Minimal Trellises for non-Degenerate and Degenerate Decoding of Quantum Stabilizer Codes
<|reference_start|>Minimal Trellises for non-Degenerate and Degenerate Decoding of Quantum Stabilizer Codes: 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.<|reference_end|>
@article{stylianou2024minimal,
    title={Minimal Trellises for non-Degenerate and Degenerate Decoding of Quantum
  Stabilizer Codes},
    author={Evagoras Stylianou, Vladimir Sidorenko, Christian Deppe and Holger
  Boche},
    journal={arXiv preprint arXiv:2410.07897},
    year={2024},
    archivePrefix={arXiv},
    eprint={2410.07897},
    primaryClass={cs.IT math.IT}
}