Spatially Coupled MacKay-Neal/Hsu-Anastasopoulos CSS Codes Achieve the Quantum-Erasure Hashing Bound by Seeded BP Decoding
Abstract
We study hard-erasure belief-propagation (BP), understood as the sum-product algorithm on erasure messages, for a punctured MacKay-Neal/Hsu-Anastasopoulos (MN/HA) representation of Calderbank-Shor-Steane (CSS) codes. For every integer degree triple 2≤ jZ<jX<k≤60, we prove that the constituent potential thresholds are jZ/k and 1−jX/k. The proof combines classical HA-MN duality with a completed exact rational certificate for all 1711 MN degree pairs in this range. The certificate establishes strict positivity on every physical nontrivial fixed-point branch, at every erasure probability in [0,1], and has an independent integer-arithmetic verifier. A reduction of the Z-side recursion and a fixed-channel vector-potential argument then prove seeded density-evolution convergence below the smaller constituent threshold for sufficiently large coupling width and an ideal seed interval at least that wide. Under jZ+jX=k, this threshold equals the quantum-erasure hashing-bound parameter determined by the CSS design rate. In particular, the degree triple (4,8,12) has potential threshold 1/3 without a positivity hypothesis. The numerical example at erasure probability $0.3325$ reproduces the resulting decoding waves. The theorem uses an ideal auxiliary-message seed; its finite-code realization and logical block-error convergence are separate questions.
Related
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BibTeX
@misc{kasai2026spatially,
title = {{Spatially Coupled MacKay-Neal/Hsu-Anastasopoulos CSS Codes Achieve the Quantum-Erasure Hashing Bound by Seeded BP Decoding}},
author = {Kenta Kasai},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2606.32001},
eprint = {2606.32001},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.32001},
}
RIS
TY - UNPB
TI - Spatially Coupled MacKay-Neal/Hsu-Anastasopoulos CSS Codes Achieve the Quantum-Erasure Hashing Bound by Seeded BP Decoding
AU - Kenta Kasai
PY - 2026
DA - 2026/09/13
PB - arXiv
DO - 10.48550/arxiv.2606.32001
UR - https://arxiv.org/abs/2606.32001
AB - We study hard-erasure belief-propagation (BP), understood as the sum-product algorithm on erasure messages, for a punctured MacKay-Neal/Hsu-Anastasopoulos (MN/HA) representation of Calderbank-Shor-Steane (CSS) codes. For every integer degree triple 2≤ j_Z<j_X<k≤60, we prove that the constituent potential thresholds are j_Z/k and 1−j_X/k. The proof combines classical HA-MN duality with a completed exact rational certificate for all 1711 MN degree pairs in this range. The certificate establishes strict positivity on every physical nontrivial fixed-point branch, at every erasure probability in [0,1], and has an independent integer-arithmetic verifier. A reduction of the Z-side recursion and a fixed-channel vector-potential argument then prove seeded density-evolution convergence below the smaller constituent threshold for sufficiently large coupling width and an ideal seed interval at least that wide. Under j_Z+j_X=k, this threshold equals the quantum-erasure hashing-bound parameter determined by the CSS design rate. In particular, the degree triple (4,8,12) has potential threshold 1/3 without a positivity hypothesis. The numerical example at erasure probability $0.3325$ reproduces the resulting decoding waves. The theorem uses an ideal auxiliary-message seed; its finite-code realization and logical block-error convergence are separate questions.
ER -