We study the performance of medium-length quantum LDPC (QLDPC) codes in the depolarizing channel. Only degenerate codes with the maximal stabilizer weight much smaller than their minimum distance are considered. It is shown that with the help of OSD-like post-processing the performance of the standard belief propagation (BP) decoder on many QLDPC codes can be improved by several orders of magnitude. Using this new BP-OSD decoder we study the performance of several known classes of degenerate QLDPC codes including hypergraph product codes, hyperbicycle codes, homological product codes, and Haah's cubic codes. We also construct several interesting examples of short generalized bicycle codes. Some of them have an additional property that their syndromes are protected by small BCH codes, which may be useful for the fault-tolerant syndrome measurement. We also propose a new large family of QLDPC codes that contains the class of hypergraph product codes, where one of the used parity-check matrices is square. It is shown that in some cases such codes have better performance than hypergraph product codes. Finally, we demonstrate that the performance of the proposed BP-OSD decoder for some of the constructed codes is better than for a relatively large surface code decoded by a near-optimal decoder.

中文翻译：

---

具有良好有限长度性能的退化量子 LDPC 码

我们研究了去极化信道中中等长度量子 LDPC (QLDPC) 码的性能。仅考虑最大稳定器重量远小于其最小距离的退化代码。结果表明，在类似 OSD 的后处理的帮助下，标准置信传播 (BP) 解码器在许多 QLDPC 码上的性能可以提高几个数量级。使用这种新的 BP-OSD 解码器，我们研究了几种已知类型的退化 QLDPC 码的性能，包括超图乘积码、超自行车码、同调乘积码和 Haah 三次码。我们还构建了几个有趣的短广义自行车代码示例。他们中的一些人有一个额外的属性，即他们的综合症受到小 BCH 代码的保护，这可能对容错综合症测量有用。我们还提出了一个新的大型 QLDPC 代码系列，其中包含超图乘积代码类，其中使用的奇偶校验矩阵之一是正方形。结果表明，在某些情况下，此类代码比超图乘积代码具有更好的性能。最后，我们证明了所提出的 BP-OSD 解码器对某些构造代码的性能优于由近乎最优解码器解码的相对较大的表面代码。