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Quaternary-Binary Message-Passing Decoder for Quantum LDPC Codes

Dimitris Chytas, Nithin Raveendran, Asit Kumar Pradhan, Bane V. Vasic

Published Dec 4, 2023• 0 citations
Publisher

Abstract

We introduce a low-complexity message-passing quantum error correction algorithm for decoding Quantum Low-Density Parity-Check (QLDPC) stabilizer codes. The proposed decoder operates on the quaternary stabilizer graph but only exchanges binary messages. This leads to a significantly reduced complexity compared to other quaternary belief propagation (BP) algorithms that pass floating-point messages. The efficacy of the proposed decoder is evaluated by providing decoding examples, performance metrics using Monte-Carlo simulations, and complexity analysis. Despite its reduced complexity, the performance loss of the proposed decoder is modest compared to floating-point parallel quaternary decoders for a Calderbank-Shor-Steane (CSS) code family. In particular, experiments obtained over the [[1054, 140, 20]] lifted product (LP) Tanner code demonstrated that for low error rates (< 0.01), the proposed quaternary-binary message-passing decoder approaches the performance of quaternary BP by converging in almost the same number of iterations while requiring less complex operations. Additionally, for non-CSS codes, our decoder performs similarly as quaternary floating-point decoders despite its lower complexity.

DOI
10.1109/globecom54140.2023.10436874

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We introduce a low-complexity message-passing quantum error correction algorithm for decoding Quantum Low-Density Parity-Check (QLDPC) stabilizer codes. The proposed decoder operates on the quaternary stabilizer graph but only exchanges binary messages. This leads to a significantly reduced complexity compared to other quaternary belief propagation (BP) algorithms that pass floating-point messages. The efficacy of the proposed decoder is evaluated by providing decoding examples, performance metrics using Monte-Carlo simulations, and complexity analysis. Despite its reduced complexity, the performance loss of the proposed decoder is modest compared to floating-point parallel quaternary decoders for a Calderbank-Shor-Steane (CSS) code family. In particular, experiments obtained over the [[1054, 140, 20]] lifted product (LP) Tanner code demonstrated that for low error rates (< 0.01), the proposed quaternary-binary message-passing decoder approaches the performance of quaternary BP by converging in almost the same number of iterations while requiring less complex operations. Additionally, for non-CSS codes, our decoder performs similarly as quaternary floating-point decoders despite its lower complexity.
Codes based on sparse matrices have good performance and can be efficiently decoded by belief-propagation (BP). Decoding binary stabilizer codes needs a quaternary BP for (additive) codes over GF(4), which has a higher check-node complexity compared to a binary BP for codes over GF(2). Moreover, BP decoding of stabilizer codes suffers a performance loss from the short cycles in the underlying Tanner graph. In this paper, we propose a refined BP algorithm for decoding quantum codes by passing scalar messages. For a given error syndrome, this algorithm decodes to the same output as the conventional quaternary BP but with a check-node complexity the same as binary BP. As every message is a scalar, the message normalization can be naturally applied to improve the performance. Another observation is that the message-update schedule affects the BP decoding performance against short cycles. We show that running BP with message normalization according to a serial schedule (or other schedules) may significantly improve the decoding performance and error-floor in computer simulation.
Quantum low-density parity-check (QLDPC) codes provide a practical balance between error-correction capability and implementation complexity in quantum error correction (QEC). In this paper, we propose an algebraic construction based on dyadic matrices for designing both classical and quantum LDPC codes. The method first generates classical binary quasi-dyadic LDPC codes whose Tanner graphs have girth 6. It is then extended to the Calderbank-Shor-Steane (CSS) framework, where the two component parity-check matrices are built to satisfy the compatibility condition required by the recently introduced CAMEL-ensemble quaternary belief propagation decoder. This compatibility condition ensures that all unavoidable cycles of length 4 are assembled in a single variable node, allowing the mitigation of their detrimental effects by decimating that variable node.
Quantum stabilizer codes constructed from sparse matrices have good performance and can be efficiently decoded by belief propagation (BP). A conventional BP decoding algorithm treats binary stabilizer codes as additive codes over GF(4). This algorithm has a relatively complex process of handling check-node messages, which incurs higher decoding complexity. Moreover, BP decoding of a stabilizer code usually suffers a performance loss due to the many short cycles in the underlying Tanner graph. In this paper, we propose a refined BP decoding algorithm for quantum codes with complexity roughly the same as binary BP. For a given error syndrome, this algorithm decodes to the same output as the conventional quaternary BP but the passed node-to-node messages are single-valued, unlike the quaternary BP, where multivalued node-to-node messages are required. Furthermore, the techniques of message strength normalization can naturally be applied to these single-valued messages to improve the performance. Another observation is that the message-update schedule affects the performance of BP decoding against short cycles. We show that running BP with message strength normalization according to a serial schedule (or other schedules) may significantly improve the decoding performance and error floor in computer simulation.
Quantum low-density parity-check (QLDPC) codes are promising candidates for error correction in quantum computers. One of the major challenges in implementing QLDPC codes in quantum computers is the lack of a universal decoder. In this work, we first propose to decode QLDPC codes with a belief propagation (BP) decoder operating on overcomplete check matrices. Then, we extend the neural BP (NBP) decoder, which was originally studied for suboptimal binary BP decoding of QLPDC codes, to quaternary BP decoders. Numerical simulation results demonstrate that both approaches as well as their combination yield a low-latency, high-performance decoder for several short to moderate length QLDPC codes.
Quantum low-density parity-check (QLDPC) codes have emerged as a promising technique for quantum error correction. A variety of decoders have been proposed for QLDPC codes and many utilize belief propagation (BP) decoding in some fashion. However, the use of BP decoding for degenerate QLDPC codes is known to have issues with convergence. These issues are typically attributed to short cycles in the Tanner graph and error patterns with the same syndrome due to code degeneracy. In this work, we propose a decoder for QLDPC codes based on BP guided decimation (BPGD), which has been previously studied for constraint satisfaction and lossy compression problems. This decimation process is applicable to both binary and quaternary BP and it involves sequentially freezing the value of the most reliable qubits to encourage BP convergence. We find that BPGD significantly reduces the BP failure rate due to non-convergence, achieving performance on par with BP with ordered statistics decoding and BP with stabilizer inactivation, without the need to solve systems of linear equations. To explore how and why BPGD improves performance, we discuss several interpretations of BPGD and their connection to BP syndrome decoding.
Low complexity decoding algorithms are necessary to meet data rate requirements in excess of 1 Tbps. In this paper, we study one and two bit message passing algorithms for belief propagation decoding of low-density parity-check (LDPC) codes and analyze them by density evolution. The variable nodes (VNs) exploit soft information from the channel output. To decrease the data flow, the messages exchanged between check nodes (CNs) and VNs are represented by one or two bits. The newly proposed quaternary message passing (QMP) algorithm is compared asymptotically and in finite length simulations to binary message passing (BMP) and ternary message passing (TMP) for spectrally efficient communication with higher-order modulation and probabilistic amplitude shaping (PAS). To showcase the potential for high throughput forward error correction, spatially coupled LDPC codes and a target spectral efficiency (SE) of 3 bits/QAM symbol are considered. Gains of about 0.7 dB and 0.1 dB are observed compared to BMP and TMP, respectively. The gap to unquantized belief propagation (BP) decoding is reduced to about 0.75 dB. For smaller code rates, the gain of QMP compared to TMP is more pronounced and amounts to 0.24 dB in the considered example.