Quantum LDPC codes at the Pauli hashing bound

Problem

For a probability vector \(\mathbf p=(p_I,p_X,p_Y,p_Z)\), the qubit Pauli channel is

\begin{equation} \Lambda_{\mathbf p}(\rho) =p_I\rho+p_XX\rho X+p_YY\rho Y+p_ZZ\rho Z, \qquad p_I+p_X+p_Y+p_Z=1, \tag{1} \end{equation}

where every \(p_i\ge0\), \(I\) is the identity, and \(X,Y,Z\) are the Pauli operators. The qubit depolarizing channel is the specialization of Eq. (1) given by

\begin{equation} \mathcal D_p(\rho) =(1-p)\rho+\frac p3\bigl(X\rho X+Y\rho Y+Z\rho Z\bigr), \qquad 0\le p\le1. \tag{2} \end{equation}

Thus Eq. (2) has \(\mathbf p=(1-p,p/3,p/3,p/3)\). For the general channel in Eq. (1), define the hashing rate by

\begin{equation} R_{\mathrm{hash}}(\mathbf p) :=\max\{0,1-H(\mathbf p)\}, \qquad H(\mathbf p):=-\sum_{i\in\{I,X,Y,Z\}}p_i\log_2p_i, \qquad 0\log_2 0:=0. \tag{3} \end{equation}

A family of stabilizer codes \([[n,k_n]]\) is quantum low-density parity-check (LDPC) if it has stabilizer generators of uniformly bounded weight and each qubit participates in a uniformly bounded number of generators. Does there exist, for a given \(\mathbf p\), such a family with decoding error tending to zero under \(\Lambda_{\mathbf p}^{\otimes n}\) and asymptotic transmission rate \(R:=\liminf_{n\to\infty}k_n/n\) satisfying \(R\ge R_{\mathrm{hash}}(\mathbf p)\) from Eq. (3)?

Source

Contributor: Bikun Li.

Progress

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. (3) 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. (3) [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

Polar codes attain the hashing rate without LDPC constraints; the other results separately address threshold, finite-block performance, or distance. The open task is a quantum LDPC theorem for Eq. (1) with vanishing decoding error and rate at least Eq. (3).

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

Quantum LDPC codes at the Pauli hashing bound

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