Quantum LDPC and High-Rate CSS Codes from Fair-Density Parity-Check Codes
Abstract
We construct quantum LDPC (qLDPC) and high-rate CSS codes from our recently introduced classical fair-density parity-check (FDPC) codes. To this end, we introduce a structured sparsification of FDPC parity-check matrices, which reduces their check weights while preserving the underlying combinatorial structure and distance guarantees. Combined with the hypergraph-product construction, this yields finite-length qLDPC codes with analytically controlled blocklength, dimension, certified distance, and stabilizer weight. For quantum blocklengths N<105, the constructions introduced here span guaranteed rates from approximately 0.35% to 25.8% and certified quantum distances from $12$ to $69$, with stabilizer weights between $8$ and $16$. In the large-blocklength regime, allowing the FDPC order and sparsified check weight to scale moderately with blocklength yields a family of high-rate CSS codes, which we refer to as quantum FDPC (qFDPC) codes, with rate RQ and minimum distance D satisfying RQ = 1−O(1/(loglog N)), D=Ω(N1/4), and stabilizer weight O(log Nloglog N). Finally, the analytically available FDPC weight distribution provides explicit information about the logical operators of the resulting hypergraph-product codes. Over the quantum erasure channel, this structure yields rigorous first-order maximum likelihood (ML) expressions and a higher-order weight-distribution approximation to the ML logical block error probability. This enables estimation of finite-length operating points and the onset of the error-floor regime. To the best of our knowledge, beyond surface-code-type constructions, this is the first finite-rate qLDPC framework to provide both finite-length certified minimum-distance information and an analytical characterization of low-weight logical multiplicities.
Related
- Breakeven demonstration of quantum low-density parity-check codes
- Low-Density Parity-Check Codes of High Girth from Permutation Polynomials
- Optimal-Access Cooperative MSR Codes: Parity-Check Matrix Construction And a Unified Transformation
- A geometric approach to the density of rank-metric codes
- Log Density Spectrum Based Decoding of Polar Codes for Channels with Strong Phase Noise
BibTeX
@misc{mahdavifar2026quantum,
title = {{Quantum LDPC and High-Rate CSS Codes from Fair-Density Parity-Check Codes}},
author = {Hessam Mahdavifar},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.01181},
eprint = {2609.01181},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.01181},
}
RIS
TY - UNPB
TI - Quantum LDPC and High-Rate CSS Codes from Fair-Density Parity-Check Codes
AU - Hessam Mahdavifar
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.01181
UR - https://arxiv.org/abs/2609.01181
AB - We construct quantum LDPC (qLDPC) and high-rate CSS codes from our recently introduced classical fair-density parity-check (FDPC) codes. To this end, we introduce a structured sparsification of FDPC parity-check matrices, which reduces their check weights while preserving the underlying combinatorial structure and distance guarantees. Combined with the hypergraph-product construction, this yields finite-length qLDPC codes with analytically controlled blocklength, dimension, certified distance, and stabilizer weight. For quantum blocklengths N<10^5, the constructions introduced here span guaranteed rates from approximately 0.35% to 25.8% and certified quantum distances from $12$ to $69$, with stabilizer weights between $8$ and $16$. In the large-blocklength regime, allowing the FDPC order and sparsified check weight to scale moderately with blocklength yields a family of high-rate CSS codes, which we refer to as quantum FDPC (qFDPC) codes, with rate R_Q and minimum distance D satisfying R_Q = 1−O(1/(loglog N)), D=Ω(N^1/4), and stabilizer weight O(log Nloglog N). Finally, the analytically available FDPC weight distribution provides explicit information about the logical operators of the resulting hypergraph-product codes. Over the quantum erasure channel, this structure yields rigorous first-order maximum likelihood (ML) expressions and a higher-order weight-distribution approximation to the ML logical block error probability. This enables estimation of finite-length operating points and the onset of the error-floor regime. To the best of our knowledge, beyond surface-code-type constructions, this is the first finite-rate qLDPC framework to provide both finite-length certified minimum-distance information and an analytical characterization of low-weight logical multiplicities.
ER -