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IEEE Communications Letters

BP-LCGCD: A Gaussian-Elimination-Free and High-Performance Decoder for Surface Codes

Qianfan Wang, Jifan Liang, Lvzhou Li, Linqi Song, Xiao Ma

Published Dec 22, 2025• 1 citation
Publisher

Abstract

Belief propagation (BP) combined with ordered statistics decoding (OSD) can achieve near-optimal logical error rates for surface codes. However, OSD requires high-latency and unstable-complexity Gaussian elimination (GE), limiting its practicality. In this paper, we propose BP-LCGCD, a GE-free and high-performance decoder that replaces the GE-based OSD with the GE-free LC-GCD. Moreover, in contrast to the original BP-OSD, which adopts a single fixed normalization factor α, we further propose a list-parameterized variant, BP-LCGCD+ α, which performs multiple BP decodings with different α to generate diverse posterior LLRs. We present complexity analysis, demonstrating that at low physical error rates, the average decoding complexity of the proposed algorithm approaches that of standard BP. Simulation results demonstrate that BP-LCGCD achieves logical error rates close to BP-OSD, while BP-LCGCD+α nearly matches the performance of the BP-OSD. They also show that both proposed decoders significantly outperform standard BP and minimum-weight perfect matching (MWPM) in terms of logical error rate and threshold.

DOI
10.1109/lcomm.2025.3646724

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Belief propagation (BP) combined with ordered statistics decoding (OSD) can achieve near-optimal logical error rates for surface codes. However, OSD requires high-latency and unstable-complexity Gaussian elimination (GE), limiting its practicality. In this paper, we propose BP-LCGCD, a GE-free and high-performance decoder that replaces the GE-based OSD with the GE-free LC-GCD. Moreover, in contrast to the original BP-OSD, which adopts a single fixed normalization factor α, we further propose a list-parameterized variant, BP-LCGCD+ α, which performs multiple BP decodings with different α to generate diverse posterior LLRs. We present complexity analysis, demonstrating that at low physical error rates, the average decoding complexity of the proposed algorithm approaches that of standard BP. Simulation results demonstrate that BP-LCGCD achieves logical error rates close to BP-OSD, while BP-LCGCD+α nearly matches the performance of the BP-OSD. They also show that both proposed decoders significantly outperform standard BP and minimum-weight perfect matching (MWPM) in terms of logical error rate and threshold.
Quantum error correction (QEC) requires accurate and efficient decoders, yet belief propagation (BP), despite its linear decoding complexity, often provides insufficient logical accuracy on surface codes. We propose Logical Neural Belief Propagation (L-NBP), a BP-based neural decoder that redirects the decoding objective from physical-level to logical-level decoding. L-NBP uses a neural BP (NBP) module to produce posterior beliefs, which a logical classifier transforms into a continuous-valued soft syndrome for logical-operator prediction. Trained end-to-end by backpropagation, the NBP module learns soft syndromes that are favorable for logical classification. On surface codes, L-NBP matches or outperforms BP with ordered-statistics decoding (BP-OSD) and minimum-weight perfect matching (MWPM) while retaining the linear complexity of BP, and achieves a threshold of 17.5%17.5\%17.5% under depolarizing noise. Under circuit-level noise, L-NBP matches the accuracy of BP-OSD on the distance-999 surface code while requiring only 0.2%0.2\%0.2% of its complexity.
Belief propagation (BP) combined with ordered statistics decoding (OSD) achieves near-optimal frame error rate (FER) for short LDPC codes, but the extensive re-encoding overhead of OSD limits its practicality. To address this, we propose the BPORBOSD, a low-complexity and high-performance decoder that replaces conventional OSD with the ordered reliability bits OSD (ORBOSD), where the test error patterns are produced by the ORB technique. Moreover, in contrast to conventional BP-OSD, which directly adopts the soft information from standard BP into OSD after BP fails, we employ a modified BP scheme with a normalization factor α to produce soft reliability metrics better suited for ORBOSD. We derive the decoder’s complexity and demonstrate that, in the high-SNR regime, the average decoding complexity of BPORBOSD approaches that of standard BP. Simulation results confirm that BP-ORBOSD achieves FERs close to those of conventional BPOSD and the finite-length bound while significantly reducing the average number of re-encodings.
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Quantum error-correcting codes (QECCs) are necessary for fault-tolerant quantum computation. Surface codes are a class of topological QECCs that have attracted significant attention due to their exceptional error-correcting capabilities and easy implementation. In the decoding process of surface codes, the syndromes are crucial for error correction, however, they are not always correctly measured. Most of the existing decoding algorithms for surface codes need extra measurements to correct syndromes with errors, which implies a potential increase in inference complexity and decoding latency. In this paper, we propose a high-performance list decoding algorithm for surface codes with erroneous syndromes, where syndrome soft information is incorporated in the decoding, allowing qubits and syndrome to be recovered without needing extra measurements. Precisely, we first use belief propagation (BP) decoding for pre-processing with syndrome soft information, followed by ordered statistics decoding (OSD) for post-processing to list and recover both qubits and syndromes. Numerical results demonstrate that our proposed algorithm efficiently recovers erroneous syndromes and significantly improves the decoding performance of surface codes with erroneous syndromes compared to minimum-weight perfect matching (MWPM), BP and original BP-OSD algorithms.
Quantum error correction (QEC) for fault-tolerant quantum computing requires a balanced decoding solution that offers high performance, low complexity, and low latency. However, the de facto standard - belief propagation (BP) combined with ordered statistics decoding (OSD) - suffers from excessive iterations in the BP stage and high complexity in the OSD stage. To address these challenges, we propose an evolutionary BP (EBP) decoder optimized via a differential evolution (DE) algorithm. By leveraging the gradient-free nature of DE, we enable end-to-end optimization of the EBP+OSD structure to maximize overall performance. In addition, a multi-objective selection rule is introduced to suppress frequent OSD activation, significantly reducing complexity overhead. Simulation results on surface codes and quantum low-density parity-check (QLDPC) codes demonstrate that EBP+OSD simultaneously achieves both superior decoding performance and substantially lower complexity compared to conventional BP+OSD, particularly in stringent low-latency regimes.
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.