", "created": "2026-09-02", "updated": "2026-09-04", "createdAt": "2026-09-02T23:12:23.000Z", "updatedAt": "2026-09-04T23:38:14.000Z", "sha256": "3ba94d2d57fe2e5745bd0c93b9413df3e77a6e6537417711c5e274ef1212735e" }, { "id": "op_c37650bfb81dbfc6", "ulid": "01M1Q787QR08CREPZSZYDXBTGN", "aliases": [ "op_c37650bfb81dbfc6", "01M1Q787QR08CREPZSZYDXBTGN", "op-c37650bfb81dbfc6" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 1, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-04T22:04:59Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "source-stated", "posed": null, "areaIds": [ "quantum-resource-theory" ], "topicIds": [ "local-unitary-equivalence" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [] }, "title": "Minimum LU–LC counterexample for graph states", "status": "Solved", "fields": [ "Quantum Resource Theory" ], "topics": [ "Local unitary equivalence" ], "tags": [ "Quantum Resource Theory", "Local unitary equivalence" ], "statement": "What is the least number of qubits for which two graph states can be locally unitary equivalent without being locally Clifford equivalent? For a simple graph $G=(V,E)$ with $V=\\{1,\\ldots,n\\}$, define its graph state by\n\n \\begin{equation}\n \\lvert G\\rangle\n :=\\left(\\prod_{\\{u,v\\}\\in E}\\mathrm{CZ}_{uv}\\right)\n \\lvert+\\rangle^{\\otimes n},\n \\qquad\n \\lvert+\\rangle:=\\frac{\\lvert0\\rangle+\\lvert1\\rangle}{\\sqrt2}.\n\\tag{1}\n\\end{equation} \nFor graphs $G$ and $H$ on $n$ vertices, use the state convention in Eq. (1) and write\n\n \\begin{equation}\n \\begin{aligned}\n G\\sim_{\\mathrm{LU}}H\n &\\iff\n \\lvert H\\rangle=e^{i\\phi}\n \\left(\\bigotimes_{j=1}^{n}U_j\\right)\\lvert G\\rangle\n &&\\text{for some }U_j\\in U(2),\\ \\phi\\in\\mathbb R,\\\\\n G\\sim_{\\mathrm{LC}}H\n &\\iff\n \\lvert H\\rangle=e^{i\\theta}\n \\left(\\bigotimes_{j=1}^{n}C_j\\right)\\lvert G\\rangle\n &&\\text{for some }C_j\\in\\mathcal C_1,\\ \\theta\\in\\mathbb R,\n \\end{aligned}\n\\tag{2}\n\\end{equation} \nwhere $\\mathcal C_1$ is the single-qubit Clifford group. Since $\\mathcal C_1\\subset U(2)$, LC equivalence implies LU equivalence. Define the minimum counterexample size by\n\n \\begin{equation}\n n_{\\min}\n :=\\min\\left\\{n:\\text{there exist $n$-vertex graphs $G,H$ with\n $G\\sim_{\\mathrm{LU}}H$ and $G\\not\\sim_{\\mathrm{LC}}H$}\\right\\}.\n\\tag{3}\n\\end{equation} \nDetermine the integer in Eq. (3).", "url": "https://qiqc-op.com/problem/op_c37650bfb81dbfc6/", "json": "https://qiqc-op.com/api/problems/op_c37650bfb81dbfc6.json", "tex": "https://qiqc-op.com/problem/op_c37650bfb81dbfc6/op_c37650bfb81dbfc6.tex", "created": "2026-09-02", "updated": "2026-09-04", "createdAt": "2026-09-02T23:12:23.000Z", "updatedAt": "2026-09-04T23:38:14.000Z", "sha256": "b192e3c0a2b91e3a7597bf526bc0908ece01a2a8e0e25967a7dbcc8fddf6d1f4" }, { "id": "op_523ed75735cfe6c3", "ulid": "01M1HME78053BRQ9RY6RY1YHSW", "aliases": [ "op_523ed75735cfe6c3", "01M1HME78053BRQ9RY6RY1YHSW", "op-523ed75735cfe6c3", "ruskai-2007-convex-decompositions-cpt-maps" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 2, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-04T22:04:59Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "source-stated", "posed": null, "areaIds": [ "quantum-communication" ], "topicIds": [ "quantum-channel-structure" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [] }, "title": "Equal-weight low-Choi-rank decompositions of quantum channels", "status": "Unsolved", "fields": [ "Quantum Communication" ], "topics": [ "Quantum channel structure" ], "tags": [ "Quantum Communication", "Quantum channel structure" ], "statement": "Can every finite-dimensional quantum channel be written as the uniform mixture of $d_B$ channels whose Choi ranks are at most the input dimension? Let $A$ and $B$ have dimensions $d_A$ and $d_B$, respectively, and let $\\Phi:\\mathcal L(A)\\to\\mathcal L(B)$ be completely positive and trace preserving. In a fixed orthonormal basis of $A$, define its Choi operator by\n\n \\begin{equation}\n J(\\Phi)\n :=\\sum_{i,j=1}^{d_A}\n \\lvert i\\rangle\\!\\langle j\\rvert_A\\otimes\n \\Phi\\!\\left(\\lvert i\\rangle\\!\\langle j\\rvert_A\\right),\n\\tag{1}\n\\end{equation} \nThe rank of the operator in Eq. (1) is independent of the chosen basis. Determine whether every $\\Phi$ admits completely positive trace-preserving maps $\\Phi_1,\\ldots,\\Phi_{d_B}:\\mathcal L(A)\\to\\mathcal L(B)$ satisfying\n\n \\begin{equation}\n \\Phi=\\frac1{d_B}\\sum_{r=1}^{d_B}\\Phi_r,\n \\qquad\n \\operatorname{rank}J(\\Phi_r)\\leq d_A\n \\quad\\text{for every }r\\in\\{1,\\ldots,d_B\\}.\n\\tag{2}\n\\end{equation} \nThus Eq. (2) requires both the prescribed input-dimension rank bound and exactly equal mixing weights.", "url": "https://qiqc-op.com/problem/op_523ed75735cfe6c3/", "json": "https://qiqc-op.com/api/problems/op_523ed75735cfe6c3.json", "tex": "https://qiqc-op.com/problem/op_523ed75735cfe6c3/op_523ed75735cfe6c3.tex", "created": "2026-09-02", "updated": "2026-09-04", "createdAt": "2026-09-02T21:32:33.000Z", "updatedAt": "2026-09-04T23:38:14.000Z", "sha256": "cb3400470092b23678eb57e92b232d75c9edb61038a1e81f7faa2ffeb97b024b" }, { "id": "op_8c6e6d0cc3d28e86", "ulid": "01M1Q787QRMK8JJ7BH5J7A8VXS", "aliases": [ "op_8c6e6d0cc3d28e86", "01M1Q787QRMK8JJ7BH5J7A8VXS", "op-8c6e6d0cc3d28e86" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 1, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-04T22:04:59Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "source-stated", "posed": null, "areaIds": [ "quantum-communication" ], "topicIds": [ "quantum-recovery", "quantum-relative-entropy", "matrix-and-entropy-inequalities" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [] }, "title": "Umegaki relative entropy of local recovery", "status": "Solved", "fields": [ "Quantum Communication" ], "topics": [ "Quantum recovery", "Quantum relative entropy", "Matrix and entropy inequalities" ], "tags": [ "Quantum Communication", "Quantum recovery", "Quantum relative entropy", "Matrix and entropy inequalities" ], "statement": "Does the conditional mutual information of every finite-dimensional tripartite state dominate its Umegaki relative entropy of local recovery? For a state $\\rho_{ABC}$, define\n\n \\begin{equation}\n I(A:C\\mid B)_\\rho\n :=S(AB)_\\rho+S(BC)_\\rho-S(B)_\\rho-S(ABC)_\\rho.\n\\tag{1}\n\\end{equation} \nWith $D(\\tau\\Vert\\omega):=\\operatorname{Tr}[\\tau(\\log\\tau-\\log\\omega)]$ when $\\operatorname{supp}\\tau\\subseteq\\operatorname{supp}\\omega$, the question is whether the quantity in Eq. (1) always satisfies\n\n \\begin{equation}\n I(A:C\\mid B)_\\rho\n \\stackrel{?}{\\ge}\n \\min_{\\mathcal R_{B\\to BC}}\n D\\!\\left(\n \\rho_{ABC}\n \\middle\\Vert\n (\\operatorname{id}_A\\otimes\\mathcal R_{B\\to BC})(\\rho_{AB})\n \\right),\n\\tag{2}\n\\end{equation} \nwhere the minimum in Eq. (2) is over all completely positive trace-preserving recovery maps $\\mathcal R_{B\\to BC}$.", "url": "https://qiqc-op.com/problem/op_8c6e6d0cc3d28e86/", "json": "https://qiqc-op.com/api/problems/op_8c6e6d0cc3d28e86.json", "tex": "https://qiqc-op.com/problem/op_8c6e6d0cc3d28e86/op_8c6e6d0cc3d28e86.tex", "created": "2026-09-02", "updated": "2026-09-04", "createdAt": "2026-09-02T21:32:33.000Z", "updatedAt": "2026-09-04T23:38:14.000Z", "sha256": "bf81b93ccfc414106425ab6ded6509fa1815d1a412e2b75142725992e7e86a68" }, { "id": "op_ad05396ff490713c", "ulid": "01M1HME780XSRZ7K9HQJSZ176R", "aliases": [ "op_ad05396ff490713c", "01M1HME780XSRZ7K9HQJSZ176R", "op-ad05396ff490713c", "ruskai-2007-werner-holevo-channel-multiplicativity" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 2, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-04T22:04:59Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "source-stated", "posed": null, "areaIds": [ "quantum-communication" ], "topicIds": [ "additivity-and-regularization", "quantum-channel-structure", "matrix-and-entropy-inequalities" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [] }, "title": "Multiplicativity for polarized Werner–Holevo channels", "status": "Unsolved", "fields": [ "Quantum Communication" ], "topics": [ "Additivity and regularization", "Quantum channel structure", "Matrix and entropy inequalities" ], "tags": [ "Quantum Communication", "Additivity and regularization", "Quantum channel structure", "Matrix and entropy inequalities" ], "statement": "For every integer $d\\geq3$, every $x\\in(0,1)$, and every $10$, define the Rényi entropy and the corresponding minimum output entropy by\n\n \\begin{equation}\n S_p(\\sigma)\n :=\\begin{cases}\n \\displaystyle\\frac{1}{1-p}\\log_2\\operatorname{Tr}(\\sigma^p),\n &p\\neq1,\\\\[2mm]\n -\\operatorname{Tr}(\\sigma\\log_2\\sigma),&p=1,\n \\end{cases}\n \\qquad\n S_{p,\\min}(\\Phi)\n :=\\min_{\\substack{\\rho\\succeq0\\\\\\operatorname{Tr}\\rho=1}}\n S_p\\!\\left(\\Phi(\\rho)\\right).\n\\tag{1}\n\\end{equation} \nWith the quantities in Eq. (1), determine whether there are $p>0$, an integer $m\\geq3$, and a channel $\\Phi$ such that\n\n \\begin{equation}\n S_{p,\\min}(\\Phi^{\\otimes n})\n =nS_{p,\\min}(\\Phi)\n \\quad\\text{for every }1\\leq n0}\n \\frac{(\\lambda_j-\\lambda_k)^2}{\\lambda_j+\\lambda_k}\n \\left\\lvert\\langle j\\rvert H_R\\lvert k\\rangle\\right\\rvert^2.\n\\tag{3}\n\\end{equation} \nEquation (3) fixes the normalization of the coherence measure used in Eq. (2). Determine Eq. (2), up to matching bounds, for every Gibbs-preserving $\\Phi$ and every $\\varepsilon$. Give the analogous tight deterministic work cost when an ideal battery begins and ends in sharp energy states and the battery’s energy loss is charged as work. Which structural features of $\\Phi$ determine whether the exact costs are finite, and what is their sharp divergence as $\\varepsilon\\downarrow0$ when they are infinite at zero error?", "url": "https://qiqc-op.com/problem/op_56578531a2c610fe/", "json": "https://qiqc-op.com/api/problems/op_56578531a2c610fe.json", "tex": "https://qiqc-op.com/problem/op_56578531a2c610fe/op_56578531a2c610fe.tex", "created": "2026-09-01", "updated": "2026-09-04", "createdAt": "2026-09-01T05:13:32.000Z", "updatedAt": "2026-09-04T23:38:14.000Z", "sha256": "390ae506e1c4e23a508a94227c27f382f1613c0575f4df1bcd9a3a7877a47b65" }, { "id": "op_56cf60cdecde44f7", "ulid": "01M1HME780CC21XQAXBTWRJRCY", "aliases": [ "op_56cf60cdecde44f7", "01M1HME780CC21XQAXBTWRJRCY", "op-56cf60cdecde44f7", "v2-collective-cost-of-tensor-power-state-preparation", "open-problem-v2-problem-16" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 2, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-04T22:04:59Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "source-stated", "posed": null, "areaIds": [ "quantum-algorithm" ], "topicIds": [ "quantum-state-preparation", "quantum-circuit-complexity", "additivity-and-regularization" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [] }, "title": "Collective cost of tensor-power state preparation", "status": "Unsolved", "fields": [ "Quantum algorithm" ], "topics": [ "Quantum state preparation", "Quantum circuit complexity", "Additivity and regularization" ], "tags": [ "Quantum algorithm", "Quantum state preparation", "Quantum circuit complexity", "Additivity and regularization" ], "statement": "Determine the asymptotic weighted circuit cost of preparing tensor powers of a known pure state, and characterize when collective preparation is cheaper per copy than independent preparation. On an arbitrary qubit register, allow Pauli-product rotations $e^{i\\phi P}$ with $\\phi\\in\\mathbb{R}$ and $P\\in\\{I,X,Y,Z\\}^{\\otimes q}$, assigning each such gate the cost $|\\phi|$. For a known $m$-qubit state $\\lvert\\psi\\rangle$, let $U(\\boldsymbol\\phi,\\boldsymbol P):=\\prod_{j=1}^r e^{i\\phi_jP_j}$ for a finite sequence of angles and Pauli products. Define the phase-independent exact $n$-copy cost by\n\n \\begin{equation}\n C_n(\\psi)\n :=\\inf\\left\\{\n \\sum_{j=1}^{r}|\\phi_j|:\n \\begin{array}{l}\n r\\in\\mathbb N_0,\\ \\phi_j\\in\\mathbb R,\\ \n P_j\\in\\{I,X,Y,Z\\}^{\\otimes mn}\\ (1\\le j\\le r),\\\\\n U(\\boldsymbol\\phi,\\boldsymbol P)\n \\lvert0^{mn}\\rangle\\!\\langle0^{mn}\\rvert\n U(\\boldsymbol\\phi,\\boldsymbol P)^\\dagger\n =(\\lvert\\psi\\rangle\\!\\langle\\psi\\rvert)^{\\otimes n}\n \\end{array}\n \\right\\}.\n\\tag{1}\n\\end{equation} \nDetermine the growth of Eq. (1) with $n$ and characterize the states for which the regularized cost\n\n \\begin{equation}\n C_\\infty(\\psi)\n :=\\lim_{n\\to\\infty}\\frac{C_n(\\psi)}{n}\n =\\inf_{n\\ge1}\\frac{C_n(\\psi)}{n}\n\\tag{2}\n\\end{equation} \nsatisfies $C_\\infty(\\psi)0:\\beta(L)>1\\right\\}\n \\quad\\text{is unbounded}.\n\\tag{2}\n\\end{equation} \nEquation (2) asks for direct, unfiltered two-setting violation by the vacuum restrictions to the two local algebras.", "url": "https://qiqc-op.com/problem/op_660dbca2258b1a9b/", "json": "https://qiqc-op.com/api/problems/op_660dbca2258b1a9b.json", "tex": "https://qiqc-op.com/problem/op_660dbca2258b1a9b/op_660dbca2258b1a9b.tex", "created": "2026-09-01", "updated": "2026-09-04", "createdAt": "2026-09-01T05:13:32.000Z", "updatedAt": "2026-09-04T23:38:14.000Z", "sha256": "f624d9bc39dc6bc675499dc23ae310014f496126226d4bc35ed0fed418774a9f" }, { "id": "op_6cb323ea3ec0b70e", "ulid": "01M1HME7808X29G1W7P0AC8QWZ", "aliases": [ "op_6cb323ea3ec0b70e", "01M1HME7808X29G1W7P0AC8QWZ", "op-6cb323ea3ec0b70e", "v2-trace-exponential-lower-bound-for-matrix-word-averages", "open-problem-v2-problem-34" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 2, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-04T22:04:59Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "source-stated", "posed": null, "areaIds": [ "quantum-communication" ], "topicIds": [ "matrix-and-entropy-inequalities" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [] }, "title": "Trace-exponential lower bound for matrix-word averages", "status": "Unsolved", "fields": [ "Quantum Communication" ], "topics": [ "Matrix and entropy inequalities" ], "tags": [ "Quantum Communication", "Matrix and entropy inequalities" ], "statement": "Is the normalized trace average of all words in two positive-definite matrices always bounded below by the corresponding trace exponential? Let $A,B\\in M_d(\\mathbb C)$ be positive definite, let $n,m\\geq1$, and let $\\mathcal W_{n,m}$ be the set of words containing exactly $n$ letters $A$ and $m$ letters $B$. Define the normalized average by\n\n \\begin{equation}\n p_{n,m}(A,B)\n :=\\frac{1}{\\binom{n+m}{n}}\n \\sum_{W\\in\\mathcal W_{n,m}}\\operatorname{Tr}W(A,B).\n\\tag{1}\n\\end{equation} \nThe question is whether the quantity in Eq. (1) satisfies\n\n \\begin{equation}\n p_{n,m}(A,B)\n \\geq \\operatorname{Tr}\\exp\\!\\bigl(n\\log A+m\\log B\\bigr)\n\\tag{2}\n\\end{equation} \nfor every finite $d$ and all $n,m\\geq1$. A positive-semidefinite extension is obtained, whenever the limit exists, by applying Eq. (2) to $A+\\varepsilon I$ and $B+\\varepsilon I$ and then taking $\\varepsilon\\downarrow0$.", "url": "https://qiqc-op.com/problem/op_6cb323ea3ec0b70e/", "json": "https://qiqc-op.com/api/problems/op_6cb323ea3ec0b70e.json", "tex": "https://qiqc-op.com/problem/op_6cb323ea3ec0b70e/op_6cb323ea3ec0b70e.tex", "created": "2026-09-01", "updated": "2026-09-04", "createdAt": "2026-09-01T05:13:32.000Z", "updatedAt": "2026-09-04T23:38:14.000Z", "sha256": "19ded7c04baf36c08172309af36c2a3aab4ab7c7fabf6390d60b4db4686fe68b" }, { "id": "op_74c4ffb5b042baf2", "ulid": "01M1HME780V1V27MDXXVA3SCW5", "aliases": [ "op_74c4ffb5b042baf2", "01M1HME780V1V27MDXXVA3SCW5", "op-74c4ffb5b042baf2", "v2-npt-bound-entanglement-and-the-rank-five-frontier", "open-problem-v2-problem-10" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 2, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-04T22:04:59Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "source-stated", "posed": null, "areaIds": [ "quantum-resource-theory" ], "topicIds": [ "entanglement-distillation", "bound-entanglement", "local-operations-and-classical-communication" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [] }, "title": "NPT bound entanglement and the rank-five frontier", "status": "Unsolved", "fields": [ "Quantum Resource Theory" ], "topics": [ "Entanglement distillation", "Bound entanglement", "Local operations and classical communication" ], "tags": [ "Quantum Resource Theory", "Entanglement distillation", "Bound entanglement", "Local operations and classical communication" ], "statement": "Does there exist a finite-dimensional bipartite state with non-positive partial transpose (NPT) that is undistillable by local operations and classical communication (LOCC)? For a density operator $\\rho_{AB}$, partial transposition on $B$ is denoted by $\\rho_{AB}^{T_B}:=(\\operatorname{id}_A\\otimes T_B)(\\rho_{AB})$. The finite-copy criterion makes the counterexample sought by the general problem precise:\n\n \\begin{equation}\n \\exists\\,\\rho_{AB}:\\quad\n \\rho_{AB}^{T_B}\\not\\succeq0,\n \\qquad\n \\forall\\,n\\geq1\\ \\ \\forall\\,\\lvert\\psi_n\\rangle\n \\text{ with }\\operatorname{SR}_{A^n:B^n}(\\lvert\\psi_n\\rangle)\\leq2,\n \\quad\n \\langle\\psi_n\\rvert\n (\\rho_{AB}^{T_B})^{\\otimes n}\n \\lvert\\psi_n\\rangle\\geq0.\n\\tag{1}\n\\end{equation} \nHere $\\operatorname{SR}_{A^n:B^n}$ is Schmidt rank across the indicated cut. A state satisfying Eq. (1) would be NPT bound entangled; equivalently, the question is whether every NPT state instead has a negative expectation on some Schmidt-rank-at-most-two vector at some finite copy number.\n\nThe nested low-rank frontier asks whether the counterexample in Eq. (1) can additionally satisfy\n\n \\begin{equation}\n \\operatorname{rank}(\\rho_{AB})=5.\n\\tag{2}\n\\end{equation} \nThus Eq. (2) asks for the lowest rank not already excluded by known one-copy results.", "url": "https://qiqc-op.com/problem/op_74c4ffb5b042baf2/", "json": "https://qiqc-op.com/api/problems/op_74c4ffb5b042baf2.json", "tex": "https://qiqc-op.com/problem/op_74c4ffb5b042baf2/op_74c4ffb5b042baf2.tex", "created": "2026-09-01", "updated": "2026-09-04", "createdAt": "2026-09-01T05:13:32.000Z", "updatedAt": "2026-09-04T23:38:14.000Z", "sha256": "b738b6b12332d7f6ed75eab94469b4749445e1d141f7beb17aa8e9c0ba039396" }, { "id": "op_7a17786328198e24", "ulid": "01M1HME7805PVXXPJE60E83TJ5", "aliases": [ "op_7a17786328198e24", "01M1HME7805PVXXPJE60E83TJ5", "op-7a17786328198e24", "v2-minimal-support-frontier-for-absolutely-maximally-entangled-states", "open-problem-v2-problem-42" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 2, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-04T22:04:59Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "source-stated", "posed": null, "areaIds": [ "quantum-resource-theory", "quantum-error-correction" ], "topicIds": [ "absolutely-maximally-entangled-states", "quantum-coding-theory" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [] }, "title": "Minimal-support frontier for absolutely maximally entangled states", "status": "Unsolved", "fields": [ "Quantum Resource Theory", "Quantum Error Correction" ], "topics": [ "Absolutely maximally entangled states", "Quantum coding theory" ], "tags": [ "Quantum Resource Theory", "Quantum Error Correction", "Absolutely maximally entangled states", "Quantum coding theory" ], "statement": "Determine, for every integer $d\\geq 2$, the largest number $\\mathcal N(d)$ of $d$-level parties that admit an absolutely maximally entangled state of minimal computational-basis support. To make this precise, let $m=\\lfloor N/2\\rfloor$ and write a normalized state as \\(|\\psi\\rangle=\\sum_{\\boldsymbol{x}\\in[d]^N}c_{\\boldsymbol{x}}\n|\\boldsymbol{x}\\rangle\\), where $[d]=\\{0,\\ldots,d-1\\}$. The state is $\\operatorname{AME}(N,d)$ when every subsystem $S\\subseteq\\{1,\\ldots,N\\}$ with $|S|\\leq m$ has reduced state\n\n \\begin{equation}\n \\operatorname{Tr}_{S^{\\mathrm c}}|\\psi\\rangle\\!\\langle\\psi|\n =\\frac{I_{d^{|S|}}}{d^{|S|}}.\n\\tag{1}\n\\end{equation} \nCondition (1) implies the computational-basis support bound\n\n \\begin{equation}\n \\bigl|\\operatorname{supp}_{\\mathrm{comp}}(\\psi)\\bigr|\n :=\\bigl|\\{\\boldsymbol{x}:c_{\\boldsymbol{x}}\\neq0\\}\\bigr|\n \\geq d^{m}.\n\\tag{2}\n\\end{equation} \nMinimal support means equality in (2), and the frontier to be determined is\n\n \\begin{equation}\n \\mathcal N(d):=\\max\\bigl\\{N\\geq2:\\text{a minimal-support }\n \\operatorname{AME}(N,d)\\text{ state exists}\\bigr\\}.\n\\tag{3}\n\\end{equation} \nEquivalently, for any alphabet $\\mathcal A$ of size $d$, the existence event in (3) is characterized by\n\n \\begin{equation}\n \\begin{split}\n &\\text{a minimal-support }\\operatorname{AME}(N,d)\\text{ state exists}\n \\\\\n &\\quad\\Longleftrightarrow\\quad\n \\exists\\,\\mathcal C\\subseteq\\mathcal A^N:\\quad\n |\\mathcal C|=d^{\\lfloor N/2\\rfloor},\\qquad\n \\min_{\\substack{\\boldsymbol{x},\\boldsymbol{y}\\in\\mathcal C\\\\\n \\boldsymbol{x}\\neq\\boldsymbol{y}}}\n d_{\\mathrm H}(\\boldsymbol{x},\\boldsymbol{y})\n =\\left\\lceil\\frac N2\\right\\rceil+1.\n \\end{split}\n\\tag{4}\n\\end{equation} \nThus (4) asks for general, possibly nonlinear, MDS codes rather than only linear codes [GAL+15], [Ber19].", "url": "https://qiqc-op.com/problem/op_7a17786328198e24/", "json": "https://qiqc-op.com/api/problems/op_7a17786328198e24.json", "tex": "https://qiqc-op.com/problem/op_7a17786328198e24/op_7a17786328198e24.tex", "created": "2026-09-01", "updated": "2026-09-04", "createdAt": "2026-09-01T05:13:32.000Z", "updatedAt": "2026-09-04T23:38:14.000Z", "sha256": "7db451a6105bad825c2c3ca66855dbfcbb77f05683d40f6162635fe8e5f5c2f3" }, { "id": "op_7ceb25c3fe5c9403", "ulid": "01M1HME780CAT6PDF6X8ZEV5VP", "aliases": [ "op_7ceb25c3fe5c9403", "01M1HME780CAT6PDF6X8ZEV5VP", "op-7ceb25c3fe5c9403", "v2-lockability-of-two-way-distillable-entanglement", "open-problem-v2-problem-21" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 2, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-04T22:04:59Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "source-stated", "posed": null, "areaIds": [ "quantum-resource-theory" ], "topicIds": [ "entanglement-distillation", "entanglement-measures", "local-operations-and-classical-communication" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [] }, "title": "Lockability of two-way distillable entanglement", "status": "Unsolved", "fields": [ "Quantum Resource Theory" ], "topics": [ "Entanglement distillation", "Entanglement measures", "Local operations and classical communication" ], "tags": [ "Quantum Resource Theory", "Entanglement distillation", "Entanglement measures", "Local operations and classical communication" ], "statement": "Can discarding one local qubit reduce two-way distillable entanglement by an arbitrarily large amount? Let $D_{\\leftrightarrow}(A:B)_\\rho$ denote the asymptotic singlet-distillation rate of $\\rho_{AB}$ under local operations and unrestricted two-way classical communication. The question is whether there are finite-dimensional states $\\rho^{(r)}_{A_ra:B_r}$, with $\\dim a=2$, such that\n\n \\begin{equation}\n D_{\\leftrightarrow}(A_ra:B_r)_{\\rho^{(r)}}\n -D_{\\leftrightarrow}(A_r:B_r)_{\\operatorname{Tr}_a\\rho^{(r)}}\n \\xrightarrow[r\\to\\infty]{}\\infty.\n\\tag{1}\n\\end{equation} \nEquation (1) requires an unbounded loss while the discarded subsystem has fixed dimension two.", "url": "https://qiqc-op.com/problem/op_7ceb25c3fe5c9403/", "json": "https://qiqc-op.com/api/problems/op_7ceb25c3fe5c9403.json", "tex": "https://qiqc-op.com/problem/op_7ceb25c3fe5c9403/op_7ceb25c3fe5c9403.tex", "created": "2026-09-01", "updated": "2026-09-04", "createdAt": "2026-09-01T05:13:32.000Z", "updatedAt": "2026-09-04T23:38:14.000Z", "sha256": "9dc0355fd485edd655e472d447964573f7c861e1814188171762581d8dcd9e32" }, { "id": "op_96cf7aa1c1be9be2", "ulid": "01M1HME780TJ3Z7X332QY1GWFR", "aliases": [ "op_96cf7aa1c1be9be2", "01M1HME780TJ3Z7X332QY1GWFR", "op-96cf7aa1c1be9be2", "v2-polynomial-shared-resource-lower-bounds-for-routing", "open-problem-v2-problem-38" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 2, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-04T22:04:59Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "source-stated", "posed": null, "areaIds": [ "quantum-communication", "quantum-cryptography" ], "topicIds": [ "quantum-communication-complexity", "position-based-quantum-cryptography" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [] }, "title": "Polynomial shared-resource lower bounds for routing", "status": "Unsolved", "fields": [ "Quantum Communication", "Quantum Cryptography" ], "topics": [ "Quantum communication complexity", "Position-based quantum cryptography" ], "tags": [ "Quantum Communication", "Quantum Cryptography", "Quantum communication complexity", "Position-based quantum cryptography" ], "statement": "Does an explicit total Boolean family require polynomial shared-state cost for bounded-error one-round $f$-routing? More precisely, do there exist constants $c>0$ and $\\varepsilon>0$ and a sequence\n\n \\begin{equation}\n f_n:\\{0,1\\}^n\\times\\{0,1\\}^n\\longrightarrow\\{0,1\\}\n\\tag{1}\n\\end{equation} \nsuch that one uniform deterministic algorithm computes $f_n(x,y)$ from $(n,x,y)$ in time polynomial in $n$, and every routing protocol for the map in Eq. (1) with worst-case diamond-norm error at most $\\varepsilon$ has cost at least $n^c$?\n\nIn an $f$-routing protocol, Alice receives $x$ and an unknown qubit $Q$, Bob receives $y$, and they may share an arbitrary state $\\rho_{LR}$ before the inputs arrive. After one simultaneous message in each direction, Alice must recover $Q$ when $f(x,y)=0$, and Bob must recover it when $f(x,y)=1$. Message sizes and local operations are unrestricted. Measure only the shared state by\n\n \\begin{equation}\n E_{\\mathrm{dim}}(\\rho_{LR})\n :=\\log_2\\min\\!\\left\\{\n \\operatorname{rank}\\rho_L,\\operatorname{rank}\\rho_R\n \\right\\}.\n\\tag{2}\n\\end{equation} \nThe target is a family in Eq. (1) for which every valid protocol satisfies $E_{\\mathrm{dim}}(\\rho_{LR})\\geq n^c$, with the cost defined in Eq. (2).", "url": "https://qiqc-op.com/problem/op_96cf7aa1c1be9be2/", "json": "https://qiqc-op.com/api/problems/op_96cf7aa1c1be9be2.json", "tex": "https://qiqc-op.com/problem/op_96cf7aa1c1be9be2/op_96cf7aa1c1be9be2.tex", "created": "2026-09-01", "updated": "2026-09-04", "createdAt": "2026-09-01T05:13:32.000Z", "updatedAt": "2026-09-04T23:38:14.000Z", "sha256": "11b03aeb502213456dc98f282f8ace797e3ac3859f84de5fd519842dc3167244" }, { "id": "op_a9d8fc135ea71ca1", "ulid": "01M1HME780PPH667FYKZK7VTV6", "aliases": [ "op_a9d8fc135ea71ca1", "01M1HME780PPH667FYKZK7VTV6", "op-a9d8fc135ea71ca1", "v2-extensible-causality-and-process-purification", "open-problem-v2-problem-35" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 2, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-04T22:04:59Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "source-stated", "posed": null, "areaIds": [ "quantum-resource-theory" ], "topicIds": [ "indefinite-causal-order" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [] }, "title": "Extensible causality and process purification", "status": "Unsolved", "fields": [ "Quantum Resource Theory" ], "topics": [ "Indefinite causal order" ], "tags": [ "Quantum Resource Theory", "Indefinite causal order" ], "statement": "Is every finite-dimensional extensibly causal process matrix purifiable? Call an $N$-laboratory process matrix $W$ extensibly causal if $W\\otimes\\rho_R$ generates only causal correlations for every ancillary input system $R$, every joint ancillary state $\\rho_R$, and every choice of local instruments. Call $W$ purifiable if there exist auxiliary global past and future systems, a fixed pure state on the auxiliary past, and a pure process $S$ such that inserting the fixed state into $S$ and discarding the auxiliary future yields $W$. If $\\mathrm{EC}_N$ and $\\mathrm{Pur}_N$ denote the corresponding classes, the question is whether\n\n \\begin{equation}\n \\mathrm{EC}_N\\subseteq\\mathrm{Pur}_N\n\\tag{1}\n\\end{equation} \nholds for every $N$ and every choice of finite local dimensions. In particular, does the inclusion in Eq. (1) hold for two laboratories?", "url": "https://qiqc-op.com/problem/op_a9d8fc135ea71ca1/", "json": "https://qiqc-op.com/api/problems/op_a9d8fc135ea71ca1.json", "tex": "https://qiqc-op.com/problem/op_a9d8fc135ea71ca1/op_a9d8fc135ea71ca1.tex", "created": "2026-09-01", "updated": "2026-09-04", "createdAt": "2026-09-01T05:13:32.000Z", "updatedAt": "2026-09-04T23:38:14.000Z", "sha256": "46547037fd0882069ad911bffa12acaa3cee53f0c1f38d7cc2a9c66b532e85a0" }, { "id": "op_b063bdae4363cda8", "ulid": "01M1HME780N8J8JBTFX6CSPACD", "aliases": [ "op_b063bdae4363cda8", "01M1HME780N8J8JBTFX6CSPACD", "op-b063bdae4363cda8", "v2-absolute-separability-from-spectra", "open-problem-v2-problem-15" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 2, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-04T22:04:59Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "source-stated", "posed": null, "areaIds": [ "quantum-resource-theory" ], "topicIds": [ "quantum-separability" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [] }, "title": "Absolute separability from spectra", "status": "Unsolved", "fields": [ "Quantum Resource Theory" ], "topics": [ "Quantum separability" ], "tags": [ "Quantum Resource Theory", "Quantum separability" ], "statement": "Characterize the spectra of bipartite states that remain separable under every global unitary, and decide whether absolute separability equals absolute positivity under partial transpose in higher local dimensions. Let $2\\le m\\le n$ and let $\\lambda=(\\lambda_1,\\ldots,\\lambda_{mn})$ be a decreasing probability vector. Define the absolutely separable spectra by\n\n \\begin{equation}\n \\operatorname{ASEP}_{m,n}\n :=\\left\\{\n \\lambda:\n U\\operatorname{diag}(\\lambda)U^\\dagger\n \\text{ is separable on }\\mathbb{C}^m\\otimes\\mathbb{C}^n\n \\text{ for every }U\\in\\mathrm{U}(mn)\n \\right\\}.\n\\tag{1}\n\\end{equation} \nThe relaxation by positivity under partial transpose is\n\n \\begin{equation}\n \\operatorname{APPT}_{m,n}\n :=\\left\\{\n \\lambda:\n \\left(U\\operatorname{diag}(\\lambda)U^\\dagger\\right)^{T_B}\\succeq0\n \\text{ for every }U\\in\\mathrm{U}(mn)\n \\right\\}.\n\\tag{2}\n\\end{equation} \nThe task is to characterize the set in Eq. (1) and, for $3\\le m\\le n$, decide whether it equals the set in Eq. (2).", "url": "https://qiqc-op.com/problem/op_b063bdae4363cda8/", "json": "https://qiqc-op.com/api/problems/op_b063bdae4363cda8.json", "tex": "https://qiqc-op.com/problem/op_b063bdae4363cda8/op_b063bdae4363cda8.tex", "created": "2026-09-01", "updated": "2026-09-04", "createdAt": "2026-09-01T05:13:32.000Z", "updatedAt": "2026-09-04T23:38:14.000Z", "sha256": "ef98586ecee8e68161db1bfb2edc60d11d13db8d967880226e59c6a47d4dc9df" }, { "id": "op_b1b41f737f9b4aeb", "ulid": "01M1HME780G3PDHGCZMWV9MP71", "aliases": [ "op_b1b41f737f9b4aeb", "01M1HME780G3PDHGCZMWV9MP71", "op-b1b41f737f9b4aeb", "v2-tough-error-models", "open-problem-v2-problem-14" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 2, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-04T22:04:59Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "source-stated", "posed": null, "areaIds": [ "quantum-error-correction" ], "topicIds": [ "quantum-coding-theory" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [] }, "title": "Tough error models", "status": "Unsolved", "fields": [ "Quantum Error Correction" ], "topics": [ "Quantum coding theory" ], "tags": [ "Quantum Error Correction", "Quantum coding theory" ], "statement": "Determine $c(e,n)$ and construct tough error models in the sense of [KW05]. Let $H=\\mathbb{C}^n$, and let an $e$-dimensional error model be a complex linear subspace $E\\subseteq\\operatorname{End}(H)$ with $1\\le e\\le n^2$. A linear subspace $C\\subseteq H$ corrects $E$ exactly when, for every $A,B\\in E$, there is a scalar $\\lambda(A,B)\\in\\mathbb{C}$ such that\n\n \\begin{equation}\n P_C A^\\dagger B P_C=\\lambda(A,B)P_C,\n\\tag{1}\n\\end{equation} \nwhere $P_C$ is the orthogonal projector onto $C$. Define the guaranteed code dimension by\n\n \\begin{equation}\n c(e,n)\n :=\\min_{\\substack{E\\subseteq\\operatorname{End}(H)\\\\\n \\dim E=e}}\n \\max\\left\\{\n \\dim C:\n \\begin{array}{l}\n C\\subseteq H\\text{ is a linear subspace, and}\\\\\n \\forall A,B\\in E\\ \\exists\\lambda(A,B)\\in\\mathbb{C}:\\\n P_C A^\\dagger B P_C=\\lambda(A,B)P_C\n \\end{array}\n \\right\\}.\n\\tag{2}\n\\end{equation} \nDetermine $c(e,n)$ in Eq. (2), exactly or with asymptotically matching bounds, and exhibit explicit tough error models whose largest correcting code is close to this value.", "url": "https://qiqc-op.com/problem/op_b1b41f737f9b4aeb/", "json": "https://qiqc-op.com/api/problems/op_b1b41f737f9b4aeb.json", "tex": "https://qiqc-op.com/problem/op_b1b41f737f9b4aeb/op_b1b41f737f9b4aeb.tex", "created": "2026-09-01", "updated": "2026-09-04", "createdAt": "2026-09-01T05:13:32.000Z", "updatedAt": "2026-09-04T23:38:14.000Z", "sha256": "d78663d5aee0484987f372be5e421ca735b9301c5ff421763cb18cec2caf8463" }, { "id": "op_b54ea1af90e24eaa", "ulid": "01M1HME7802V93CEFVEFG215MP", "aliases": [ "op_b54ea1af90e24eaa", "01M1HME7802V93CEFVEFG215MP", "op-b54ea1af90e24eaa", "v2-semialgebraicity-of-closed-quantum-correlations", "open-problem-v2-problem-31" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 2, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-04T22:04:59Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "source-stated", "posed": null, "areaIds": [ "quantum-resource-theory" ], "topicIds": [ "bell-nonlocality" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [] }, "title": "Semialgebraicity of closed quantum correlations", "status": "Unsolved", "fields": [ "Quantum Resource Theory" ], "topics": [ "Bell nonlocality" ], "tags": [ "Quantum Resource Theory", "Bell nonlocality" ], "statement": "Is the closure of the finite-dimensional tensor-product correlation set semialgebraic in every fixed bipartite Bell scenario? Fix positive integers $n_A,n_B,m_A,m_B$. Inputs satisfy $1\\leq x\\leq n_A$ and $1\\leq y\\leq n_B$, while outputs satisfy $1\\leq a\\leq m_A$ and $1\\leq b\\leq m_B$. The set $\\mathcal C_q(n_A,n_B,m_A,m_B)$ consists of behaviors\n\n \\begin{equation}\n p(a,b\\mid x,y)\n =\\operatorname{Tr}\\!\\left[\n \\rho\\bigl(E_a^x\\otimes F_b^y\\bigr)\n \\right]\n\\tag{1}\n\\end{equation} \nobtained from arbitrary finite-dimensional local Hilbert spaces, a density operator $\\rho$, and local POVMs satisfying\n\n \\begin{equation}\n E_a^x\\succeq0,\\qquad F_b^y\\succeq0,\\qquad\n \\sum_{a=1}^{m_A}E_a^x=I,\\qquad\n \\sum_{b=1}^{m_B}F_b^y=I.\n\\tag{2}\n\\end{equation} \nEquation (2) fixes the measurement class in Eq. (1). Define the closed correlation set by\n\n \\begin{equation}\n \\mathcal C_{qa}(n_A,n_B,m_A,m_B)\n :=\\overline{\\mathcal C_q(n_A,n_B,m_A,m_B)}.\n\\tag{3}\n\\end{equation} \nFor every fixed tuple, is the set in Eq. (3) a finite Boolean combination of polynomial equalities and inequalities with real coefficients? The coefficients may be arbitrary real numbers, and the description need not be one conjunction of weak inequalities.", "url": "https://qiqc-op.com/problem/op_b54ea1af90e24eaa/", "json": "https://qiqc-op.com/api/problems/op_b54ea1af90e24eaa.json", "tex": "https://qiqc-op.com/problem/op_b54ea1af90e24eaa/op_b54ea1af90e24eaa.tex", "created": "2026-09-01", "updated": "2026-09-04", "createdAt": "2026-09-01T05:13:32.000Z", "updatedAt": "2026-09-04T23:38:14.000Z", "sha256": "c87c816a95d3e81cec6bbd651c4711ca68dd775face0ae3594197a336c677806" }, { "id": "op_b93eb7197c926d9e", "ulid": "01M1HME780NTMHKFB95TSXKKFW", "aliases": [ "op_b93eb7197c926d9e", "01M1HME780NTMHKFB95TSXKKFW", "op-b93eb7197c926d9e", "v2-distillable-entanglement-of-bell-diagonal-states", "open-problem-v2-problem-3" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 2, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-04T22:04:59Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "source-stated", "posed": null, "areaIds": [ "quantum-resource-theory" ], "topicIds": [ "entanglement-distillation", "bell-diagonal-states", "local-operations-and-classical-communication" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [] }, "title": "Distillable entanglement of Bell-diagonal states", "status": "Unsolved", "fields": [ "Quantum Resource Theory" ], "topics": [ "Entanglement distillation", "Bell-diagonal states", "Local operations and classical communication" ], "tags": [ "Quantum Resource Theory", "Entanglement distillation", "Bell-diagonal states", "Local operations and classical communication" ], "statement": "What is the distillable entanglement $D(\\rho_{\\mathbf p})$ under local operations and classical communication (LOCC) for the Bell-diagonal state\n\n \\begin{equation}\n \\rho_{\\mathbf p}\n =p_I\\lvert\\Phi^+\\rangle\\!\\langle\\Phi^+\\rvert\n +p_X\\lvert\\Psi^+\\rangle\\!\\langle\\Psi^+\\rvert\n +p_Y\\lvert\\Psi^-\\rangle\\!\\langle\\Psi^-\\rvert\n +p_Z\\lvert\\Phi^-\\rangle\\!\\langle\\Phi^-\\rvert?\n\\tag{1}\n\\end{equation} \nWhat is this LOCC protocol?\n\nThe Bell states in Eq. (1) are defined by\n\n \\begin{equation}\n \\lvert\\Phi^\\pm\\rangle:=\\frac{\\lvert00\\rangle\\pm\\lvert11\\rangle}{\\sqrt2},\n \\qquad\n \\lvert\\Psi^\\pm\\rangle:=\\frac{\\lvert01\\rangle\\pm\\lvert10\\rangle}{\\sqrt2}.\n\\tag{2}\n\\end{equation} \nEquation (2) fixes the phase convention. Assume $\\sum_i p_i=1$ and $p_I\\ge1/2\\ge p_i>0$ for $i\\in\\{X,Y,Z\\}$.", "url": "https://qiqc-op.com/problem/op_b93eb7197c926d9e/", "json": "https://qiqc-op.com/api/problems/op_b93eb7197c926d9e.json", "tex": "https://qiqc-op.com/problem/op_b93eb7197c926d9e/op_b93eb7197c926d9e.tex", "created": "2026-09-01", "updated": "2026-09-04", "createdAt": "2026-09-01T05:13:32.000Z", "updatedAt": "2026-09-04T23:38:14.000Z", "sha256": "ea595505f1fa555a63361688bdd61a94113c5529d00cf875ec8755cba8070e6d" }, { "id": "op_b952ec2dd8246a10", "ulid": "01M1HME780SNG2DQVEGCDB0XSK", "aliases": [ "op_b952ec2dd8246a10", "01M1HME780SNG2DQVEGCDB0XSK", "op-b952ec2dd8246a10", "v2-secret-key-from-every-entangled-state", "open-problem-v2-problem-20" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 2, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-04T22:04:59Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "source-stated", "posed": null, "areaIds": [ "quantum-cryptography", "quantum-resource-theory" ], "topicIds": [ "secret-key-distillation", "bound-entanglement", "local-operations-and-classical-communication" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [] }, "title": "Secret key from every entangled state", "status": "Unsolved", "fields": [ "Quantum Cryptography", "Quantum Resource Theory" ], "topics": [ "Secret-key distillation", "Bound entanglement", "Local operations and classical communication" ], "tags": [ "Quantum Cryptography", "Quantum Resource Theory", "Secret-key distillation", "Bound entanglement", "Local operations and classical communication" ], "statement": "Does every finite-dimensional entangled bipartite state have positive asymptotic distillable secret key? For a state $\\rho_{AB}$, with an adversary holding a purification, define its distillable-key rate by\n\n \\begin{equation}\n K_D(\\rho_{AB})\n :=\\sup\\left\\{R\\geq0:\n \\begin{array}{l}\n \\text{there are LOPC protocols $\\Lambda_n$ and private states\n $\\gamma^{(n)}_{K_n}$, with $K_n=2^{\\lfloor nR\\rfloor}$, such that}\\\\[-1mm]\n \\displaystyle\n \\lim_{n\\to\\infty}\n \\left\\|\\Lambda_n(\\rho_{AB}^{\\otimes n})\n -\\gamma^{(n)}_{K_n}\\right\\|_1=0\n \\end{array}\n \\right\\},\n\\tag{1}\n\\end{equation} \nwhere each target $\\gamma^{(n)}_{K_n}$ may have its own shield system and has key registers of dimension $K_n$. With the operational convention in Eq. (1), the question is whether the implication\n\n \\begin{equation}\n \\rho_{AB}\\ \\text{entangled}\n \\quad\\Longrightarrow\\quad\n K_D(\\rho_{AB})>0\n\\tag{2}\n\\end{equation} \nholds for every finite-dimensional $\\rho_{AB}$.", "url": "https://qiqc-op.com/problem/op_b952ec2dd8246a10/", "json": "https://qiqc-op.com/api/problems/op_b952ec2dd8246a10.json", "tex": "https://qiqc-op.com/problem/op_b952ec2dd8246a10/op_b952ec2dd8246a10.tex", "created": "2026-09-01", "updated": "2026-09-04", "createdAt": "2026-09-01T05:13:32.000Z", "updatedAt": "2026-09-04T23:38:14.000Z", "sha256": "253f6a421a284a7d140d87d0a05db8512ef0fe6f9e533a15930da1fb44461b48" }, { "id": "op_c9c62042b15fcb06", "ulid": "01M1HME780CSV212H08EXN5XFK", "aliases": [ "op_c9c62042b15fcb06", "01M1HME780CSV212H08EXN5XFK", "op-c9c62042b15fcb06", "v2-maximum-number-of-mutually-unbiased-bases", "open-problem-v2-problem-44" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 2, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-04T22:04:59Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "source-stated", "posed": null, "areaIds": [ "quantum-metrology" ], "topicIds": [ "mutually-unbiased-bases" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [] }, "title": "Maximum number of mutually unbiased bases", "status": "Unsolved", "fields": [ "Quantum metrology" ], "topics": [ "Mutually unbiased bases" ], "tags": [ "Quantum metrology", "Mutually unbiased bases" ], "statement": "For each integer $d\\geq2$, determine the maximum number $\\mu(d)$ of pairwise mutually unbiased orthonormal bases of $\\mathbb C^d$. Two orthonormal bases $\\mathcal B_r=\\{|e_i^{(r)}\\rangle\\}_{i=1}^{d}$ and $\\mathcal B_s=\\{|e_j^{(s)}\\rangle\\}_{j=1}^{d}$ are mutually unbiased when\n\n \\begin{equation}\n \\bigl|\\langle e_i^{(r)}|e_j^{(s)}\\rangle\\bigr|^2=\\frac1d\n \\qquad\\text{for every }i,j\\in\\{1,\\ldots,d\\}.\n\\tag{1}\n\\end{equation} \nThus the extremal quantity defined by Eq. (1) is\n\n \\begin{equation}\n \\mu(d):=\\max\\left\\{m:\\text{there exist $m$ orthonormal bases of\n $\\mathbb C^d$ that are pairwise mutually unbiased}\\right\\}.\n\\tag{2}\n\\end{equation} \nDetermine Eq. (2) in the non-prime-power regime, in particular for $d\\in\\{6,10,12,14,15\\}$.", "url": "https://qiqc-op.com/problem/op_c9c62042b15fcb06/", "json": "https://qiqc-op.com/api/problems/op_c9c62042b15fcb06.json", "tex": "https://qiqc-op.com/problem/op_c9c62042b15fcb06/op_c9c62042b15fcb06.tex", "created": "2026-09-01", "updated": "2026-09-04", "createdAt": "2026-09-01T05:13:32.000Z", "updatedAt": "2026-09-04T23:38:14.000Z", "sha256": "d1fe5e267bff02485fb928a865954ddb31215d1300555b00f2e9689cd33a5fda" }, { "id": "op_ccd560469b815618", "ulid": "01M1HME780VWYDSN7KF3J06003", "aliases": [ "op_ccd560469b815618", "01M1HME780VWYDSN7KF3J06003", "op-ccd560469b815618", "v2-optimal-cglmp-measurements-for-a-maximally-entangled-state", "open-problem-v2-problem-24" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 2, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-04T22:04:59Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "source-stated", "posed": null, "areaIds": [ "quantum-resource-theory", "quantum-metrology" ], "topicIds": [ "bell-nonlocality" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [] }, "title": "Optimal CGLMP measurements for a maximally entangled state", "status": "Unsolved", "fields": [ "Quantum Resource Theory", "Quantum metrology" ], "topics": [ "Bell nonlocality" ], "tags": [ "Quantum Resource Theory", "Quantum metrology", "Bell nonlocality" ], "statement": "For every $d\\geq3$, do the standard Fourier–phase measurements optimize the CGLMP violation of the maximally entangled state among all projective $d$-outcome measurements? Fix the state\n\n \\begin{equation}\n \\lvert\\Phi_d\\rangle\n =\\frac{1}{\\sqrt d}\\sum_{j=0}^{d-1}\\lvert j,j\\rangle.\n\\tag{1}\n\\end{equation} \nEquation (1) is held fixed; the shared state is not part of the optimization.\n\nFor outcomes in $\\mathbb Z_d$, let $[t]_d\\in\\{0,\\ldots,d-1\\}$ be the residue of $t$ and define the CGLMP functional by\n\n \\begin{equation}\n \\begin{aligned}\n B_d={}&\\mathbb E([A_0-B_0]_d)+\\mathbb E([B_0-A_1]_d)\\\\\n &+\\mathbb E([A_1-B_1]_d)+\\mathbb E([B_1-A_0-1]_d).\n \\end{aligned}\n\\tag{2}\n\\end{equation} \nLocal behaviors satisfy $B_d\\geq d-1$ under the convention in Eq. (2). The candidate measurements have bases\n\n \\begin{equation}\n \\begin{aligned}\n \\lvert a;x\\rangle\n &=\\frac{1}{\\sqrt d}\\sum_{j=0}^{d-1}\n \\exp\\!\\left(\\frac{2\\pi i}{d}j(a+\\alpha_x)\\right)\\lvert j\\rangle,\\\\\n \\lvert b;y\\rangle\n &=\\frac{1}{\\sqrt d}\\sum_{j=0}^{d-1}\n \\exp\\!\\left(\\frac{2\\pi i}{d}j(-b+\\beta_y)\\right)\\lvert j\\rangle,\n \\end{aligned}\n\\tag{3}\n\\end{equation} \nwith $\\alpha_0=0$, $\\alpha_1=-1/2$, $\\beta_0=1/4$, and $\\beta_1=3/4$. The question is whether the bases in Eq. (3) minimize Eq. (2) on Eq. (1), up to symmetries of the state and functional.", "url": "https://qiqc-op.com/problem/op_ccd560469b815618/", "json": "https://qiqc-op.com/api/problems/op_ccd560469b815618.json", "tex": "https://qiqc-op.com/problem/op_ccd560469b815618/op_ccd560469b815618.tex", "created": "2026-09-01", "updated": "2026-09-04", "createdAt": "2026-09-01T05:13:32.000Z", "updatedAt": "2026-09-04T23:38:14.000Z", "sha256": "5c31df2d745e64f6745cb23846584fe916f1e7d228dc6a01fe7b7f63b9bd6ddb" }, { "id": "op_dcea1e5e3032b8c5", "ulid": "01M1HME780803VM2A45MA86N41", "aliases": [ "op_dcea1e5e3032b8c5", "01M1HME780803VM2A45MA86N41", "op-dcea1e5e3032b8c5", "v2-quantum-violations-of-bipartite-bell-facets", "open-problem-v2-problem-23" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 2, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-04T22:04:59Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "source-stated", "posed": null, "areaIds": [ "quantum-resource-theory" ], "topicIds": [ "bell-nonlocality" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [] }, "title": "Quantum violations of bipartite Bell facets", "status": "Unsolved", "fields": [ "Quantum Resource Theory" ], "topics": [ "Bell nonlocality" ], "tags": [ "Quantum Resource Theory", "Bell nonlocality" ], "statement": "Must every nontrivial facet Bell inequality in a finite bipartite scenario have a quantum violation? Let $\\mathcal L$ be the local polytope, and let $\\mathcal Q$ consist of behaviors realizable as\n\n \\begin{equation}\n p(a,b\\mid x,y)\n =\\operatorname{Tr}\\!\\left[\\rho_{AB}\n \\bigl(M_x^a\\otimes N_y^b\\bigr)\\right],\n\\tag{1}\n\\end{equation} \nwhere $\\rho_{AB}$ is a finite-dimensional state and $\\{M_x^a\\}_a$ and $\\{N_y^b\\}_b$ are local POVMs. Equation (1) defines the quantum set used below.\n\nFor a linear functional $F(p)=\\sum_{a,b,x,y}c_{abxy}p(a,b\\mid x,y)$, suppose $F(p)\\leq\\beta_{\\mathrm L}$ supports a facet of $\\mathcal L$ and is not a positivity facet within the normalization and no-signalling affine hull. Is it necessarily true that\n\n \\begin{equation}\n \\sup_{p\\in\\mathcal Q}F(p)>\\beta_{\\mathrm L}?\n\\tag{2}\n\\end{equation} \nEquation (2) asks whether the bipartite local and quantum sets can share a nontrivial facet-supporting hyperplane.", "url": "https://qiqc-op.com/problem/op_dcea1e5e3032b8c5/", "json": "https://qiqc-op.com/api/problems/op_dcea1e5e3032b8c5.json", "tex": "https://qiqc-op.com/problem/op_dcea1e5e3032b8c5/op_dcea1e5e3032b8c5.tex", "created": "2026-09-01", "updated": "2026-09-04", "createdAt": "2026-09-01T05:13:32.000Z", "updatedAt": "2026-09-04T23:38:14.000Z", "sha256": "3c7a0c16421ccaafbebf573fa13f4c09e3436b62227ec2b1368bf468839fec00" }, { "id": "op_e2149f4ced34d1a8", "ulid": "01M1HME780JCWZMCJMARNSAXH3", "aliases": [ "op_e2149f4ced34d1a8", "01M1HME780JCWZMCJMARNSAXH3", "op-e2149f4ced34d1a8", "v2-relative-entropy-of-entanglement-for-two-qubits", "open-problem-v2-problem-12" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 2, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-04T22:04:59Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "source-stated", "posed": null, "areaIds": [ "quantum-resource-theory" ], "topicIds": [ "entanglement-measures", "quantum-separability", "quantum-relative-entropy" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [] }, "title": "Relative entropy of entanglement for two qubits", "status": "Unsolved", "fields": [ "Quantum Resource Theory" ], "topics": [ "Entanglement measures", "Quantum separability", "Quantum relative entropy" ], "tags": [ "Quantum Resource Theory", "Entanglement measures", "Quantum separability", "Quantum relative entropy" ], "statement": "Find a closed formula for the relative entropy of entanglement of every two-qubit density operator $\\rho$, including an explicit closest separable state. With $\\operatorname{Sep}(\\mathbb{C}^2:\\mathbb{C}^2)$ denoting the two-qubit separable states, the quantity is\n\n \\begin{equation}\n E_R(\\rho)\n :=\\min_{\\sigma\\in\\operatorname{Sep}(\\mathbb{C}^2:\\mathbb{C}^2)}\n D(\\rho\\Vert\\sigma),\n \\qquad\n D(\\rho\\Vert\\sigma)\n :=\\begin{cases}\n \\operatorname{Tr}\\!\\left[\\rho(\\log\\rho-\\log\\sigma)\\right],\n &\\operatorname{supp}\\rho\\subseteq\\operatorname{supp}\\sigma,\\\\\n +\\infty,&\\text{otherwise}.\n \\end{cases}\n\\tag{1}\n\\end{equation} \nIn the finite branch of Eq. (1), the trace is evaluated on $\\operatorname{supp}\\rho$, with $0\\log 0:=0$. The formula sought must determine at least one minimizing state $\\sigma_\\rho$ for every $\\rho$.", "url": "https://qiqc-op.com/problem/op_e2149f4ced34d1a8/", "json": "https://qiqc-op.com/api/problems/op_e2149f4ced34d1a8.json", "tex": "https://qiqc-op.com/problem/op_e2149f4ced34d1a8/op_e2149f4ced34d1a8.tex", "created": "2026-09-01", "updated": "2026-09-04", "createdAt": "2026-09-01T05:13:32.000Z", "updatedAt": "2026-09-04T23:38:14.000Z", "sha256": "6849f44dc32ae3cf3dbe2a2ab33680b0ef3f828ec233dd0a9ece0ce10bb4d899" }, { "id": "op_eb5ca2d40deb7a38", "ulid": "01M1HME780637N8XEVFA4V7B4N", "aliases": [ "op_eb5ca2d40deb7a38", "01M1HME780637N8XEVFA4V7B4N", "op-eb5ca2d40deb7a38", "v2-square-root-remainder-in-generalized-quantum-equipartition", "open-problem-v2-problem-48" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 2, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-04T22:04:59Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "source-stated", "posed": null, "areaIds": [ "quantum-resource-theory", "quantum-communication" ], "topicIds": [ "quantum-hypothesis-testing", "quantum-relative-entropy", "one-shot-and-finite-blocklength-bounds" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [] }, "title": "Square-root remainder in generalized quantum equipartition", "status": "Unsolved", "fields": [ "Quantum Resource Theory", "Quantum Communication" ], "topics": [ "Quantum hypothesis testing", "Quantum relative entropy", "One-shot and finite-blocklength bounds" ], "tags": [ "Quantum Resource Theory", "Quantum Communication", "Quantum hypothesis testing", "Quantum relative entropy", "One-shot and finite-blocklength bounds" ], "statement": "Let $H$ be a $d$-dimensional Hilbert space. For each $n$, let $\\mathcal A_n\\subseteq\\mathcal D(H^{\\otimes n})$ and $\\mathcal B_n\\subseteq\\mathcal L(H^{\\otimes n})_+$ be nonempty compact, convex, permutation-invariant sets, each closed under tensor products. For a positive-operator set $\\mathcal C$, define its positive polar by\n\n \\begin{equation}\n \\mathcal C_+^\\circ\n :=\\{X\\geq0:\\operatorname{Tr}(XY)\\leq1\n \\text{ for every }Y\\in\\mathcal C\\}.\n\\tag{1}\n\\end{equation} \nAssume that both families of polars from Eq. (1) satisfy\n\n \\begin{equation}\n (\\mathcal A_m)_+^\\circ\\otimes(\\mathcal A_n)_+^\\circ\n \\subseteq(\\mathcal A_{m+n})_+^\\circ,\n \\qquad\n (\\mathcal B_m)_+^\\circ\\otimes(\\mathcal B_n)_+^\\circ\n \\subseteq(\\mathcal B_{m+n})_+^\\circ.\n\\tag{2}\n\\end{equation} \nIn addition to Eq. (2), assume that some constant $C<\\infty$ satisfies\n\n \\begin{equation}\n D_{\\max}(\\rho_n\\|\\sigma_n)\\leq Cn,\n \\qquad \\log_2\\operatorname{Tr}\\sigma_n\\leq Cn\n\\tag{3}\n\\end{equation} \nfor all $\\rho_n\\in\\mathcal A_n$ and $\\sigma_n\\in\\mathcal B_n$. The linear growth condition in Eq. (3) uses $D_{\\max}(\\rho\\|\\sigma):=\\inf\\{\\lambda:\\rho\\leq2^\\lambda\\sigma\\}$. For positive operators $\\rho$ and $\\sigma$, define\n\n \\begin{equation}\n \\begin{aligned}\n D(\\rho\\|\\sigma)\n &:=\\operatorname{Tr}[\\rho(\\log_2\\rho-\\log_2\\sigma)],\\\\\n D_H^\\varepsilon(\\rho\\|\\sigma)\n &:=-\\log_2\\inf_{\\substack{0\\leq Q\\leq I\\\\\n \\operatorname{Tr}(Q\\rho)\\geq1-\\varepsilon}}\n \\operatorname{Tr}(Q\\sigma),\n \\end{aligned}\n\\tag{4}\n\\end{equation} \nwhere $D(\\rho\\|\\sigma)=+\\infty$ unless $\\operatorname{supp}\\rho\\subseteq\\operatorname{supp}\\sigma$. Define the divergences between the two sets from Eq. (4) by\n\n \\begin{equation}\n \\begin{aligned}\n D(\\mathcal A_n\\|\\mathcal B_n)\n &:=\\inf_{\\rho_n\\in\\mathcal A_n,\\,\\sigma_n\\in\\mathcal B_n}\n D(\\rho_n\\|\\sigma_n),\\\\\n D_H^\\varepsilon(\\mathcal A_n\\|\\mathcal B_n)\n &:=\\inf_{\\rho_n\\in\\mathcal A_n,\\,\\sigma_n\\in\\mathcal B_n}\n D_H^\\varepsilon(\\rho_n\\|\\sigma_n),\n \\end{aligned}\n\\tag{5}\n\\end{equation} \nThe two quantities in Eq. (5) determine the first- and finite-blocklength orders of interest. Set\n\n \\begin{equation}\n D^\\infty(\\mathcal A\\|\\mathcal B)\n :=\\lim_{n\\to\\infty}\\frac1nD(\\mathcal A_n\\|\\mathcal B_n).\n\\tag{6}\n\\end{equation} \nDoes every fixed $\\varepsilon\\in(0,1)$ admit a constant $K_\\varepsilon<\\infty$ such that, for all sufficiently large $n$,\n\n \\begin{equation}\n \\left|D_H^\\varepsilon(\\mathcal A_n\\|\\mathcal B_n)\n -nD^\\infty(\\mathcal A\\|\\mathcal B)\\right|\n \\leq K_\\varepsilon\\sqrt n?\n\\tag{7}\n\\end{equation}", "url": "https://qiqc-op.com/problem/op_eb5ca2d40deb7a38/", "json": "https://qiqc-op.com/api/problems/op_eb5ca2d40deb7a38.json", "tex": "https://qiqc-op.com/problem/op_eb5ca2d40deb7a38/op_eb5ca2d40deb7a38.tex", "created": "2026-09-01", "updated": "2026-09-04", "createdAt": "2026-09-01T05:13:32.000Z", "updatedAt": "2026-09-04T23:38:14.000Z", "sha256": "171da1fd1884e54c156b6b4cd9636d47bbe1eaf8bddaa87ccf3b961e7ac87627" }, { "id": "op_ebee7d5c442d81a4", "ulid": "01M1HME780M4B4RRABEG3RDDCZ", "aliases": [ "op_ebee7d5c442d81a4", "01M1HME780M4B4RRABEG3RDDCZ", "op-ebee7d5c442d81a4", "v2-complete-facet-descriptions-for-bell-polytopes", "open-problem-v2-problem-11" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 2, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-04T22:04:59Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "source-stated", "posed": null, "areaIds": [ "quantum-resource-theory" ], "topicIds": [ "bell-nonlocality" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [] }, "title": "Complete facet descriptions for Bell polytopes", "status": "Unsolved", "fields": [ "Quantum Resource Theory" ], "topics": [ "Bell nonlocality" ], "tags": [ "Quantum Resource Theory", "Bell nonlocality" ], "statement": "Determine complete facet descriptions for local-behavior polytopes beyond the presently solved Bell scenarios, either for a specified unresolved finite scenario or for a nontrivial infinite family with additional structure. For positive integers $N$, $M$, and $K$, let $\\mathbf{x}\\in\\{1,\\ldots,M\\}^{N}$ denote the measurement settings and $\\mathbf{a}\\in\\{1,\\ldots,K\\}^{N}$ the outcomes. The relevant local polytope is\n\n \\begin{equation}\n \\mathcal{L}_{N,M,K}\n :=\\operatorname{conv}\\!\\left\\{\n p_{\\mathbf f}:p_{\\mathbf f}(\\mathbf a\\mid\\mathbf x)\n =\\prod_{i=1}^{N}\\mathbf{1}\\!\\left\\{a_i=f_i(x_i)\\right\\},\\quad\n f_i:\\{1,\\ldots,M\\}\\to\\{1,\\ldots,K\\}\n \\right\\}.\n\\tag{1}\n\\end{equation} \nHere $\\mathbf 1\\{\\cdot\\}$ is the indicator function. The task is to characterize all facet-defining inequalities of the polytope in Eq. (1) in a chosen unresolved regime, modulo permutations of parties, settings, and outcomes and modulo liftings obtained by adjoining redundant settings or outcomes. A solution must carry a proof of completeness, not merely generate a large collection of facets.", "url": "https://qiqc-op.com/problem/op_ebee7d5c442d81a4/", "json": "https://qiqc-op.com/api/problems/op_ebee7d5c442d81a4.json", "tex": "https://qiqc-op.com/problem/op_ebee7d5c442d81a4/op_ebee7d5c442d81a4.tex", "created": "2026-09-01", "updated": "2026-09-04", "createdAt": "2026-09-01T05:13:32.000Z", "updatedAt": "2026-09-04T23:38:14.000Z", "sha256": "58ea4a157158c131a77d1c47aac98ed1934977369a4e5a1b6a56c35b30f3e3ba" }, { "id": "op_ec184fc49232a9c0", "ulid": "01M1HME78078BW0JG3X252BVX5", "aliases": [ "op_ec184fc49232a9c0", "01M1HME78078BW0JG3X252BVX5", "op-ec184fc49232a9c0", "v2-constant-trace-distance-separability-testing", "open-problem-v2-problem-36" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 2, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-04T22:04:59Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "source-stated", "posed": null, "areaIds": [ "quantum-algorithm", "quantum-resource-theory" ], "topicIds": [ "quantum-separability", "computational-complexity-and-computability" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [] }, "title": "Constant trace-distance separability testing", "status": "Unsolved", "fields": [ "Quantum algorithm", "Quantum Resource Theory" ], "topics": [ "Quantum separability", "Computational complexity and computability" ], "tags": [ "Quantum algorithm", "Quantum Resource Theory", "Quantum separability", "Computational complexity and computability" ], "statement": "What is the computational complexity of testing bipartite separability with a constant trace-distance promise gap? For local dimension $d$, define\n\n \\begin{equation}\n \\operatorname{Sep}(d,d)\n :=\\operatorname{conv}\\!\\left\\{\n \\alpha\\otimes\\beta:\n \\alpha,\\beta\\in\\mathcal D(\\mathbb C^d)\n \\right\\}.\n\\tag{1}\n\\end{equation} \nFix a constant $0<\\varepsilon_0<1$. Given a rational description of $\\rho\\in\\mathcal D(\\mathbb C^d\\otimes\\mathbb C^d)$, promised that exactly one of the following alternatives holds,\n\n \\begin{equation}\n \\begin{aligned}\n \\text{YES:}\\quad&\\rho\\in\\operatorname{Sep}(d,d),\\\\\n \\text{NO:}\\quad&\n \\inf_{\\sigma\\in\\operatorname{Sep}(d,d)}\n \\frac12\\lVert\\rho-\\sigma\\rVert_1\\geq\\varepsilon_0,\n \\end{aligned}\n\\tag{2}\n\\end{equation} \ndecide which case in Eq. (2) holds. Is there an algorithm polynomial or quasipolynomial in $d$? More generally, when the gap $\\varepsilon$ is part of the input, determine the optimal dependence of the complexity on $d$ and $\\varepsilon$.", "url": "https://qiqc-op.com/problem/op_ec184fc49232a9c0/", "json": "https://qiqc-op.com/api/problems/op_ec184fc49232a9c0.json", "tex": "https://qiqc-op.com/problem/op_ec184fc49232a9c0/op_ec184fc49232a9c0.tex", "created": "2026-09-01", "updated": "2026-09-04", "createdAt": "2026-09-01T05:13:32.000Z", "updatedAt": "2026-09-04T23:38:14.000Z", "sha256": "9cc81d7b56927be0be21604d037a60f1748ef71edd9ebe5a2c82dc23c3583e5d" }, { "id": "op_f1e5a1cad168b38b", "ulid": "01M1HME780G89RD9W1SZ24KSRP", "aliases": [ "op_f1e5a1cad168b38b", "01M1HME780G89RD9W1SZ24KSRP", "op-f1e5a1cad168b38b", "v2-honest-party-lockability-of-distillable-key", "open-problem-v2-problem-22" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 2, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-04T22:04:59Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "source-stated", "posed": null, "areaIds": [ "quantum-cryptography", "quantum-resource-theory" ], "topicIds": [ "secret-key-distillation", "local-operations-and-classical-communication" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [] }, "title": "Honest-party lockability of distillable key", "status": "Unsolved", "fields": [ "Quantum Cryptography", "Quantum Resource Theory" ], "topics": [ "Secret-key distillation", "Local operations and classical communication" ], "tags": [ "Quantum Cryptography", "Quantum Resource Theory", "Secret-key distillation", "Local operations and classical communication" ], "statement": "Can loss of one qubit held by an honest party reduce the two-way distillable secret key by an arbitrarily large amount? Let $K_D(A:B)_\\rho$ denote the asymptotic secret-key rate obtainable from $\\rho_{AB}$ by local operations and public two-way classical communication, with an adversary holding a purification. The question is whether there are finite-dimensional states $\\rho^{(r)}_{A_ra:B_r}$, with $\\dim a=2$, such that\n\n \\begin{equation}\n K_D(A_ra:B_r)_{\\rho^{(r)}}\n -K_D(A_r:B_r)_{\\operatorname{Tr}_a\\rho^{(r)}}\n \\xrightarrow[r\\to\\infty]{}\\infty.\n\\tag{1}\n\\end{equation} \nEquation (1) is the $AB$-locking question: the discarded qubit belongs to Alice rather than being transferred to the adversary.", "url": "https://qiqc-op.com/problem/op_f1e5a1cad168b38b/", "json": "https://qiqc-op.com/api/problems/op_f1e5a1cad168b38b.json", "tex": "https://qiqc-op.com/problem/op_f1e5a1cad168b38b/op_f1e5a1cad168b38b.tex", "created": "2026-09-01", "updated": "2026-09-04", "createdAt": "2026-09-01T05:13:32.000Z", "updatedAt": "2026-09-04T23:38:14.000Z", "sha256": "38ae8492207dc7e9ecacf77d76ac43b27992399446fb6ad59228d5332ea07194" }, { "id": "op_f60b9a99d7945e3b", "ulid": "01M1HME780ZVGZ03D56MC08Z5J", "aliases": [ "op_f60b9a99d7945e3b", "01M1HME780ZVGZ03D56MC08Z5J", "op-f60b9a99d7945e3b", "v2-real-four-quhex-absolutely-maximally-entangled-state", "open-problem-v2-problem-41" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 2, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-04T22:04:59Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "source-stated", "posed": null, "areaIds": [ "quantum-resource-theory" ], "topicIds": [ "absolutely-maximally-entangled-states" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [] }, "title": "Real four-quhex absolutely maximally entangled state", "status": "Unsolved", "fields": [ "Quantum Resource Theory" ], "topics": [ "Absolutely maximally entangled states" ], "tags": [ "Quantum Resource Theory", "Absolutely maximally entangled states" ], "statement": "Does there exist an $\\operatorname{AME}(4,6)$ state whose coefficients are real in a product basis? Fix $[6]=\\{0,\\ldots,5\\}$ and ask for a normalized state\n\n \\begin{equation}\n |\\psi\\rangle\n =\\sum_{i,j,k,l\\in[6]}T_{ijkl}|i\\rangle_A|j\\rangle_B\n |k\\rangle_C|l\\rangle_D,\n \\qquad\n T_{ijkl}\\in\\mathbb{R},\n \\qquad\n \\sum_{i,j,k,l\\in[6]}T_{ijkl}^{2}=1 .\n\\tag{1}\n\\end{equation} \nThe state in (1) must have maximally mixed reductions on every pair of parties: for each $S\\subseteq\\{A,B,C,D\\}$ with $|S|=2$,\n\n \\begin{equation}\n \\rho_S:=\\operatorname{Tr}_{S^{c}}|\\psi\\rangle\\!\\langle\\psi|\n =\\frac{I_{36}}{36} .\n\\tag{2}\n\\end{equation} \nEquivalently, define the $36\\times36$ flattening $U$ and its reshuffling $U^R$ and second-factor partial transpose $U^\\Gamma$ by\n\n \\begin{equation}\n U_{(i,j),(k,l)}:=6T_{ijkl},\n \\qquad\n (U^R)_{(i,k),(j,l)}:=U_{(i,j),(k,l)},\n \\qquad\n (U^\\Gamma)_{(i,j),(k,l)}:=U_{(i,l),(k,j)} .\n\\tag{3}\n\\end{equation} \nHere the three matrices in (3) are the coefficient flattenings for the bipartitions $AB|CD$, $AC|BD$, and $AD|CB$, up to the displayed ordering of tensor factors. Consequently, (2) is equivalent to the orthogonal $2$-unitarity conditions\n\n \\begin{equation}\n U\\in O(36),\n \\qquad\n U^R\\in O(36),\n \\qquad\n U^\\Gamma\\in O(36) .\n\\tag{4}\n\\end{equation} \nThus the problem is also to construct an orthogonal $2$-unitary matrix of order $36$, or prove that none exists.", "url": "https://qiqc-op.com/problem/op_f60b9a99d7945e3b/", "json": "https://qiqc-op.com/api/problems/op_f60b9a99d7945e3b.json", "tex": "https://qiqc-op.com/problem/op_f60b9a99d7945e3b/op_f60b9a99d7945e3b.tex", "created": "2026-09-01", "updated": "2026-09-04", "createdAt": "2026-09-01T05:13:32.000Z", "updatedAt": "2026-09-04T23:38:14.000Z", "sha256": "3f95665e401452db524da8541117dfae70fa0609c65a9528a9edae810528849b" }, { "id": "op_fcd21a1a5021e464", "ulid": "01M1HME780H9TAVH85TF8KJDS5", "aliases": [ "op_fcd21a1a5021e464", "01M1HME780H9TAVH85TF8KJDS5", "op-fcd21a1a5021e464", "v2-capacity-achieving-codes-for-amplitude-damping", "open-problem-v2-problem-2" ], "metadata": { "type": "Problem", "schemaVersion": "1.0", "revision": 2, "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1", "createdAt": "2026-09-04T22:04:59Z", "role": "primary", "parentProblemId": null, "parentClauseId": null, "origin": "source-stated", "posed": null, "areaIds": [ "quantum-error-correction", "quantum-communication" ], "topicIds": [ "quantum-capacity", "quantum-coding-theory", "decoding-algorithms" ], "keywords": [], "difficulty": "unrated", "verificationCost": "unrated", "relatedProblemIds": [] }, "title": "Capacity-achieving codes for amplitude damping", "status": "Solved", "fields": [ "Quantum Error Correction", "Quantum Communication" ], "topics": [ "Quantum capacity", "Quantum coding theory", "Decoding algorithms" ], "tags": [ "Quantum Error Correction", "Quantum Communication", "Quantum capacity", "Quantum coding theory", "Decoding algorithms" ], "statement": "What is the constructive quantum code for achieving the quantum capacity $\\mathcal{Q}(\\mathcal A_p)$ of the qubit amplitude-damping channel\n\n \\begin{equation}\n \\mathcal A_p(\\rho)=A_0\\rho A_0^\\dagger+A_1\\rho A_1^\\dagger,\n \\qquad 0\\le p\\le1?\n\\tag{1}\n\\end{equation} \nThe operators appearing in Eq. (1) are\n\n \\begin{equation}\n \\begin{aligned}\n A_0&=\\lvert0\\rangle\\!\\langle0\\rvert\n +\\sqrt{1-p}\\,\\lvert1\\rangle\\!\\langle1\\rvert\n =\\begin{pmatrix}1&0\\\\0&\\sqrt{1-p}\\end{pmatrix},\\\\\n A_1&=\\sqrt p\\,\\lvert0\\rangle\\!\\langle1\\rvert\n =\\begin{pmatrix}0&\\sqrt p\\\\0&0\\end{pmatrix}.\n \\end{aligned}\n\\tag{2}\n\\end{equation} \nEquation (2) uses $p$ as the decay probability of the excited state.", "url": "https://qiqc-op.com/problem/op_fcd21a1a5021e464/", "json": "https://qiqc-op.com/api/problems/op_fcd21a1a5021e464.json", "tex": "https://qiqc-op.com/problem/op_fcd21a1a5021e464/op_fcd21a1a5021e464.tex", "created": "2026-09-01", "updated": "2026-09-04", "createdAt": "2026-09-01T05:13:32.000Z", "updatedAt": "2026-09-04T23:38:14.000Z", "sha256": "24c66f232043e284fa8f1e7b3290786307002d8494171145d7652fae884b784a" } ] }