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arXiv:2606.32001

Spatially Coupled MacKay-Neal/Hsu-Anastasopoulos CSS Codes Achieve the Quantum-Erasure Hashing Bound by Seeded BP Decoding

Kenta Kasai

quant-phcs.ITSubmitted Jun 30, 2026• Announced Jul 1, 2026• 0 citations• Revised Sep 13, 2026 (v2)
v1 · Jun 30, 2026v2 · Sep 13, 2026
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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 $2\leq j_Z<j_X<k\leq60$, 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.

Comments
32 pages, 8 figures. Completed exact positivity verification for all degree pairs 2 <= j < k <= 60, supplied a self-contained seeded-convergence proof, and clarified the finite-code model and density-evolution scope. Certificates, verification code, and reproduction data are available on GitHub and as ancillary files

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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≤602\leq j_Z<j_X<k\leq602≤jZ​<jX​<k≤60, we prove that the constituent potential thresholds are jZ/kj_Z/kjZ​/k and 1−jX/k1-j_X/k1−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][0,1][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=kj_Z+j_X=kjZ​+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)(4,8,12)(4,8,12) has potential threshold 1/31/31/3 without a positivity hypothesis. The numerical example at erasure probability 0.33250.33250.3325 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.
Kudekar et al. recently proved that for transmission over the binary erasure channel (BEC), spatial coupling of LDPC codes increases the BP threshold of the coupled ensemble to the MAP threshold of the underlying LDPC codes. One major drawback of the capacity-achieving spatially-coupled LDPC codes is that one needs to increase the column and row weight of parity-check matrices of the underlying LDPC codes. It is proved, that Hsu-Anastasopoulos (HA) codes and MacKay-Neal (MN) codes achieve the capacity of memoryless binary-input symmetric-output channels under MAP decoding with bounded column and row weight of the parity-check matrices. The HA codes and the MN codes are dual codes each other. The aim of this paper is to present an empirical evidence that spatially-coupled MN (resp. HA) codes with bounded column and row weight achieve the capacity of the BEC. To this end, we introduce a spatial coupling scheme of MN (resp. HA) codes. By density evolution analysis, we will show that the resulting spatially-coupled MN (resp. HA) codes have the BP threshold close to the Shannon limit.
We construct asymptotically good nested Calderbank-Shor-Steane (CSS) code pairs from Hsu-Anastasopoulos codes and MacKay-Neal codes. For fixed balanced triples with even component degrees and jZ≥4j_Z\ge4jZ​≥4, we prove that the coding rate stays bounded away from zero and that the relative distances on both sides stay bounded away from zero with probability tending to one as the blocklength grows. Moreover, within an explicit low-degree search window, we prove that all 56 balanced triples in that window with even common row degree, column degree at least four, and positive design quantum rate attain the classical Gilbert-Varshamov (GV) bound on both constituent sides, and consequently the CSS GV bound at fixed finite degree.
MacKay-Neal (MN) and Hsu-Anastasopoulos (HA) low-density parity-check (LDPC) codes are known to achieve the capacity of memoryless binary-input symmetric-output channels under maximum likelihood (ML) decoding with bounded col- umn and row weight in their associated parity-check matrices. Recently, Kasai and Sakaniwa showed that spatially coupled (SC) versions of the MN and HA LDPC codes have belief propagation (BP) iterative decoding thresholds that approach capacity on the binary erasure channel (BEC) as the coupling length increases. In this paper, we extend the results of Kasai and Sakaniwa to the additive white Gaussian noise (AWGN) channel and show that the thresholds of the SC-MN and SC-HA ensembles approach capacity with bounded density as the coupling length increases, i.e., the number of edges per information bit approaches a finite value as the estimated BP threshold approaches the Shannon limit. We also perform an asymptotic weight enumerator analysis and show that, provided the density parameters are chosen to be sufficiently large, the SC-MN and SC-HA ensembles are asymptotically good. Further, for certain selections of parameters, some of these ensembles are shown to have both excellent thresholds and good distance properties.
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Several notions of code products are known in quantum error correction, such as hypergraph products, homological products, lifted products, balanced products, to name a few. In this paper we introduce a new product code construction which is a natural generalization of classical product codes to quantum codes: starting from a set of component Calderbank-Shor-Steane (CSS) codes, a larger CSS code is obtained where both X parity checks and Z parity checks are associated to classical product codes. We deduce several properties of product CSS codes from the properties of the component codes, including bounds to the code distance, and show that built-in redundancies in the parity checks result in so-called meta-checks which can be exploited to correct syndrome read-out errors. We then specialize to the case of single-parity-check (SPC) product codes which in the classical domain are a common choice for constructing product codes. Logical error rate simulations of a SPC 3 -fold product CSS code having parameters [[512,174,8]] are shown under both a maximum likelihood decoder for the erasure channel and belief propagation decoding for depolarizing noise. We compare the results with other codes of comparable length and dimension, including a code from the family of asymptotically good Tanner codes. We observe that our reference product CSS code outperforms all the other examined codes.