Quantum error-correcting codes (QECCs) are necessary for fault-tolerant quantum computation. Surface codes are a class of topological QECCs that have attracted significant attention due to their exceptional error-correcting capabilities and easy implementation. In the decoding process of surface codes, the syndromes are crucial for error correction, however, they are not always correctly measured. Most of the existing decoding algorithms for surface codes need extra measurements to correct syndromes with errors, which implies a potential increase in inference complexity and decoding latency. In this paper, we propose a high-performance list decoding algorithm for surface codes with erroneous syndromes, where syndrome soft information is incorporated in the decoding, allowing qubits and syndrome to be recovered without needing extra measurements. Precisely, we first use belief propagation (BP) decoding for pre-processing with syndrome soft information, followed by ordered statistics decoding (OSD) for post-processing to list and recover both qubits and syndromes. Numerical results demonstrate that our proposed algorithm efficiently recovers erroneous syndromes and significantly improves the decoding performance of surface codes with erroneous syndromes compared to minimum-weight perfect matching (MWPM), BP and original BP-OSD algorithms.

翻译：量子纠错码（QECCs）是实现容错量子计算所必需的。表面码是一类拓扑量子纠错码，因其优异的纠错能力和易于实现的特点而备受关注。在表面码的译码过程中，综合症对于纠错至关重要，然而它们并非总能被正确测量。现有的大多数表面码译码算法需要额外的测量来纠正含错误的综合症，这意味着推理复杂度和译码延迟可能增加。本文针对含错误综合症的表面码提出了一种高性能列表译码算法，该算法在译码过程中融入了综合症软信息，从而无需额外测量即可恢复量子比特和综合症。具体而言，我们首先利用置信传播（BP）译码结合综合症软信息进行预处理，随后通过有序统计译码（OSD）进行后处理，以列表方式恢复量子比特和综合症。数值结果表明，与最小权重完美匹配（MWPM）、BP以及原始BP-OSD算法相比，我们提出的算法能有效恢复错误综合症，并显著提升了含错误综合症表面码的译码性能。