Erasures are the primary type of errors in physical systems dominated by leakage errors. While quantum error correction (QEC) using stabilizer codes can combat these error, the question of achieving near-capacity performance with explicit codes and efficient decoders remains a challenge. Quantum decoding is a classical computational problem that decides what the recovery operation should be based on the measured syndromes. For QEC, using an accurate decoder with the shortest possible runtime will minimize the degradation of quantum information while awaiting the decoder's decision. We examine the quantum erasure decoding problem for general stabilizer codes and present decoders that not only run in linear-time but are also accurate. We achieve this by exploiting the symmetry of degenerate errors. Numerical evaluations show near maximum-likelihood decoding for various codes, achieving capacity performance with topological codes and near-capacity performance with non-topological codes. We furthermore explore the potential of our decoders to handle other error models, such as mixed erasure and depolarizing errors, and also local deletion errors via concatenation with permutation invariant codes.

翻译：擦除是泄漏误差主导的物理系统中的主要错误类型。虽然使用稳定子码的量子纠错（QEC）可以对抗这些错误，但如何通过显式编码和高效解码器实现接近容量的性能仍然是一个挑战。量子解码是一个经典计算问题，它根据测量的校验子决定恢复操作。对于QEC，使用尽可能短运行时间的精确解码器将最小化在等待解码器决策期间量子信息的退化。我们研究了通用稳定子码的量子擦除解码问题，并提出不仅运行时间为线性且精确的解码器。我们通过利用退化错误的对称性实现了这一点。数值评估表明，对于各种编码，该解码器接近最大似然解码，对于拓扑编码实现了容量性能，对于非拓扑编码实现了接近容量的性能。此外，我们还探索了我们的解码器处理其他错误模型的潜力，例如混合擦除与去极化错误，以及通过与置换不变码级联处理局部删除错误。