Quantum stabilizer codes often struggle with syndrome errors due to measurement imperfections. Typically, multiple rounds of syndrome extraction are employed to ensure reliable error information. In this paper, we consider phenomenological decoding problems, where data qubit errors may occur between extractions, and each measurement can be faulty. We introduce generalized quantum data-syndrome codes along with a generalized check matrix that integrates both quaternary and binary alphabets to represent diverse error sources. This results in a Tanner graph with mixed variable nodes, enabling the design of belief propagation (BP) decoding algorithms that effectively handle phenomenological errors. Importantly, our BP decoders are applicable to general sparse quantum codes. Through simulations, we achieve an error threshold of more than 3\% for quantum memory protected by rotated toric codes, using solely BP without post-processing. Our results indicate that $d$ rounds of syndrome extraction are sufficient for a toric code of distance $d$. We observe that at high error rates, fewer rounds of syndrome extraction tend to perform better, while more rounds improve performance at lower error rates. Additionally, we propose a method to construct effective redundant stabilizer checks for single-shot error correction. Our simulations show that BP decoding remains highly effective even with a high syndrome error rate.

翻译：量子稳定子码常因测量不完善而面临校验子误差的挑战。通常需采用多轮校验子提取以确保可靠的误差信息。本文研究现象学解码问题，其中数据量子比特误差可能在提取间期发生，且每次测量均可能出错。我们提出广义量子数据-校验子码及对应的广义校验矩阵，该矩阵整合四进制与二进制字母表以表征不同误差源，从而构建具有混合变量节点的Tanner图，使得能设计有效处理现象学误差的置信传播解码算法。值得注意的是，我们的BP解码器适用于一般稀疏量子码。通过仿真实验，在仅使用BP解码而无后处理的情况下，受旋转环面码保护的量子存储器实现了超过3%的误差阈值。研究结果表明，距离为d的环面码仅需d轮校验子提取即可。我们观察到在高误差率条件下，较少轮次的校验子提取往往表现更优，而在低误差率时增加轮次可提升性能。此外，我们提出一种为单次误差校正构建有效冗余稳定子校验的方法。仿真表明即使在校验子误差率较高时，BP解码仍能保持卓越效能。