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arXiv:2502.19971

Efficient and Universal Neural-Network Decoder for Stabilizer-Based Quantum Error Correction

Gengyuan Hu, Wanli Ouyang, Chao‐Yang Lu, Lin Chen, Han-Sen Zhong

Published Feb 27, 2025• 0 citations
arXivHTMLPDF

Abstract

Scaling quantum computing to practical applications necessitates reliable quantum error correction. Although numerous correction codes have been proposed, the overall correction efficiency critically limited by the decode algorithms. We introduce GraphQEC, a code-agnostic decoder leveraging machine-learning on the graph structure of stabilizer codes with linear time complexity. GraphQEC demonstrates unprecedented accuracy and efficiency across all tested code families, including surface codes, color codes, and quantum low-density parity-check (QLDPC) codes. For instance, on a distance-12 QLDPC code, GraphQEC achieves a logical error rate of $9.55 \times 10^{-5}$, an 18-fold improvement over the previous best specialized decoder's $1.74 \times 10^{-3}$ under $p=0.005$ physical error rates, while maintaining $157μ$s/cycle decoding speed. Our approach represents the first universal solution for real-time quantum error correction across arbitrary stabilizer codes.

DOI
10.48550/arxiv.2502.19971

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Scaling quantum computing to practical applications necessitates reliable quantum error correction. Although numerous correction codes have been proposed, the overall correction efficiency critically limited by the decode algorithms. We introduce GraphQEC, a code-agnostic decoder leveraging machine-learning on the graph structure of stabilizer codes with linear time complexity. GraphQEC demonstrates unprecedented accuracy and efficiency across all tested code families, including surface codes, color codes, and quantum low-density parity-check (QLDPC) codes. For instance, on a distance-12 QLDPC code, GraphQEC achieves a logical error rate of 9.55×10−59.55 \times 10^{-5}9.55×10−5, an 18-fold improvement over the previous best specialized decoder's 1.74×10−31.74 \times 10^{-3}1.74×10−3 under p=0.005p=0.005p=0.005 physical error rates, while maintaining 157μ157μ157μs/cycle decoding speed. Our approach represents the first universal solution for real-time quantum error correction across arbitrary stabilizer codes.
Quantum computing requires effective error correction strategies to mitigate noise and decoherence. Quantum Low-Density Parity-Check (QLDPC) codes have emerged as a promising solution for scalable Quantum Error Correction (QEC) applications by supporting constant-rate encoding and a sparse parity-check structure. However, decoding QLDPC codes via traditional approaches such as Belief Propagation (BP) suffers from poor convergence in the presence of short cycles. Machine learning techniques like Graph Neural Networks (GNNs) utilize learned message passing over their node features; however, they are restricted to pairwise interactions on Tanner graphs, which limits their ability to capture higher-order correlations. In this work, we propose HyperNQ, the first Hypergraph Neural Network (HGNN)- based QLDPC decoder that captures higher-order stabilizer constraints by utilizing hyperedges-thus enabling highly expressive and compact decoding. We use a two-stage message passing scheme and evaluate the decoder over the pseudo-threshold region. Below the pseudo-threshold mark, HyperNQ improves the Logical Error Rate (LER) up to 84% over BP and 50% over GNN-based strategies, demonstrating enhanced performance over the existing state-of-the-art decoders.
Fault-tolerant quantum computing will require error rates far below those achievable with physical qubits. Quantum error correction (QEC) bridges this gap, but depends on decoders being simultaneously fast, accurate, and scalable. This combination of requirements remains unmet by a machine-learning decoder, nor by any decoder for promising resource-efficient codes such as the color code. Here we introduce AlphaQubit 2, a neural-network decoder that achieves near-optimal logical error rates for both surface and color codes at scale under realistic noise. For the color code, it is orders of magnitude faster than other high-accuracy decoders. We demonstrate real-time decoding faster than 1μs per cycle on commercial accelerators: for the surface code to distance 11, with better accuracy than leading real-time decoders; and the first real-time decoding of the color code to distance 9. These results support the practical application of a wider class of promising QEC codes, and establish a credible path towards high-accuracy, real-time neural decoding at the scales required for fault-tolerant quantum computation.
Quantum error correction (QEC) is essential for scalable quantum computing. However, it requires classical decoders that are fast and accurate enough to keep pace with quantum hardware. While quantum low-density parity-check codes have recently emerged as a promising route to efficient fault tolerance, current decoding algorithms do not allow one to realize the full potential of these codes in practical settings. Here, we introduce a convolutional neural network decoder that exploits the geometric structure of QEC codes, and use it to probe a novel "waterfall" regime of error suppression, demonstrating that the logical error rates required for large-scale fault-tolerant algorithms are attainable with modest code sizes at current physical error rates, and with latencies within the real-time budgets of several leading hardware platforms. For example, for the [144,12,12][144, 12, 12][144,12,12] Gross code, the decoder achieves logical error rates up to ∼17\sim 17∼17x below existing decoders - reaching logical error rates ∼10−10\sim 10^{-10}∼10−10 at physical error p=0.1%p=0.1\%p=0.1% - with 3-5 orders of magnitude higher throughput. This decoder also produces well-calibrated confidence estimates that can significantly reduce the time overhead of repeat-until-success protocols. Taken together, these results suggest that the space-time costs associated with fault-tolerant quantum computation may be significantly lower than previously anticipated.
Quantum computers can solve selected problems more efficiently than classical computers, but current devices are limited by high physical error rates. Quantum Error Correction (QEC) mitigates this by encoding many physical qubits into a logical qubit, with the surface code among the most widely studied approaches. Since syndrome measurements are produced continuously, the decoder must process them fast enough to avoid becoming a system bottleneck, making real-time decoding essential for fault-tolerant quantum computing. While most state-of-the-art real-time decoders rely on Minimum-Weight Perfect Matching (MWPM), we instead use a high-accuracy Graph Neural Network (GNN) that trades higher computational cost for lower logical error rates. To make this GNN practical for real-time decoding, we apply algorithm-hardware co-design. We reduce complexity through hardware-guided pruning and retraining, producing two hardware-friendly models that reduce parameter count by 3.1×3.1\times3.1× and 6.5×6.5\times6.5×. These target, respectively, an average decoding latency of one syndrome cycle and a worst-case latency within one syndrome cycle. We further reduce hardware cost using input-graph filtering and post-training quantization. Based on these optimized models, we propose an FPGA-based architecture for low-latency inference and real-time decoding. Evaluated on surface codes up to distance 7 under circuit-level noise at physical error rate p=10−3p=10^{-3}p=10−3, our decoder outperforms MWPM in decoding accuracy for both average-latency and max-latency settings. It reduces logical error rate by 40% at 1μs1μs1μs average latency, and by 13% under a strict 1μs1μs1μs deadline.
Fault-tolerant quantum computation critically depends on the ability to detect and correct physical qubit errors faster than they accumulate. The surface code (SC) has emerged as the most experimentally promising quantum error correction (QEC) framework due to its high threshold and local stabilizer structure, yet its practical performance is bottlenecked by decoder efficiency. Traditional decoding approaches such as minimum-weight perfect matching (MWPM) achieve near-optimal accuracy under independent noise but struggle with correlated errors, circuit-level noise, and real-time latency constraints. This paper investigates how neural network (NN)-based decoders reduce logical error rates in SC quantum systems by learning complex syndrome-to-correction mappings from data. We propose a hybrid convolutional-recurrent neural architecture that processes stabilizer measurement syndromes across both spatial and temporal dimensions, enabling the decoder to capture correlated error patterns inherent in realistic hardware. Systematic simulations under depolarizing, biased, and circuit-level noise models demonstrate that the proposed neural decoder achieves a logical error rate reduction of up to 41.3% compared to MWPM at a physical error rate of p = 0.008, while maintaining sub-millisecond decoding latency compatible with superconducting qubit cycle times. These results establish that data-driven neural decoders represent a scalable and hardware-compatible pathway toward achieving the logical error rates required for practical fault-tolerant quantum computing.
To leverage the full potential of quantum error-correcting stabilizer codes it is crucial to have an efficient and accurate decoder. Accurate, maximum likelihood, decoders are computationally very expensive whereas decoders based on more efficient algorithms give sub-optimal performance. In addition, the accuracy will depend on the quality of models and estimates of error rates for idling qubits, gates, measurements, and resets, and will typically assume symmetric error channels. In this work, we explore a model-free, data-driven, approach to decoding, using a graph neural network (GNN). The decoding problem is formulated as a graph classification task in which a set of stabilizer measurements is mapped to an annotated detector graph for which the neural network predicts the most likely logical error class. We show that the GNN-based decoder can outperform a matching decoder for circuit level noise on the surface code given only the simulated data, while the matching decoder is given full information of the underlying error model. Although training is computationally demanding, inference is fast and scales approximately linearly with the space-time volume of the code. We also find that we can use large, but more limited, datasets of real experimental data for the repetition code, giving decoding accuracies that are on par with minimum weight perfect matching. The results show that a purely data-driven approach to decoding may be a viable future option for practical quantum error correction, which is competitive in terms of speed, accuracy, and versatility.