空间耦合的MacKay-Neal/Hsu-Anastasopoulos CSS码通过种子BP解码达到量子擦除哈希界
Spatially Coupled MacKay-Neal/Hsu-Anastasopoulos CSS Codes Achieve the Quantum-Erasure Hashing Bound by Seeded BP Decoding
AI总结：
本文在密度演化层面证明，对于X/Z等速率族，基于有限度因子图的种子BP解码能够达到量子擦除信道的哈希界。
摘要：
在经典稀疏图编码中，空间耦合是一种机制，通过该机制置信传播（BP）解码达到未耦合系统的最大后验（MAP）或面积阈值性能。由于MacKay-Neal/Hsu-Anastasopoulos（MN/HA）打孔稀疏集成在MAP解码下达到容量，自然要问空间耦合的MN/HA型Calderbank-Shor-Steane（CSS）码是否能在种子BP解码下达到量子擦除信道上的哈希界。我们在硬擦除CSS解码的密度演化（DE）层面回答这个问题。在擦除坐标上，两个二进制泡利分量仍未解决，等价地，擦除量子比特由四个泡利可能性表示。我们首先通过稀疏打孔矩阵和相应的密集奇偶校验矩阵定义CSS集成。对于固定的有限Z侧、X侧和校验度数，然后推导出五消息未耦合DE递归，将其分解为Z侧和X侧组成系统，并定义两个组成势。分别对两个组成应用耦合向量势方法，证明在所得有限度因子图上的种子BP解码达到Z侧度数比和X侧补度数比中的较小者。在X/Z等速率特化中，其中Z侧和X侧组成设计速率相等，该BP阈值是由设计速率确定的哈希界信道参数。因此，本文给出了一个DE层面的证明，即具有有限度因子图的种子BP解码达到了X/Z等速率族的哈希界。有限长度BP集中性、块错误收敛以及理想DE种子的有限码实现是单独的问题。
英文摘要：
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.
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