Abstract Finding efficient decoders for quantum error correcting codes adapted to realistic experimental noise in fault-tolerant devices represents a significant challenge. In this paper we introduce several decoding algorithms complemented by deep neural decoders (DND) and apply them to analyze several fault-tolerant error correction (EC) protocols such as the surface code as well as Steane and Knill EC. Our methods require no knowledge of the underlying noise model afflicting the quantum device making them appealing for real-world experiments. Our analysis is based on a full circuit-level noise model. It considers both distance-three and five codes, and is performed near the codes pseudo-threshold regime. Training DND in low noise rate regimes appears to be a challenging machine learning endeavour. We provide a detailed description of our neural network architectures and training methodology. We then discuss both the advantages and limitations of DND. Lastly, we provide a rigorous analysis of the decoding runtime of trained DND and compare our methods with anticipated gate times in future quantum devices. Given the broad applications of our decoding schemes, we believe that the methods presented in this paper could have practical applications for near term fault-tolerant experiments.
Abstract Finding efficient decoders for quantum error correcting codes adapted to realistic experimental noise in fault-tolerant devices represents a significant challenge. In this paper we introduce several decoding algorithms complemented by deep neural decoders (DND) and apply them to analyze several fault-tolerant error correction (EC) protocols such as the surface code as well as Steane and Knill EC. Our methods require no knowledge of the underlying noise model afflicting the quantum device making them appealing for real-world experiments. Our analysis is based on a full circuit-level noise model. It considers both distance-three and five codes, and is performed near the codes pseudo-threshold regime. Training DND in low noise rate regimes appears to be a challenging machine learning endeavour. We provide a detailed description of our neural network architectures and training methodology. We then discuss both the advantages and limitations of DND. Lastly, we provide a rigorous analysis of the decoding runtime of trained DND and compare our methods with anticipated gate times in future quantum devices. Given the broad applications of our decoding schemes, we believe that the methods presented in this paper could have practical applications for near term fault-tolerant experiments.
Abstract. Suppressing qubit errors is one of the main challenges in building fault-tolerant quantum computers, requiring quantum error correction to protect fragile quantum information during computation cycles. The surface code is a leading approach because of its high error tolerance and compatibility with realistic hardware, but it requires a decoder that can accurately and quickly use the syndrome measurements to discern whether a logical error has occurred. This study investigates whether a feedforward neural-network can serve as a practical decoder for the surface code under circuit-level noise. Using the Stim software package, this study generates surface code circuits at multiple code distances and physical error rates and compiles them into detector samplers, then trains a PyTorch neural-network to predict whether a logical observable flip occurred from the syndrome data. The neural-network decoder is compared against Minimum-Weight Perfect Matching (MWPM), focusing on metrics including logical error rate and inference speed. The results show that the neural-network decoder performs competitively with MWPM for code distances 3 and 5 and provides inference time advantages at higher physical error rates as its inference time remains relatively constant over increasing physical error rates. However, MWPM outperforms the neural-network at distance 7, suggesting that scalability remains a major limitation for neural-network decoders. These results show that neural-networks are a promising approach for tackling quantum error correction, although future work is still needed to improve these models for real systems at larger code distances.
The practical realization of fault-tolerant quantum computing critically hinges on the capacity of quantum error correction (QEC) schemes to robustly mitigate physical noise. Among the foremost candidates, the surface code is distinguished by its high threshold and local stabilizer checks. In recent years, neural network-based decoders, particularly those leveraging deep reinforcement learning (RL) and convolutional architectures, have demonstrated competitive performance against algorithmic staples such as Minimum Weight Perfect Matching (MWPM) and Union-Find. However, the vast majority of these learned decoders are trained and evaluated under idealized phenomenological or circuit-level depolarizing noise models. Real-world quantum processors exhibit complex, compound noise profiles encompassing crosstalk, leakage, non-Markovian temporal drift, and spatially correlated errors. In this work, we empirically evaluate the robustness of state-of-the-art neural decoders against highly structured compound noise models. We introduce a comprehensive stress-testing methodology tailored to quantify the degradation of the logical error rate when decoders are subjected to out-of-distribution noise characteristics. Our empirical results reveal that while neural decoders can overfit to their training noise profile, strategically designed training curricula and architecture regularizations enable significant resilience. We provide extensive numerical benchmarks validating that learned decoders can maintain near-optimal pseudothresholds across a broad spectrum of shifting noise conditions, highlighting their readiness for real-world superconducting qubit deployments.
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.
The practical realization of fault-tolerant quantum computing critically hinges on the capacity of quantum error correction (QEC) schemes to robustly mitigate physical noise. Among the foremost candidates, the surface code is distinguished by its high threshold and local stabilizer checks. In recent years, neural network-based decoders, particularly those leveraging deep reinforcement learning (RL) and convolutional architectures, have demonstrated competitive performance against algorithmic staples such as Minimum Weight Perfect Matching (MWPM) and Union-Find. However, the vast majority of these learned decoders are trained and evaluated under idealized phenomenological or circuit-level depolarizing noise models. Real-world quantum processors exhibit complex, compound noise profiles encompassing crosstalk, leakage, non-Markovian temporal drift, and spatially correlated errors. In this work, we empirically evaluate the robustness of state-of-the-art neural decoders against highly structured compound noise models. We introduce a comprehensive stress-testing methodology tailored to quantify the degradation of the logical error rate when decoders are subjected to out-of-distribution noise characteristics. Our empirical results reveal that while neural decoders can overfit to their training noise profile, strategically designed training curricula and architecture regularizations enable significant resilience. We provide extensive numerical benchmarks validating that learned decoders can maintain near-optimal pseudothresholds across a broad spectrum of shifting noise conditions, highlighting their readiness for real-world superconducting qubit deployments.
Decoding is an essential component of quantum error correction (QEC), translating stabilizer measurement outcomes into corrective actions that suppress logical errors and preserve logical quantum information. Building fault-tolerant architectures requires increasing the code distance, which in turn places growing demands on decoding accuracy, scalability, and practical deployability. While a wide range of decoding algorithms have been proposed and demonstrated, achieving reliable, scalable, and real-time decoding remains a significant challenge. Machine-learning (ML) approaches are particularly well suited to this setting, as quantum error decoding is fundamentally a problem of processing large volumes of classical data with complex spatiotemporal correlations. This chapter surveys ML-based methods for quantum error decoding, with a focus on topological codes and an emphasis on architectural principles, practical performance, and real-time considerations. We first frame decoding as a learning problem and outline key paradigms, including discriminative, generative, and reinforcement-learning formulations. We then introduce the neural network building blocks that underpin most contemporary neural decoders and discuss how these components can be integrated to balance expressivity, scalability, and latency. Building on this architectural perspective, we review recent progress and benchmarks in neural decoding for memory experiments, and discuss real-time decoding, open challenges, and future directions toward scalable fault-tolerant quantum computing.
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.