Year: 2026
Type: preprint
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.
Source: arXiv (Cornell University)
Institution: Stanford University
Language: English
Subfield: Artificial Intelligence
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
Awards: NSF-BSF: AF: Small: Algorithms for Graph-Based Codes, AF: Small: Understanding Expansion Phenomena: Graphical, Hypergraphical, Geometric, and Quantum, CAREER: Optimal High-Dimensional Estimators Using Sum-of-Squares Proof Systems, Collaborative Research: CISE: Large: Cross-Layer Resilience to Silent Data Corruption, NSF-BSF: AF: Small: Advancing Coding Theory Through the Lens of Pseudorandomness
OpenAlex ID: W7220430075