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arXiv:2305.17505

Improved belief propagation decoding algorithm based on decoupling representation of Pauli operators for quantum LDPC codes

Zhengzhong Yi, Zhipeng Liang, Kaixin Zhong, Yulin Wu, Zhou Fang, Xuan Wang

Published May 27, 2023• 0 citations
arXivHTMLPDF

Abstract

We propose a new method called decoupling representation to represent Pauli operators as vectors over $GF(2)$, based on which we propose partially decoupled belief propagation and fully decoupled belief propagation decoding algorithm for quantum low density parity-check codes. These two algorithms have the capability to deal with the correlations between the $X$ part and the $Z$ part of the vectors in symplectic representation, which are introduced by Pauli $Y$ errors. Hence, they can not only apply to CSS codes, but also to non-CSS codes. Under the assumption that there is no measurement error, compared with traditional belief propagation algorithm in symplectic representation over $GF(2)$, within the same number of iterations, the decoding accuracy of partially decoupled belief propagation and fully decoupled belief propagation algorithm is significantly improved in pure $Y$ noise and depolarizing noise, which supports that decoding algorithms of quantum error correcting codes might have better performance in decoupling representation than in symplectic representation. The impressive performance of fully decoupled belief propagation algorithm might promote the realization of quantum error correcting codes in engineering.

DOI
10.48550/arxiv.2305.17505

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We propose a new method called decoupling representation to represent Pauli operators as vectors over GF(2)GF(2)GF(2), based on which we propose partially decoupled belief propagation and fully decoupled belief propagation decoding algorithm for quantum low density parity-check codes. These two algorithms have the capability to deal with the correlations between the XXX part and the ZZZ part of the vectors in symplectic representation, which are introduced by Pauli YYY errors. Hence, they can not only apply to CSS codes, but also to non-CSS codes. Under the assumption that there is no measurement error, compared with traditional belief propagation algorithm in symplectic representation over GF(2)GF(2)GF(2), within the same number of iterations, the decoding accuracy of partially decoupled belief propagation and fully decoupled belief propagation algorithm is significantly improved in pure YYY noise and depolarizing noise, which supports that decoding algorithms of quantum error correcting codes might have better performance in decoupling representation than in symplectic representation. The impressive performance of fully decoupled belief propagation algorithm might promote the realization of quantum error correcting codes in engineering.
Quantum low-density parity-check codes can be decoded using a syndrome based GF(4) belief propagation decoder (where GF denotes Galois field). However, the performance of this decoder is limited both by unavoidable 4-cycles in the code's factor graph and the degenerate nature of quantum errors. For the subclass of CSS codes, the number of 4-cycles can be reduced by breaking an error into an XXX and ZZZ component and decoding each with an individual GF(2)\mathrm{GF}(2)GF(2) based decoder. However, this comes at the expense of ignoring potential correlations between these two error components. We present a number of modified belief propagation decoders that address these issues. We propose a GF(2)\mathrm{GF}(2)GF(2) based decoder for CSS codes that reintroduces error correlations by reattempting decoding with adjusted error probabilities. We also propose the use of an augmented decoder, which has previously been suggested for classical binary low-density parity-check codes. This decoder iteratively reattempts decoding on factor graphs that have a subset of their check nodes duplicated. The augmented decoder can be based on a GF(4)\mathrm{GF}(4)GF(4) decoder for any code, a GF(2)\mathrm{GF}(2)GF(2) decoder for CSS code, or even a supernode decoder for a dual-containing CSS code. For CSS codes, we further propose a GF(2)\mathrm{GF}(2)GF(2) based decoder that combines the augmented decoder with error probability adjustment. We demonstrate the performance of these new decoders on a range of different codes, showing that they perform favorably compared to other decoders presented in literature.
A hybrid belief propagation (BP) decoding algorithm for low-density parity-check code is proposed to accelerate the convergence of decoding and reduce the number of decoding iterations. The hybrid scheme is presented by integrating the BP algorithm with the bit-flipping function which is used as adaptive weighted factor in the BP iteration decoding. The adaptive weighting factor is based on the average weighted reliability measure of the hard output of decoder. The proposed hybrid adaptive weighted iteration scheme can efficiently increase the convergence speed of the belief propagation decoding and therefore improve the error performance. Simulation results show that the proposed algorithm can significantly reduce the decoding complexity and achieve better error performances as compared with the sum-product algorithm.
Abstract Quantum error-correction codes are immeasurable resources for quantum computing and quantum communication. However, the existing decoders are generally incapable of checking node duplication of belief propagation (BP) on quantum low-density parity check (QLDPC) codes. Based on the probability theory in the machine learning, mathematical statistics and topological structure, a GF(4) (the Galois field is abbreviated as GF) augmented model BP decoder with Tanner graph is designed. The problem of repeated check nodes can be solved by this decoder. In simulation, when the random perturbation strength p=0.0115–0.0116 and number of attempts N = 60–70, the highest decoding efficiency of the augmented model BP decoder is obtained, and the low-loss frame error rate (FER) decreases to 7.1975 ×10−5. Hence, we design a novel augmented model decoder to compare the relationship between GF(2) and GF(4) for quantum code [[450,200]] on the depolarization channel. It can be verified that the proposed decoder provides the widely application range, and the decoding performance is better in QLDPC codes.
In this paper, we propose a belief-propagation (BP)-based decoder, termed the Multiple-Bases Belief-Propagation List Decoder (MBBP-LD), for quantum low-density parity-check (QLDPC) codes. The key idea is to generate \emph{structured decoding diversity} by constructing multiple redundant parity-check representations via cycle-free subtree decompositions of the Tanner graph, and running BP decoding in parallel across these representations. This extends the classical Multiple-Bases Belief-Propagation (MBBP) framework to the quantum setting while preserving the linear-time complexity and efficiency of standard BP decoding, and avoids the need for super-linear post-processing. Simulation results demonstrate that MBBP-LD improves upon existing BP-based decoders, including BP with ordered statistics decoding (BP-OSD) and belief propagation with guided decimation (BPGD) across several QLDPC codes, while requiring substantially fewer total BP iterations. For bivariate bicycle codes [[144,12,12]][[144,12,12]][[144,12,12]] and [[288,12,18]][[288,12,18]][[288,12,18]], MBBP-LD achieves up to 20%20\%20% reduction in error rate compared to BPGD and up to 30%30\%30% compared to BP-OSD in the low- and moderate-error regimes. For the larger B1 code [[882,24,18≤d≤24]][[882, 24, 18 \leq d \leq 24]][[882,24,18≤d≤24]], MBBP-LD attains comparable or improved performance relative to BPGD while maintaining BP-like decoding latency under parallel implementation.
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 low-density parity-check (LDPC) codes are a promising family of quantum error-correcting codes for fault tolerant quantum computing with low overhead. Decoding quantum LDPC codes on quantum erasure channels has received more attention recently due to advances in erasure conversion for various types of qubits including neutral atoms, trapped ions, and superconducting qubits. Belief propagation with guided decimation (BPGD) decoding of quantum LDPC codes has demonstrated good performance in bit-flip and depolarizing noise. In this work, we apply BPGD decoding to quantum erasure channels. Using a natural modification, we show that BPGD offers competitive performance on quantum erasure channels for multiple families of quantum LDPC codes. Furthermore, we show that the performance of BPGD decoding on erasure channels can sometimes be improved significantly by either adding damping or adjusting the initial channel log-likelihood ratio for bits that are not erased. More generally, our results demonstrate BPGD is an effective general-purpose solution for erasure decoding across the quantum LDPC landscape.