Quantum low-density parity-check codes are a promising approach to fault-tolerant quantum computation, offering potential advantages in rate and decoding efficiency. In this work, we introduce quantum Margulis codes, a new class of QLDPC codes derived from Margulis' classical LDPC construction via the two-block group algebra framework. We show that quantum Margulis codes, unlike bivariate bicycle codes which require ordered statistics decoding for effective error correction, can be efficiently decoded using a standard min-sum decoder with linear complexity, when decoded under the code capacity noise model. This is attributed to their Tanner graph structure, which does not exhibit group symmetry, thereby mitigating the well-known problem of error degeneracy in QLDPC decoding. To further enhance performance, we propose an algorithm for constructing 2BGA codes with controlled girth, ensuring a minimum girth of 6 or 8, and use it to generate several quantum Margulis codes of length 240 and 642. We validate our approach through numerical simulations, demonstrating that quantum Margulis codes behave significantly better than BB codes in the error floor region, under min-sum decoding.

翻译：量子低密度奇偶校验码是容错量子计算的一种有前景的方案，在码率和译码效率方面具有潜在优势。本文通过双块群代数框架，基于Margulis经典LDPC构造引入了一类新型QLDPC码——量子Margulis码。研究表明，与需要有序统计译码才能实现有效纠错的二元自行车码不同，量子Margulis码在码容量噪声模型下可采用标准最小和译码器进行高效译码，其复杂度为线性。这一优势归因于其Tanner图结构缺乏群对称性，从而缓解了QLDPC译码中知名的错误简并问题。为进一步提升性能，我们提出了一种可控围长（保证最小围长为6或8）的2BGA码构造算法，并利用该算法生成了多个长度为240和642的量子Margulis码。通过数值仿真验证，在最小和译码下，量子Margulis码在错误平层区域的表现显著优于二元自行车码。