Minimal Trellises for non-Degenerate and Degenerate Decoding of Quantum Stabilizer Codes
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| Main Authors: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866916431928492032 |
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| author | Stylianou, Evagoras Sidorenko, Vladimir Deppe, Christian Boche, Holger |
| author_facet | Stylianou, Evagoras Sidorenko, Vladimir Deppe, Christian Boche, Holger |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_07897 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Minimal Trellises for non-Degenerate and Degenerate Decoding of Quantum Stabilizer Codes Stylianou, Evagoras Sidorenko, Vladimir Deppe, Christian Boche, Holger Information Theory 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. |
| title | Minimal Trellises for non-Degenerate and Degenerate Decoding of Quantum Stabilizer Codes |
| topic | Information Theory |
| url | https://arxiv.org/abs/2410.07897 |