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法，在码率和解码效率方面具有潜在优势。本研究通过双块群代数框架，从马古利斯经典LDPC构造中推导出一类新型QLDPC码——量子马古利斯码。我们证明，与需要有序统计解码才能有效纠错的双变量自行车码不同，量子马古利斯码在码容量噪声模型下可通过标准最小和译码器高效解码，且其复杂度呈线性。这源于其坦纳图结构不具有群对称性，从而缓解了QLDPC解码中著名的误差简并问题。为进一步提升性能，我们提出一种构造可控围长（保证最小围长为6或8）的2BGA码算法，并利用该算法生成多个长度为240和642的量子马古利斯码。通过数值模拟验证，量子马古利斯码在最小和译码下的错误平层区域表现显著优于BB码。