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International Symposium on Turbo Codes and Iterative Information Processing

Analysis of Syndrome-Based Iterative Decoder Failure of QLDPC Codes

Kirsten D. Morris, Tefjol Pllaha, Christine A. Kelley

Published Sep 4, 2023• 2 citations
Publisher

Abstract

Iterative decoder failures of quantum low density parity check (QLDPC) codes are attributed to substructures in the code’s graph, known as trapping sets, as well as degenerate errors that can arise in quantum codes. Failure inducing sets are subsets of codeword coordinates that, when initially in error, lead to decoding failure in a trapping set. In this paper we examine the failure inducing sets of QLDPC codes under syndrome-based iterative decoding, and their connection to absorbing sets in classical LDPC codes.

DOI
10.1109/istc57237.2023.10273573

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Iterative decoder failures of quantum low density parity check (QLDPC) codes are attributed to substructures in the code’s graph, known as trapping sets, as well as degenerate errors that can arise in quantum codes. Failure inducing sets are subsets of codeword coordinates that, when initially in error, lead to decoding failure in a trapping set. In this paper we examine the failure inducing sets of QLDPC codes under syndrome-based iterative decoding, and their connection to absorbing sets in classical LDPC codes.
Iterative decoder failures of quantum low density parity check (QLDPC) codes are attributed to substructures in the code's graph, known as trapping sets, as well as degenerate errors that can arise in quantum codes. Failure inducing sets are subsets of codeword coordinates that, when initially in error, lead to decoding failure in a trapping set. The purpose of this paper is to examine failure inducing sets of QLDPC codes under syndrome-based iterative decoding. As for classical LDPC codes, we show that absorbing sets play a central role in understanding decoder failures. Raveendran and Vasic initiated the study of quantum trapping sets, where beyond the classical-type trapping sets, they identified rigid symmetric structures (a.k.a symmetric stabilizers) responsible for degenerate errors. In this paper, we show that this behavior is part of a much more general phenomenon that can be described by the absorbing set framework.
Iterative decoder failures of quantum low density parity check (QLDPC) codes are attributed to substructures in the code's graph, known as trapping sets, as well as degenerate errors that can arise in quantum codes. Failure inducing sets are subsets of codeword coordinates that, when initially in error, lead to decoding failure in a trapping set. The purpose of this paper is to examine failure inducing sets of QLDPC codes under syndrome-based iterative decoding. As for classical LDPC codes, we show that absorbing sets play a central role in understanding decoder failures. Raveendran and Vasić [11] initiated the study of quantum trapping sets, where beyond the classical-type trapping sets, they identified rigid symmetric structures (a.k.a. symmetric stabilizers) responsible for degenerate errors. In this paper, we show that this behavior is part of a much more general phenomenon that can be described by the absorbing set framework.
Iterative decoders for finite length quantum low-density parity-check (QLDPC) codes are impacted by short cycles, detrimental graphical configurations known as trapping sets (TSs) present in a code graph as well as symmetric degeneracy of errors. In this paper, we develop a systematic methodology by which quantum trapping sets (QTSs) can be defined and categorized according to their topological structure. Conventional definition of a TS from classical error correction is generalized to address the syndrome decoding scenario for QLDPC codes. We show that QTS information can be used to design better QLDPC code and decoder. For certain finite-length QLDPC codes, frame error rate improvements of two orders of magnitude in the error floor regime are demonstrated without needing any post-processing steps.
Quantum low-density parity-check (QLDPC) codes with asymptotically non-zero rates are prominent candidates for achieving fault-tolerant quantum computation, primarily due to the low operational depth of their syndrome-measurement circuits. Numerous studies advocate the necessity of fast decoders to fully harness the capabilities of QLDPC codes, thus driving the focus towards designing low-complexity iterative decoders. However, empirical investigations indicate that such iterative decoders are susceptible to having a high error floor when decoding QLDPC codes. The main objective of this paper is to analyze the decoding failures of thehypergraph-product(HGP) andlifted-product(LP) codes and to design decoders that mitigate these failures, thus achieving a reduced error floor. The suboptimal performance of these codes can predominantly be ascribed to two structural phenomena: (1) stabilizer-induced trapping sets (TS), which correspond to stabilizer-induced subgraphs in the Tanner graphs, and (2) classical trapping sets (TS), which originate from the classical codes used in the construction of HGP and LP codes. The dynamics of stabilizer-induced TSs are examined, and a straightforward modification of iterative decoders is proposed to circumvent these TSs. Moreover, this work proposes a systematic methodology for designing decoders that can circumvent the classical TSs in both HGP and LP codes by deriving them from decoders capable of avoiding the TSs in the parent classical LDPC codes. When decoders that can avoid stabilizer-induced TSs are run in parallel with those that can mitigate the effect of classical TSs, the logical error rate improves significantly in the error-floor region.
Iterative decoders for finite length quantum low-density parity-check (QLDPC) codes are attractive because their hardware complexity scales only linearly with the number of physical qubits. However, they are impacted by short cycles, detrimental graphical configurations known as trapping sets (TSs) present in a code graph as well as symmetric degeneracy of errors. These factors significantly degrade the decoder decoding probability performance and cause so-called error floor. In this paper, we establish a systematic methodology by which one can identify and classify quantum trapping sets (QTSs) according to their topological structure and decoder used. The conventional definition of a TS from classical error correction is generalized to address the syndrome decoding scenario for QLDPC codes. We show that the knowledge of QTSs can be used to design better QLDPC codes and decoders. Frame error rate improvements of two orders of magnitude in the error floor regime are demonstrated for some practical finite-length QLDPC codes without requiring any post-processing.
Adding redundancy to the parity-check matrix representation has been shown to improve iterative decoder performance in the classical setting due to the consequent removal of small stopping sets or other structures that adversely affect the performance. In this paper, we examine the effect of redundancy on iterative syndrome decoder performance, with the goal of improving decoding for quantum LDPC codes. In particular, we provide a sufficient condition for the graph representation of a code to guarantee that the syndrome decoder converges to the input syndrome for any given input error.