发表机构
Northeastern University（东北大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文从经典FDPC码构造qLDPC与高码率CSS码，通过结构化稀疏化与超图乘积得到可控参数的有限长码，提出qFDPC码，且是首个兼具有限长认证距离与低权重逻辑多重性解析表征的非表面码类qLDPC框架。
AI 中文摘要
我们从近期提出的经典公平密度奇偶校验（FDPC）码出发，构造量子低密度奇偶校验（qLDPC）码与高码率CSS码。为此，我们引入FDPC奇偶校验矩阵的结构化稀疏化方法，该方法在保留底层组合结构与距离保障的同时降低校验权重。结合超图乘积构造，这可得到具有解析可控块长、维度、认证距离与稳定子权重的有限长qLDPC码。对于量子块长N<10^5，本文引入的构造覆盖了约0.35%至25.8%的保障码率，以及12至69的认证量子距离，稳定子权重介于8至16之间。在大块长区域，允许FDPC阶数与稀疏化校验权重随块长适度缩放，可得到一类高码率CSS码，我们称之为量子FDPC（qFDPC）码，其码率R_Q与最小距离D满足R_Q=1-O(1/log log N)、D=Ω(N^{1/4})，稳定子权重为O(log N log log N)。最后，可解析获取的FDPC权重分布为所得超图乘积码的逻辑算子提供了明确信息。在量子删除信道上，该结构给出了严格的一阶最大似然（ML）表达式，以及ML逻辑块错误概率的高阶权重分布近似，这使得有限长工作点与错误平层区域的起始点可被估计。据我们所知，除表面码类型构造外，这是首个有限码率qLDPC框架，既提供有限长认证最小距离信息，又实现了低权重逻辑多重性的解析表征。
英文摘要
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 \[ R_Q = 1-O\left(\frac{1}{\log\log N}\right), \ \ D=Ω(N^{1/4}), \] and stabilizer weight $O(\log N\log\log N)$. 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.
