Minimal Trellises for non-Degenerate and Degenerate Decoding of Quantum Stabilizer Codes

TL;DR

This paper develops minimal trellis structures for efficient decoding of quantum stabilizer codes, significantly reducing complexity for both non-degenerate and degenerate cases using novel algorithms and approaches.

Contribution

It introduces new methods for constructing minimal trellises for quantum codes, including merging, Shannon-product, and BCJR-Wolf techniques, reducing decoding complexity.

Findings

Decoding complexity is reduced by a factor of approximately n.

New trellis construction methods improve degenerate decoding efficiency.

Application to CSS codes demonstrates practical complexity reduction.

Abstract

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.