Random Quantum LDPC Codes Approaching the Gilbert-Varshamov Bound

2026-10-01

AI Summary

This study addresses the challenge of controlling degeneracy in low-density parity-check (LDPC) codes that approach the quantum Gilbert–Varshamov (GV) bound. To this end, this work proposes a quantum adaptation of Thommesen’s concatenation method, leveraging ensembles of random stabilizer codes and coordinate concatenation. By integrating the expander-based distance amplification technique introduced by Alon et al., the proposed framework achieves constructions with low degeneracy. The primary contribution lies in the successful construction of high-rate, low-weight p-ary quantum LDPC codes whose minimum distance approaches the quantum GV bound with high probability. These codes exhibit performance matching that of random stabilizer codes while closely approaching both the erasure channel capacity and the hashing bound for Pauli channels, thereby offering a promising pathway toward practical, near-optimal quantum error correction.

Abstract

We show that for any $R,ε>0$ and any prime $p\geq 2$, there exists an infinite family of $p$-ary quantum low-density parity-check (QLDPC) codes, rate $R$, checks of weight $O_ε(1)$, and normalized distance at least $δ_{\mathrm{GV}}(p,R)-ε$. Here, $δ_{\mathrm{GV}}(p,R)$ denotes the quantum Gilbert-Varshamov (GV) bound for $p$-ary stabilizer codes. In fact, we construct an ensemble of such QLDPC codes, such that a random code from this ensemble is close to the GV bound with high probability. Moreover, this ensemble matches the performance of random stabilizer codes on several quantum channels. Specifically, it approaches the quantum capacity of the erasure channel and the hashing bound for memoryless Pauli channels, including the depolarizing channel. A significant challenge in working with QLDPC codes is that they are necessarily \emph{degenerate}, i.e., contain many low-weight stabilizers. A key contribution of our work is a construction of QLDPC codes with quantitative control on their degeneracy. These codes are obtained by combining known constructions of asymptotically good QLDPC codes with the expander-based distance amplification procedure of Alon, Edmonds, and Luby [FOCS'95]. Our random ensemble is constructed by starting with these low-degeneracy QLDPC codes near the quantum Singleton bound and concatenating each coordinate with a random inner code. This can be viewed as a quantum analogue of Thommesen's construction, and as an LDPC version of a result of Ouyang, with an appropriately designed outer code.

Problem

• Quantum LDPC codes
• Gilbert-Varshamov bound
• degeneracy
• quantum capacity
• stabilizer codes

Innovation

• Quantum LDPC codes
• Gilbert-Varshamov bound
• Degeneracy control
• Distance amplification
• Code concatenation