Quantum LDPC codes at the Pauli hashing bound

Problem

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?

The channel and target rate are

\begin{equation} \Lambda_{\mathbf p}(\rho) :=\sum_{P\in\{I,X,Y,Z\}}p_P P\rho P, \tag{1} \end{equation}

\begin{equation} R_{\mathrm{hash}}(\mathbf p):=\max\{0,1-H(\mathbf p)\}, \qquad H(\mathbf p):=-\sum_{P\in\{I,X,Y,Z\}}p_P\log_2p_P, \qquad 0\log_2 0:=0. \tag{2} \end{equation}

For 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

\begin{equation} \liminf_{n\to\infty}\frac{k_n}{n}\ge R_{\mathrm{hash}}(\mathbf p), \qquad F_e\!\left(2^{-k_n}I, \mathcal D_n\circ\Lambda_{\mathbf p}^{\otimes n}\circ\mathcal E_n \right)\longrightarrow1. \tag{3} \end{equation}

Here \(F_e\) 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.

Source

Contributor: Bikun Li.

Progress

Exact attainment is impossible for the dephasing subfamily \(\mathbf p=(1-q,0,0,q)\) with \(0<q<1/2\) at any fixed maximum generator weight \(w\). The following entropy argument was given in the catalog review [Guo26]. Let \(G\subset\mathbb F_2^{2n}\) be the binary stabilizer space, \(r_x:=\dim\pi_X(G)\) and \(r_z:=\dim\ker(\pi_X|_G)\), where \(\pi_X(x,z)=x\). Rank-nullity gives \(r_x+r_z=n-k\) for every stabilizer code, including non-CSS codes. For the independent phase-error vector \(E\), choose \(r_x\) independent syndrome bits \(S\) from the bounded-weight generators. With \(h_2\) the binary entropy, put

\begin{equation} h_w:=h_2\!\left(\frac{1-(1-2q)^w}{2}\right)<1, \qquad H(S)\le r_x h_w. \tag{4} \end{equation}

For a fixed syndrome, pure phase errors have \(2^k\) logical classes \(L\). Optimal recovery for stabilizer encodings and Pauli noise is syndrome measurement followed by maximum-likelihood logical-class correction [FSW08], Appendix, Theorem 3. Thus entanglement fidelity tending to one implies \(H(L|S)=o(n)\) by Fano’s inequality. For fixed \((S,L)\) the errors form a coset of the pure-Z stabilizer subgroup. Its \(r_z\) pivot coordinates determine the remaining error, giving \(H(E|S,L)\le r_z h_2(q)\). Consequently Eq. (4) yields

\begin{equation} nh_2(q)\le r_xh_w+r_zh_2(q)+o(n)\le(n-k)h_w+o(n), \qquad R\le1-\frac{h_2(q)}{h_w}<1-h_2(q). \tag{5} \end{equation}

Equation (5) accounts for stabilizer degeneracy and uses no CSS assumption or bound on the weight of products of generators. For \(q=0.1\) and \(w=6\) it gives \(R\le0.5062401\), below hashing \(0.5310044\). This does not establish the same obstruction at every other Pauli parameter.
The weaker existence question with a prescribed gap \(\delta>0\) and LDPC bounds allowed to depend on \((\mathbf p,\delta)\) has an affirmative answer by the following concatenation consequence of established coding theorems. Let \(C=R_{\mathrm{hash}}(\mathbf p)>\delta\); otherwise the zero-rate code suffices. Quantum Tanner codes have outer rate \(R_o\) arbitrarily close to one and relative distance \(\alpha>0\), with fixed check weight and degree [LZ22], Theorems 17–18. Choose \(R_o\) with \(R_o(C-\delta/2)>C-\delta\). Stabilizer hashing achievability [Ham02], Theorem 1 and its proof, supplies a fixed inner \([[m,\ell]]\) code of rate at least \(C-\delta/2\) with Pauli recovery and logical block-error probability \(\epsilon<\alpha/4\). Take \(\ell\) copies of an outer \([[N,K,d\ge\alpha N]]\) code. For each outer coordinate, encode its \(\ell\) qubits, one from each copy, in one inner block. The residual inner noise is a logical Pauli channel, independently across the \(N\) blocks. Every failed block affects at most one qubit of each outer copy. A Chernoff bound therefore makes the probability of \(\alpha N/2\) or more failed blocks vanish; below this threshold all outer copies are recoverable. The resulting parameters obey

\begin{equation} n=mN,\qquad k=\ell K,\qquad \frac{k}{n}\ge\frac\ell m R_o>C-\delta. \tag{6} \end{equation}

Equation (6) gives reliable transmission. Lifted outer checks have weight at most \(m w_o\), while each physical qubit has degree at most \(v_i+\ell v_o\), where \(w_o,v_o\) and \(v_i\) are the fixed outer weight, outer degree, and inner degree. Hence this is a quantum LDPC family for each fixed gap. This deduction does not bound the constants uniformly as \(\delta\to0\) or provide a practically useful degree bound; it is not the exact-attainment claim in Eq. (3).
The XZZX surface code is an explicit quantum LDPC family whose numerically estimated code-capacity threshold closely matches the zero-rate hashing threshold for every single-qubit Pauli channel; for depolarizing noise its reported threshold is \(18.7(1)\%\). It encodes only \(O(1)\) qubits into \(n=O(d^2)\) qubits, however, so its asymptotic rate is zero [BAT+21].
Kasai reported a nonbinary quantum LDPC code of rate \(1/3\) with \(312{,}000\) physical and \(104{,}000\) logical qubits, attaining frame-error rate \(10^{-4}\) at depolarizing error probability \(p=9.45\%\). This is strong finite-length numerical progress, but at that \(p\) Eq. (2) gives \(R_{\mathrm{hash}}\simeq0.399>1/3\), and no asymptotic capacity-achieving theorem is proved [Kas25].
More recently, fixed-degree quantum LDPC ensembles were proved to have nonvanishing rate and relative distance with high probability; selected finite degree choices attain the CSS Gilbert–Varshamov distance bound. These distance guarantees do not by themselves establish reliable Pauli channel transmission at the rate in Eq. (2) [Kas26].
Quantum polar codes attain the hashing rate without establishing the LDPC constraints. The entanglement-assisted CSS construction of Renes, Dupuis, and Renner has net rate \(1-H(\mathbf p)\) and efficient operations for Pauli channels, but can consume preshared ebits [RDR12]. The later unassisted two-level CSS construction has vanishing error and rate at least \(\max\{0,1-H(\mathbf p)\}\), with \(O(n\log n)\) encoding and decoding once its frozen sets are specified; efficient construction of the outer frozen set was left open [RSDR15]. Neither work proves bounded check weight or bounded qubit degree.

Comment

The dephasing obstruction in Eq. (5) refutes a universal affirmative answer with exact attainment and fixed check weight. It does not classify exact attainment for all other Pauli parameters, so the parameter-dependent question remains Unsolved. Zero hashing rate permits the trivial zero-logical-qubit code. The negative argument is unpublished; its full reasoning is included above. Approaching hashing with LDPC bounds allowed to depend on a positive rate gap is a different quantifier order. The concatenation argument in Eq. (6) already establishes that weaker existence statement, so it should not be relabeled as the remaining open problem. Quantitative bounds on the degrees as a function of the gap would require additional specifications.

References

[BAT+21]: J. P. Bonilla Ataides, D. K. Tuckett, S. D. Bartlett, S. T. Flammia, and B. J. Brown, “The XZZX Surface Code,” Nature Communications 12, 2172 (2021).DOI arXiv
[Kas25]: K. Kasai, “Quantum Error Correction Exploiting Degeneracy to Approach the Hashing Bound” (2025).arXiv
[Kas26]: K. Kasai, “Finite-Degree Quantum LDPC Codes Reaching the Gilbert–Varshamov Bound” (2026).arXiv
[RDR12]: J. M. Renes, F. Dupuis, and R. Renner, “Efficient Polar Coding of Quantum Information,” Physical Review Letters 109, 050504 (2012).DOI arXiv
[RSDR15]: J. M. Renes, D. Sutter, F. Dupuis, and R. Renner, “Efficient Quantum Polar Codes Requiring No Preshared Entanglement,” IEEE Transactions on Information Theory 61, 6395–6414 (2015).DOI arXiv
[Guo26]: N. Guo, “Scientific review: confirm channel, resource and QEC statements in 10 catalog records,” issue 41, item E (2026). Unpublished argument. GitHub issue 41.link
[FSW08]: A. S. Fletcher, P. W. Shor, and M. Z. Win, “Structured Near-Optimal Channel-Adapted Quantum Error Correction,” Physical Review A 77, 012320 (2008).DOI arXiv
[LZ22]: A. Leverrier and G. Zémor, “Quantum Tanner Codes,” 2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS), 872–883 (2022).DOI arXiv
[Ham02]: M. Hamada, “Exponential Lower Bound on the Highest Fidelity Achievable by Quantum Error-Correcting Codes,” Physical Review A 65, 052305 (2002).DOI arXiv

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Bikun Li

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