Erasures are the primary type of errors in physical systems dominated by leakage errors. While quantum error correction (QEC) using stabilizer codes can combat erasure errors, it remains unknown which constructions achieve capacity performance. If such codes exist, decoders with linear runtime in the code length are also desired. In this paper, we present erasure capacity-achieving quantum codes under maximum-likelihood decoding (MLD), though MLD requires cubic runtime in the code length. For QEC, using an accurate decoder with the shortest possible runtime will minimize the degradation of quantum information while awaiting the decoder's decision. To address this, we propose belief propagation (BP) decoders that run in linear time and exploit error degeneracy in stabilizer codes, achieving capacity or near-capacity performance for a broad class of codes, including bicycle codes, product codes, and topological codes. We furthermore explore the potential of our BP decoders to handle mixed erasure and depolarizing errors, and also local deletion errors via concatenation with permutation invariant codes.

翻译：在物理系统中，擦除错误是泄漏误差主导下的主要错误类型。尽管基于稳定子码的量子纠错（QEC）能够对抗擦除错误，但哪些构造能达到容量性能仍属未知。若此类编码存在，则同时期望具有码长线性运行时间的解码器。本文中，我们提出了在最大似然解码（MLD）下达到擦除容量的量子码，尽管MLD需要码长的立方运行时间。对于量子纠错而言，使用具有最短可能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。为此，我们提出了运行时间为线性的置信传播（BP）解码器，该解码器利用稳定子码中的错误简并性，对包括自行车码、乘积码和拓扑码在内的广泛码类实现了容量或接近容量的性能。此外，我们探索了所提出的BP解码器处理混合擦除与退极化错误的潜力，以及通过与置换不变码级联来处理局部删除错误的可能性。