This paper presents a comprehensive guide to designing minimal trellises for both non-degenerate and degenerate decoding of quantum stabilizer codes. For non-degenerate decoding, various strategies are explored, leveraging insights from classical rectangular codes to minimize the complexity associated with the non-degenerate maximum likelihood error estimation using the Viterbi algorithm. Additionally, novel techniques for constructing minimal multi-goal trellises for degenerate decoding are introduced, including a merging algorithm, a Shannon-product approach, and the BCJR-Wolf method. The study establishes essential properties of multi-goal trellises and provides bounds on the decoding complexity using the sum-product Viterbi decoding algorithm. These advancements decrease the decoding complexity by a factor $\mathcal{O}(n)$, where $n$ is the code length. Finally, the paper applies these results to CSS codes and demonstrates a reduction in complexity by independently applying degenerate decoding to $X$ and $Z$ errors.

翻译：本文全面阐述了针对量子稳定子码非退化与退化解码的最小网格设计方法。对于非退化解码，研究借鉴经典矩形码的洞见，探索了多种策略以降低采用Viterbi算法进行非退化最大似然误差估计的复杂度。同时，论文提出了构建退化解码最小多目标网格的创新技术，包括合并算法、香农积方法以及BCJR-Wolf方法。本研究确立了多目标网格的基本性质，并利用和积Viterbi解码算法给出了解码复杂度的理论界。这些进展将解码复杂度降低了$\mathcal{O}(n)$倍，其中$n$为码长。最后，论文将上述成果应用于CSS码，通过独立对$X$错误和$Z$错误实施退化解码，验证了复杂度的有效降低。