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Minimal Trellises for Degenerate Decoding of Quantum Stabilizer Codes

Evagoras Stylianou, Vladimir R. Sidorenko, Christian Deppe, Holger Boche

Published Dec 8, 2024• 1 citation
Publisher

Abstract

This paper introduces several techniques for minimal trellis construction for degenerate decoding of quantum stabilizer codes, specifically the minimal multi-goal trellis for the cosets of the stabilizer group S in the normalizer group N. The methods include a merging algorithm, a Shannon-product approach, and the BCJR-Wolf method. The study establishes the necessary properties of multi-goal trellises and bounds on the decoding complexity of the minimal multi-goal trellis using the sum-product Viterbi algorithm. The proposed multi-goal trellises decrease the decoding complexity by a factor O(n), where n is the code length.

DOI
10.1109/globecom52923.2024.10901586

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This paper introduces several techniques for minimal trellis construction for degenerate decoding of quantum stabilizer codes, specifically the minimal multi-goal trellis for the cosets of the stabilizer group S in the normalizer group N. The methods include a merging algorithm, a Shannon-product approach, and the BCJR-Wolf method. The study establishes the necessary properties of multi-goal trellises and bounds on the decoding complexity of the minimal multi-goal trellis using the sum-product Viterbi algorithm. The proposed multi-goal trellises decrease the decoding complexity by a factor O(n), where n is the code length.
The size of vertex set of quantum trellises affects the efficiency of decoding algorithms,which can be improved by reducing the number of vertices.Based on the standard check matrix of stabilizer codes,an algorithm to construct trellis-oriented generators for quantum stabilizer codes is presented.By using this algorithm,the trellises with minimal vertex set can be constructed.In addition,an algorithm to construct trellises iteratively for quantum stabilizer codes is also introduced.The algorithm proposed in this paper is more efficient and less complex than the one proposed by Olliver,and so is more suitable for the applications of larger scale and stricter timeliness.
Trellis decoders are a general decoding technique first applied to qubit-based quantum error correction codes by Ollivier and Tillich in 2006. Here, we improve the scalability and practicality of their theory, show that it has strong structure, extend the results using classical coding theory as a guide, and demonstrate a canonical form from which the structural properties of the decoding graph may be computed. The resulting formalism is valid for any prime-dimensional quantum system. The modified decoder works for any stabilizer codeSSSand separates into two parts: 1) a one-time offline computation that builds a compact graphical representation of the normalizer of the code,S⊥\mathcal {S}^{\perp}S⊥and 2) a quick, parallel, online query of the resulting vertices using the Viterbi algorithm. We show the utility of trellis decoding by applying it to four high-density length-20 stabilizer codes for depolarizing noise and the well-studied Steane, rotated surface, and 4.8.8/6.6.6 color codes forZZZonly noise. Numerical simulations demonstrate a 20% improvement in the code-capacity threshold for color codes with boundaries by avoiding the mapping from color codes to surface codes. We identify trellis edge number as a key metric of difficulty of decoding, allowing us to quantify the advantage of single-axis (XXXorZZZ) decoding for Calderbank–Steane–Shor codes and block decoding for concatenated codes.
Iterative decoding techniques have become a viable alternative for constructing high performance coding systems. In particular, the recent success of turbo codes indicates that performance close to the Shannon limit may be achieved. In this thesis, it is showed that many iterative decoding algorithms are special cases of two generic algorithms, the min-sum and sum-product algorithms, which also include non-iterative algorithms such as Viterbi decoding. The min-sum and sum-product algorithms are developed and presented as generalized trellis algorithms, where the time axis of the trellis is replaced by an arbitrary graph, the "Tanner graph". With cycle-free Tanner graphs, the resulting decoding algorithms (e.g., Viterbi decoding) are maximum-likelihood but suffer from an exponentially increasing complexity. Iterative decoding occurs when the Tanner graph has cycles (e.g., turbo codes); the resulting algorithms are in general suboptimal, but significant complexity reductions are possible compared to the cycle-free case. Several performance estimates for iterative decoding are developed, including a generalization of the union bound used with Viterbi decoding and a characterization of errors that are uncorrectable after infinitely many decoding iterations.
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