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arXiv:2504.16999 · Nature Computational Science

Learning to decode logical circuits

Yiqing Zhou, Chao Wan, Yichen Xu, Jin Peng Zhou, Kilian Q. Weinberger, Eun-Ah Kim

Published Nov 4, 2025• 6 citations
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Abstract

As quantum hardware advances toward enabling error-corrected quantum circuits in the near future, the absence of an efficient polynomial-time decoding algorithm for logical circuits presents a critical bottleneck. While quantum memory decoding has been well studied, inevitable correlated errors introduced by transversal entangling logical gates prevent the straightforward generalization of quantum memory decoders. Here we introduce a data-centric, modular decoder framework, the Multi-Core Circuit Decoder (MCCD), which consists of decoder modules corresponding to each logical operation supported by the quantum hardware. The MCCD handles both single-qubit and entangling gates within a unified framework. We train MCCD using mirror-symmetric random Clifford circuits, demonstrating its ability to effectively learn correlated decoding patterns. Through extensive testing on circuits substantially deeper than those used in training, we show that MCCD maintains high logical accuracy while exhibiting competitive polynomial decoding time across increasing circuit depths and code distances. When compared with conventional decoders such as minimum weight perfect matching (MWPM), most likely error (MLE) and belief propagation with ordered statistics post-processing (BP-OSD), MCCD achieves competitive accuracy with substantially better time efficiency, particularly for circuits with entangling gates. Our approach provides a noise-model-agnostic solution to the decoding challenge in deep logical quantum circuits.

DOI
10.1038/s43588-025-00897-4

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As quantum hardware advances toward enabling error-corrected quantum circuits in the near future, the absence of an efficient polynomial-time decoding algorithm for logical circuits presents a critical bottleneck. While quantum memory decoding has been well studied, inevitable correlated errors introduced by transversal entangling logical gates prevent the straightforward generalization of quantum memory decoders. Here we introduce a data-centric, modular decoder framework, the Multi-Core Circuit Decoder (MCCD), which consists of decoder modules corresponding to each logical operation supported by the quantum hardware. The MCCD handles both single-qubit and entangling gates within a unified framework. We train MCCD using mirror-symmetric random Clifford circuits, demonstrating its ability to effectively learn correlated decoding patterns. Through extensive testing on circuits substantially deeper than those used in training, we show that MCCD maintains high logical accuracy while exhibiting competitive polynomial decoding time across increasing circuit depths and code distances. When compared with conventional decoders such as minimum weight perfect matching (MWPM), most likely error (MLE) and belief propagation with ordered statistics post-processing (BP-OSD), MCCD achieves competitive accuracy with substantially better time efficiency, particularly for circuits with entangling gates. Our approach provides a noise-model-agnostic solution to the decoding challenge in deep logical quantum circuits.
Fault-tolerant quantum computing demands decoders that are fast, accurate, and adaptable to circuit structure and realistic noise. While machine learning decoders have demonstrated impressive performance for quantum memory, their use in algorithmic decoding—where logical gates create complex error correlations—remains limited. We introduce a modular attention-based neural decoder that learns gate-induced correlations and generalizes from training on random circuits to unseen multiqubit algorithmic workloads. Our decoders achieve fast inference and logical error rates comparable to most-likely-error decoders across varied circuit depths and qubit counts. Addressing realistic noise, we incorporate loss-resolving readout, yielding substantial gains when qubit loss is present. We further show that by tailoring the decoder to the structure of the algorithm and decoding only the relevant observables, we can simplify the decoder design without sacrificing accuracy. We validate our framework on multiple error correction codes—including surface codes and two-dimensional color codes—and demonstrate state-of-the-art performance under circuit-level noise. Finally, we show that the attention mechanism learns to track physically meaningful error correlations, consistent with the expected propagation pathways through entangling gates. Enabling experimental validation of deep-circuit fault-tolerant algorithms and architectures [Bluvstein , ], these results establish neural decoders as practical, versatile, and high-performance tools for quantum computing.
Quantum error correction (QEC) is required for large-scale computation, but incurs a significant resource overhead. Recent advances have shown that by jointly decoding logical qubits in algorithms composed of transversal gates, the number of syndrome extraction rounds can be reduced by a factor of the code distance ddd, at the cost of increased classical decoding complexity. Here, we reformulate the problem of decoding transversal circuits by directly decoding relevant logical operator products as they propagate through the circuit. This procedure transforms the decoding task into one closely resembling that of a single-qubit memory propagating through time. The resulting approach leads to fast decoding and reduced problem size while maintaining high performance. Focusing on the surface code, we prove that this method enables fault-tolerant decoding with minimum-weight perfect matching, and benchmark its performance on example circuits including magic state distillation. We find that the threshold is comparable to that of a single-qubit memory, and that the total decoding run time can be, in fact, less than that of conventional lattice surgery. Our approach enables fast correlated decoding, providing a pathway to directly extend single-qubit QEC techniques to transversal algorithms.
Quantum error correction is believed to be essential for scalable quantum computation, but its implementation is challenging due to its considerable space-time overhead. Motivated by recent experiments demonstrating efficient manipulation of logical qubits using transversal gates [Bluvstein et al., Nature (London) 626, 58 (2024)NATUAS0028-083610.1038/s41586-023-06927-3], we show that the performance of logical algorithms can be substantially improved by decoding the qubits jointly to account for error propagation during transversal entangling gates. We find that such correlated decoding improves the performance of both Clifford and non-Clifford transversal entangling gates, and explore two decoders offering different computational runtimes and accuracies. In particular, by leveraging the deterministic propagation of stabilizer measurement errors, we find that correlated decoding enables the number of noisy syndrome extraction rounds between gates to be reduced from O(d) to O(1) in transversal Clifford circuits, where d is the code distance. We verify numerically that this approach substantially reduces the space-time cost of deep logical Clifford circuits. These results demonstrate that correlated decoding provides a major advantage in early fault-tolerant computation, as realized in recent experiments, and further indicate it has considerable potential to reduce the space-time cost in large-scale logical algorithms.
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.
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Efficient and accurate decoding of quantum error-correcting codes is essential for fault-tolerant quantum computation, however, it is challenging due to the degeneracy of errors, the complex code topology, and the large space for logical operators in high-rate codes. In this work, we propose a decoding algorithm utilizing generative modeling in machine learning. We employ autoregressive neural networks to learn the joint probability of logical operators and syndromes in an unsupervised manner, eliminating the need for labeled training data. The learned model can approximately perform maximum likelihood decoding by directly generating the most likely logical operators for kkk logical qubits with O(2k)\mathcal O(2k)O(2k) computational complexity. Thus, it is particularly efficient for decoding high-rate codes with many logical qubits. The proposed approach is general and applies to a wide spectrum of quantum error-correcting codes including surface codes and quantum low-density parity-check codes (qLDPC), under noise models ranging from code capacity noise to circuit level noise. We conducted extensive numerical experiments to demonstrate that our approach achieves significantly higher decoding accuracy compared to the minimum weight perfect matching and belief propagation with ordered statistics on the surface codes and high-rate quantum low-density parity-check codes. Our approach highlights generative artificial intelligence as a potential solution for the real-time decoding of realistic and high-rate quantum error correction codes.