Minimal Trellises for Decoding Quantum Stabilizer Codes

Abstract:
In 2006, Ollivier and Tillich described a trellis presentation for a quantum code with a stabilizer group S. This allowed for efficient maximum likelihood error estimation by decoding a coset of the code S⊥, the normalizer of S. The decoding can be implemented efficiently using a trellis of the coset of the code S⊥, defined by the syndrome, obtained from the quantum measurement. This semi-tutorial paper presents various approaches to designing a minimal trellis for S⊥ and its cosets, which minimizes the complexity of the maximum likelihood error estimation using the Viterbi algorithm. Most of the results in the paper were obtained directly from the theory of rectangular codes since the code S⊥ is rectangular.