Quantum stabilizer codes often face the challenge of syndrome errors due to error-prone measurements. To address this issue, multiple rounds of syndrome extraction are typically employed to obtain reliable error syndromes. In this paper, we consider phenomenological decoding problems, where data qubit errors may occur between two syndrome extractions, and each syndrome measurement can be faulty. To handle these diverse error sources, we define a generalized check matrix over mixed quaternary and binary alphabets to characterize their error syndromes. This generalized check matrix leads to the creation of a Tanner graph comprising quaternary and binary variable nodes, which facilitates the development of belief propagation (BP) decoding algorithms to tackle phenomenological errors. Importantly, our BP decoders are applicable to general sparse quantum codes. Through simulations of quantum memory protected by rotated toric codes, we demonstrates an error threshold of 3.3% in the phenomenological noise model. Additionally, we propose a method to construct effective redundant stabilizer checks for single-shot error correction. Simulations show that BP decoding performs exceptionally well, even when the syndrome error rate greatly exceeds the data error rate.

翻译：量子稳定子码常面临因易错测量导致的综合征错误挑战。为解决此问题，通常采用多轮综合征提取来获取可靠的错误综合征。本文考虑现象学解码问题，其中数据量子比特错误可能发生于两次综合征提取之间，且每次综合征测量均可能出错。为处理这些多样化错误来源，我们定义了混合四进制与二进制符号的广义校验矩阵来表征其错误综合征。该广义校验矩阵构建了包含四进制与二进制变量节点的Tanner图，从而促进开发基于置信传播（BP）的解码算法以处理现象学错误。重要的是，我们的BP解码器适用于一般稀疏量子码。通过旋转环面码保护的量子存储器仿真，我们在现象学噪声模型中展示了3.3%的错误阈值。此外，我们提出了一种为单次错误校正构建有效冗余稳定子校验的方法。仿真表明，即使综合征错误率远超数据错误率，BP解码仍能表现出优异性能。