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arXiv:2306.17142

Belief propagation as a partial decoder

Laura Caune, Reid, Brendan, Camps, Joan, Earl T. Campbell

Published Jun 29, 2023• 2 citations
arXivHTMLPDF

Abstract

One of the fundamental challenges in enabling fault-tolerant quantum computation is realising fast enough quantum decoders. We present a new two-stage decoder that accelerates the decoding cycle and boosts accuracy. In the first stage, a partial decoder based on belief propagation is used to correct errors that occurred with high probability. In the second stage, a conventional decoder corrects any remaining errors. We study the performance of our two-stage decoder with simulations using the surface code under circuit-level noise. When the conventional decoder is minimum-weight perfect matching, adding the partial decoder decreases bandwidth requirements, increases speed and improves logical accuracy. Specifically, we observe partial decoding consistently speeds up the minimum-weight perfect matching stage by between $2$x-$4$x on average depending on the parameter regime, and raises the threshold from $0.94\%$ to $1.02\%$.

DOI
10.48550/arxiv.2306.17142

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•Jun 25, 2026•0 citations
One of the fundamental challenges in enabling fault-tolerant quantum computation is realising fast enough quantum decoders. We present a new two-stage decoder that accelerates the decoding cycle and boosts accuracy. In the first stage, a partial decoder based on belief propagation is used to correct errors that occurred with high probability. In the second stage, a conventional decoder corrects any remaining errors. We study the performance of our two-stage decoder with simulations using the surface code under circuit-level noise. When the conventional decoder is minimum-weight perfect matching, adding the partial decoder decreases bandwidth requirements, increases speed and improves logical accuracy. Specifically, we observe partial decoding consistently speeds up the minimum-weight perfect matching stage by between 222x-444x on average depending on the parameter regime, and raises the threshold from 0.94%0.94\%0.94% to 1.02%1.02\%1.02%.
Quantum error correction (QEC) is critical for scalable fault-tolerant quantum computing. Topological codes, such as the toric code, offer hardware-efficient architectures but their Tanner graphs contain many girth-4 cycles that degrade the performance of belief-propagation (BP) decoding. For this reason, BP decoding is typically followed by a more complex second stage decoder such as minimum-weight perfect matching. These combined decoders achieve a remarkable performance, albeit at the cost of increased complexity. In this paper we propose two key improvements for the decoding of toric code. The first one is replacing the BP decoder by a neural BP decoder, giving rise to the neural belief-matching decoder which substantially decreases the average decoding complexity. The main drawback of this approach is the high cost associated with the training of the neural BP decoder. To address this issue, we impose a convolutional architecture on the neural BP decoder, enabling weight sharing across the spatially homogeneous structure of the code's factor graph. This design allows a model trained on a modest-size topological code to be directly transferred to much larger instances, preserving decoding quality while dramatically lowering the training burden. Our numerical experiments on toric-code lattices of various sizes demonstrate that this technique does not result in a noticeable loss in performance.
Fault tolerance is a prerequisite for scalable quantum computing. Architectures based on 2D topological codes are effective for near-term implementations of fault tolerance. To obtain high performance with these architectures, we require a decoder which can adapt to the wide variety of error models present in experiments. The typical approach to the problem of decoding the surface code is to reduce it to minimum-weight perfect matching in a way that provides a suboptimal threshold error rate, and is specialized to correct a specific error model. Recently, optimal threshold error rates for a variety of error models have been obtained by methods which do not use minimum-weight perfect matching, showing that such thresholds can be achieved in polynomial time. It is an open question whether these results can also be achieved by minimum-weight perfect matching. In this work, we use belief propagation and a novel algorithm for producing edge weights to increase the utility of minimum-weight perfect matching for decoding surface codes. This allows us to correct depolarizing errors using the rotated surface code, obtaining a threshold of 17.76 ± 0.02 % . This is larger than the threshold achieved by previous matching-based decoders ( 14.88 ± 0.02 % ), though still below the known upper bound of ∼ 18.9 % .
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.
The standard approach to Belief Propagation (BP) decoding passes messages on the Tanner graph of the quantum error-correcting (QEC) code, and is known to lack a threshold for the surface code. We propose novel BP decoders that exchange messages on the decoding graph and obtain thresholds (both for code capacity and circuit-level noise) via standalone BP for the surface code under depolarizing noise. Our approach, similarly to the minimum weight perfect matching (MWPM) decoder, is applicable to any graphlike QEC code. The thresholds observed with our decoders are close to those obtained by MWPM. This result opens the path towards scalable hardware-accelerated implementations of MWPM-compatible decoders.
Efficient and realistic error decoding is crucial for fault-tolerant quantum computation (FTQC) on near-term devices. While decoding is a classical post-processing task, its effectiveness depends on accurately modeling quantum noise, which is hardware-dependent. In particular, correlated bit-flip (XXX) and phase-flip (ZZZ) errors often arise under circuit-level noise. We introduce the Iterative Reweighting Minimum-Weight Perfect Matching (IRMWPM) decoder, which systematically incorporates such correlations to enhance quantum error correction. Our method leverages fault-detection patterns to guide reweighting: correlated XXX and ZZZ detection events are identified, and their conditional probabilities update weights on the primal and dual lattices. This iterative procedure improves handling of realistic error propagation in a hardware-agnostic yet noise-aware manner. We prove that IRMWPM converges in finite time while preserving the distance guarantee of MWPM. Numerical results under circuit-level noise show substantial improvements. For distances ≥17\geq 17≥17 and physical error rates ≤0.001\leq 0.001≤0.001, IRMWPM reduces logical error rates by over 20x with only a few iterations. It also raises the accuracy threshold from 1% to 1.16%, making it practical for near-term real-time decoding. Extrapolated estimates suggest that to reach logical error rate 10−1610^{-16}10−16, IRMWPM requires distance d=31d=31d=31, while standard MWPM needs d=50d=50d=50, implying a major reduction in qubit overhead.
Abstract One of the critical requirements for fault-tolerant quantum computation is fast and reliable decoding,Minimum-weight perfect matching (MWPM) is a reliable surface-code decoder,but it will face long decoding time when decoding instances grow. Neural-network decoders was used to solve this problem, but large models often require costly retraining when the code distance or noise model changes.In this work, we propose a two-stage decoder that combines pretrained graph neural network (GNN) residual pre-decoding with global MWPM cleanup.The GNN receives a detector graph generated by the established decoding backend and assigns confidence scores to local correction candidates.Two staggered layers of parallel local windows process confidence corrections.The residual syndrome is passed to the NVIDIA-accelerated MWPM backend.On matched batches of 32,768 samples, the complete method reduced the logical failure rate by 6.25% to 18.01% relative to Pure MWPM for code distances d=3,5,7 and 9 at physical error rates p = 0.007 to 0.013. At d=3,it decoded 3.48x – 4.38x faster over p=0.003 to 0.021. We also trained separate GNN residual pre-decoders for four circuit-level noise models over 160 code-distance/error-rate configurations and obtained 2.25x to 5.41x speedups in every case.The logical-failure benefit depended on the noise model,but the pace is all improving. Our work can accelerate surface-code decoding pace and extend to others noise models.