Abstract Neural‐network (NN) based decoders are becoming increasingly popular in the field of quantum error correction (QEC), including for decoding of state‐of‐the‐art quantum computation experiments. In this work, established interpretability methods are used from the field of machine learning, to introduce a toolbox to achieve an understanding of the underlying decoding logic of NN decoders, which have been trained but otherwise typically operate as black‐box models. To illustrate the capabilities of the employed interpretability method, based on the Shapley value approximation, an examplary case study of a NN decoder is provided that is trained for flag‐qubit based fault‐tolerant (FT) QEC with the Steane code. The interpretation of particular decoding decisions of the NN is analysed, by doing so it is revealed how the NN learns to capture fundamental structures in the information gained from syndrome and flag qubit measurements, in order to come to a FT correction decision. Further, it is showed that the understanding of how the NN obtains a decoding decision can be used on the one hand to identify flawed processing of error syndrome information by the NN, resulting in decreased decoding performance, as well as for well‐informed improvements of the NN architecture. The diagnostic capabilities of the interpretability method presented here can help ensure successful application of machine learning for decoding of QEC protocols.
Abstract Neural‐network (NN) based decoders are becoming increasingly popular in the field of quantum error correction (QEC), including for decoding of state‐of‐the‐art quantum computation experiments. In this work, established interpretability methods are used from the field of machine learning, to introduce a toolbox to achieve an understanding of the underlying decoding logic of NN decoders, which have been trained but otherwise typically operate as black‐box models. To illustrate the capabilities of the employed interpretability method, based on the Shapley value approximation, an examplary case study of a NN decoder is provided that is trained for flag‐qubit based fault‐tolerant (FT) QEC with the Steane code. The interpretation of particular decoding decisions of the NN is analysed, by doing so it is revealed how the NN learns to capture fundamental structures in the information gained from syndrome and flag qubit measurements, in order to come to a FT correction decision. Further, it is showed that the understanding of how the NN obtains a decoding decision can be used on the one hand to identify flawed processing of error syndrome information by the NN, resulting in decreased decoding performance, as well as for well‐informed improvements of the NN architecture. The diagnostic capabilities of the interpretability method presented here can help ensure successful application of machine learning for decoding of QEC protocols.
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.
The article discusses the challenge of finding an efficient decoder for quantum error correction codes for fault-tolerant experiments in quantum computing. The study aims to develop a better decoding scheme based on the flag-bridge fault tolerance experiment. The research compares two decoding algorithms, a deep neural network decoding scheme and a simple decoder, and a recurrent neural network decoding scheme based on the belief propagation algorithm variant MBP 4 algorithm. The study improved the syndrome extraction circuit based on the flag-bridge method to meet the requirements of fault-tolerant experiments better. Two decoding schemes were studied, a combination of a deep neural network and a simple decoder and a recurrent neural network structure based on the MBP 4 algorithm. The first scheme used neural networks to assist simple decoders in determining whether additional logical corrections need to be added. The second scheme used a recurrent neural network structure designed through the variant MBP 4 algorithm, along with a post-processing method to pinpoint the error qubit position for decoding. Experimental results showed that the decoding scheme developed in the study improved the pseudo-threshold by 39.52% compared to the minimum-weight perfect matching decoder. The two decoders had thresholds of approximately 15.8% and 16.4%, respectively. The study’s findings suggest that the proposed decoding schemes could improve quantum error correction and fault-tolerant experiments in quantum computing.
Quantum computation has the promise to solve problems that are not efficiently solvable by classical means. Yet, due to the fragile nature of quantum states, it is unavoidable that quantum systems are subject to decoherence, gate imperfections, and measurement errors. These problems stand as a major roadblock to the construction of large-scale fault-tolerant quantum computers. Conventional quantum error correction (QEC) codes, such as the surface code or Shor code, need to use an extensive amount of redundancy and classical control, which can be both computationally expensive and challenging to scale. In this context, we propose to investigate the use of machine learning (ML) techniques by realizing on-line identification, localization and correction of these quantum errors for fault-tolerant quantum computation by means of fault-tolerant quantum computing processes. Our model introduces a hybrids ML-QEC architecture that combines classical neural networks and decision-tree classifiers with quantum syndrome decoding. These models are trained on error syndromes produced using the approximation of common noise models (for example, depolarizing noise, dephasing and amplitude damping), simulated under different QEC schemes. The machine learning methods are trained with quantum error patterns and have a capability to rapidly infer where errors are likely to occur, as well as recommend correction operations with high confidence. As a result, the system reduces dramatically the delay and computational costs when compared to conventional decoding methods. There is also a self-adaptive learning module in the work, which recommences training on fresh quantum error distributions, meaning the system can adapt to changes in the environment (and also the decay of the hardware) as time progresses. The study further carries out ML predictions inside a real-time loop and cross-checks them with quantum state tomography and entanglement fidelity such as indicators. Experimental characterization on dedicated devices as well as noise models show that our ML-enhanced decoders reach higher fidelity and lower logical error rates compared to baseline decoders, especially in non-uniform or time-dependent noise. This work is a significant stride towards realizing practical fault-tolerant quantum computation by combining quantum error correction with the versatility and scalability of machine learning. With intelligent models to dictate learning, our proposed framework is able to eliminate a lot of trial-and-error manual processes in the conventional heuristic-based approach, and points to a potential path to achieve scalable quantum information processing platforms.
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.
Machine learning has the potential to become an important tool in quantum error correction as it allows the decoder to adapt to the error distribution of a quantum chip. An additional motivation for using neural networks is the fact that they can be evaluated by dedicated hardware which is very fast and consumes little power. Machine learning has been previously applied to decode the surface code. However, these approaches are not scalable as the training has to be redone for every system size which becomes increasingly difficult. In this work the existence of local decoders for higher dimensional codes leads us to use a low-depth convolutional neural network to locally assign a likelihood of error on each qubit. For noiseless syndrome measurements, numerical simulations show that the decoder has a threshold of around 7.1% when applied to the 4D toric code. When the syndrome measurements are noisy, the decoder performs better for larger code sizes when the error probability is low. We also give theoretical and numerical analysis to show how a convolutional neural network is different from the 1-nearest neighbor algorithm, which is a baseline machine learning method.
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.