Achievability of CSS codes performance limits over Pauli channels under separate ML decoding with bounded graphical complexity
Abstract
Capacity-approaching quantum error correction faces a fundamental tension: the required parity-check matrices need to become increasingly dense, while practical decoding and syndrome extraction favor sparse, bounded-degree structures. Reconciling these competing requirements remains an important open problem in quantum code design. We construct Calderbank-Shor-Steane (CSS) code ensembles that approach the CSS rate over memoryless Pauli channels under maximum-likelihood decoding while retaining sparse, bounded-degree graphical representations. The construction separates parity-check density from graphical complexity: although capacity-approaching components may require dense parity-check matrices, they are represented by sparse concatenated factor graphs. We first study a construction combining low-density parity-check (LDPC) and low-density generator-matrix (LDGM) component codes and numerically analyze the average weight distributions of the resulting codes. Motivated by these results, we introduce a second construction based on concatenated LDPC and LDGM codes and generalized irregular repeat-accumulate (GIRA) codes. We derive bounds on the average weight distribution of the resulting sum ensemble and use the Miller-Burshtein framework to prove simultaneous reliability of the two CSS decoding components. Consequently, the proposed ensemble achieves rates arbitrarily close to the CSS hashing bound under separate bit- and phase-error correction, while maintaining bounded graphical complexity, thus providing a foundation for future iterative decoders and indirect syndrome-extraction architectures.
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BibTeX
@misc{mercanoglu2026achievability,
title = {{Achievability of CSS codes performance limits over Pauli channels under separate ML decoding with bounded graphical complexity}},
author = {Gulnihal Mercanoglu and Achilleas Anastasopoulos and S. Sandeep Pradhan},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.02339},
eprint = {2611.02339},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.02339},
}
RIS
TY - UNPB
TI - Achievability of CSS codes performance limits over Pauli channels under separate ML decoding with bounded graphical complexity
AU - Gulnihal Mercanoglu
AU - Achilleas Anastasopoulos
AU - S. Sandeep Pradhan
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.02339
UR - https://arxiv.org/abs/2611.02339
AB - Capacity-approaching quantum error correction faces a fundamental tension: the required parity-check matrices need to become increasingly dense, while practical decoding and syndrome extraction favor sparse, bounded-degree structures. Reconciling these competing requirements remains an important open problem in quantum code design. We construct Calderbank-Shor-Steane (CSS) code ensembles that approach the CSS rate over memoryless Pauli channels under maximum-likelihood decoding while retaining sparse, bounded-degree graphical representations. The construction separates parity-check density from graphical complexity: although capacity-approaching components may require dense parity-check matrices, they are represented by sparse concatenated factor graphs. We first study a construction combining low-density parity-check (LDPC) and low-density generator-matrix (LDGM) component codes and numerically analyze the average weight distributions of the resulting codes. Motivated by these results, we introduce a second construction based on concatenated LDPC and LDGM codes and generalized irregular repeat-accumulate (GIRA) codes. We derive bounds on the average weight distribution of the resulting sum ensemble and use the Miller-Burshtein framework to prove simultaneous reliability of the two CSS decoding components. Consequently, the proposed ensemble achieves rates arbitrarily close to the CSS hashing bound under separate bit- and phase-error correction, while maintaining bounded graphical complexity, thus providing a foundation for future iterative decoders and indirect syndrome-extraction architectures.
ER -