Field
Quantum Error Correction
24 records: 23 unsolved, 1 solved. Filter the catalog by this field →
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Qutrit magic-distillation thresholds beyond the ternary Golay code
Does any finite qutrit stabilizer-code protocol distill depolarized Strange states above the ternary Golay code’s threshold?
Unsolved -
CNOT-count and depth frontier for quantum Golay state preparation
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?
Unsolved -
Eight-ququart AME state and the ((7,4,4)) ququart MDS code
Does an absolutely maximally entangled state of eight ququarts exist?
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Asymptotically good quantum locally testable codes
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?
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Optimal dimension of nine-qubit distance-three codes
Is the known non-additive \(((9,12,3))_2\) code of largest possible dimension, or does a nine-qubit code of dimension thirteen and distance three exist?
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Good quantum LDPC codes with a non-Clifford transversal CCZ gate
Do there exist triples of qubit CSS LDPC codes with linear combined rate and linear distance on which the transversal CCZ gate induces a non-Clifford logical operation?
Unsolved -
Minimal block length of permutation-invariant qubit codes
Is the least number of physical qubits on which a two-dimensional permutation-invariant code corrects arbitrary errors on \(t\) qubits equal to \(3t^2+3t+1\) for every integer \(t\geq1\)?
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Weight-four generators for an [[11,3,3]] stabilizer code
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?
Unsolved -
Optimal dimension of seven-qubit distance-two codes
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 \(26\)?
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Linear-size logarithmic-depth encoders for good quantum LDPC codes
Does some family of asymptotically good qubit CSS LDPC codes admit unitary encoding circuits with linearly many gates and logarithmic depth?
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Two-error perfect quantum codes over non-prime-power local dimensions
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?
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Distance of constant-rate quantum XYZ product codes
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?
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Minimal permutation-invariant qubit code for two errors
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?
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Square-root distance bound for weight-four stabilizer codes
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?
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Non-additive ((6,2,3)) qubit code with transversal T
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?
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Maximal length of stabilizer quantum MDS codes
Is there a stabilizer quantum maximum-distance-separable code \([[n,k,d]]_q\) with \(k\geq1\) and \(d\geq3\) whose length \(n\) exceeds the conjectured limit \(b(q,d)\) of Eq. (2)?
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Existence of a ((7,3,3)) qubit code
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?
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Quantum LDPC codes at the Pauli hashing bound
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?
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Three-dimensional self-correcting quantum memory
Does there exist a passive self-correcting quantum memory in three spatial dimensions?
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Qubit code distance bound
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
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Unclassified existence parameters for homogeneous AME states
For which of the parameter pairs specified below does an absolutely maximally entangled state exist?
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Minimal-support frontier for absolutely maximally entangled states
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.
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Tough error models
Determine \(c(e,n)\) and construct tough error models in the sense of [KW05].
Unsolved -
Capacity-achieving codes for amplitude damping
What is the constructive quantum code for achieving the quantum capacity \(\mathcal{Q}(\mathcal A_p)\) of the qubit amplitude-damping channel
Solved