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 has not yet been met by a machine-learning decoder, nor by any decoder for promising resource-efficient codes such as the colour code. Here we introduce AlphaQubit 2, a neural-network decoder that achieves near-optimal logical error rates for both surface and colour codes at large scales under realistic noise. For the colour code, it is orders of magnitude faster than other high-accuracy decoders. For the surface code, we demonstrate real-time decoding faster than 1 microsecond per cycle up to distance 11 on current commercial accelerators with better accuracy than leading real-time decoders. 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.

Real-time, scalable, and accurate decoding is a critical component for realizing a fault-tolerant quantum computer. While Transformer-based neural decoders such as \textit{AlphaQubit} have demonstrated high accuracy, the computational complexity of their core attention mechanism, which scales as $\mathcal{O}(d^4)$ with code distance $d$, results in decoding speeds insufficient for practical real-time applications. In this work, we introduce and evaluate a \textit{Mamba}-based decoder, a state-space model with $\mathcal{O}(d^2)$ complexity. In memory experiments using Sycamore hardware data, our Mamba decoder matches the performance of its Transformer-based counterpart, providing that its superior efficiency does not come at the cost of performance. Crucially, in simulated real-time scenarios that account for decoder-induced noise, the Mamba decoder significantly outperforms the Transformer, exhibiting a higher error threshold of $0.0104$ compared to $0.0097$. These results demonstrate that Mamba decoders offer a compelling balance between speed and accuracy, making them a promising architecture for scalable, real-time quantum error correction.

For reliable large-scale quantum computation, a quantum error correction (QEC) scheme must effectively resolve physical errors to protect logical information. Leveraging recent advances in deep learning, neural network-based decoders have emerged as a promising approach to enhance the reliability of QEC. We propose the Hierarchical Qubit-Merging Transformer (HQMT), a novel and general decoding framework that explicitly leverages the structural graph of stabilizer codes to learn error correlations across multiple scales. Our architecture first computes attention locally on structurally related groups of stabilizers and then systematically merges these qubit-centric representations to build a global view of the error syndrome. The proposed HQMT achieves substantially lower logical error rates for surface codes by integrating a dedicated qubit-merging layer within the transformer architecture. Across various code distances, HQMT significantly outperforms previous neural network-based QEC decoders as well as a powerful belief propagation with ordered statistics decoding (BP+OSD) baseline. This hierarchical approach provides a scalable and effective framework for surface code decoding, advancing the realization of reliable quantum computing.

Quantum error correction is crucial for large-scale quantum computing, but the absence of efficient decoders for new codes like quantum low-density parity-check (QLDPC) codes has hindered progress. Here we introduce a universal decoder based on linear attention sequence modeling and graph neural network that operates directly on any stabilizer code's graph structure. Our numerical experiments demonstrate that this decoder outperforms specialized algorithms in both accuracy and speed across diverse stabilizer codes, including surface codes, color codes, and QLDPC codes. The decoder maintains linear time scaling with syndrome measurements and requires no structural modifications between different codes. For the Bivariate Bicycle code with distance 12, our approach achieves a 39.4% lower logical error rate than previous best decoders while requiring only ~1% of the decoding time. These results provide a practical, universal solution for quantum error correction, eliminating the need for code-specific decoders.

Quantum Error Correction (QEC) is one of the fundamental problems in quantum computer systems, which aims to detect and correct errors in the data qubits within quantum computers. Due to the presence of unreliable data qubits in existing quantum computers, implementing quantum error correction is a critical step when establishing a stable quantum computer system. Recently, machine learning (ML)-based approaches have been proposed to address this challenge. However, they lack a thorough understanding of quantum error correction. To bridge this research gap, we provide a new perspective to understand machine learning-based QEC in this paper. We find that syndromes in the ancilla qubits result from errors on connected data qubits, and distant ancilla qubits can provide auxiliary information to rule out some incorrect predictions for the data qubits. Therefore, to detect errors in data qubits, we must consider the information present in the long-range ancilla qubits. To the best of our knowledge, machine learning is less explored in the dependency relationship of QEC. To fill the blank, we curate a machine learning benchmark to assess the capacity to capture long-range dependencies for quantum error correction. To provide a comprehensive evaluation, we evaluate seven state-of-the-art deep learning algorithms spanning diverse neural network architectures, such as convolutional neural networks, graph neural networks, and graph transformers. Our exhaustive experiments reveal an enlightening trend: By enlarging the receptive field to exploit information from distant ancilla qubits, the accuracy of QEC significantly improves. For instance, U-Net can improve CNN by a margin of about 50%. Finally, we provide a comprehensive analysis that could inspire future research in this field.

Quantum hardware suffers from high error rates and noise, which makes directly running applications on them ineffective. Quantum Error Correction (QEC) is a critical technique towards fault tolerance which encodes the quantum information distributively in multiple data qubits and uses syndrome qubits to check parity. Minimum-Weight-Perfect-Matching (MWPM) is a popular QEC decoder that takes the syndromes as input and finds the matchings between syndromes that infer the errors. However, there are two paramount challenges for MWPM decoders. First, as noise in real quantum systems can drift over time, there is a potential misalignment with the decoding graph's initial weights, leading to a severe performance degradation in the logical error rates. Second, while the MWPM decoder addresses independent errors, it falls short when encountering correlated errors typical on real hardware, such as those in the 2Q depolarizing channel. We propose DGR, an efficient decoding graph edge re-weighting strategy with no quantum overhead. It leverages the insight that the statistics of matchings across decoding iterations offer rich information about errors on real quantum hardware. By counting the occurrences of edges and edge pairs in decoded matchings, we can statistically estimate the up-to-date probabilities of each edge and the correlations between them. The reweighting process includes two vital steps: alignment re-weighting and correlation re-weighting. The former updates the MWPM weights based on statistics to align with actual noise, and the latter adjusts the weight considering edge correlations. Extensive evaluations on surface code and honeycomb code under various settings show that DGR reduces the logical error rate by 3.6x on average-case noise mismatch with exceeding 5000x improvement under worst-case mismatch.

Quantum computing has the potential to solve problems that are intractable for classical systems, yet the high error rates in contemporary quantum devices often exceed tolerable limits for useful algorithm execution. Quantum Error Correction (QEC) mitigates this by employing redundancy, distributing quantum information across multiple data qubits and utilizing syndrome qubits to monitor their states for errors. The syndromes are subsequently interpreted by a decoding algorithm to identify and correct errors in the data qubits. This task is complex due to the multiplicity of error sources affecting both data and syndrome qubits as well as syndrome extraction operations. Additionally, identical syndromes can emanate from different error sources, necessitating a decoding algorithm that evaluates syndromes collectively. Although machine learning (ML) decoders such as multi-layer perceptrons (MLPs) and convolutional neural networks (CNNs) have been proposed, they often focus on local syndrome regions and require retraining when adjusting for different code distances. We introduce a transformer-based QEC decoder which employs self-attention to achieve a global receptive field across all input syndromes. It incorporates a mixed loss training approach, combining both local physical error and global parity label losses. Moreover, the transformer architecture's inherent adaptability to variable-length inputs allows for efficient transfer learning, enabling the decoder to adapt to varying code distances without retraining. Evaluation on six code distances and ten different error configurations demonstrates that our model consistently outperforms non-ML decoders, such as Union Find (UF) and Minimum Weight Perfect Matching (MWPM), and other ML decoders, thereby achieving best logical error rates. Moreover, the transfer learning can save over 10x of training cost.

Quantum error-correction is a prerequisite for reliable quantum computation. Towards this goal, we present a recurrent, transformer-based neural network which learns to decode the surface code, the leading quantum error-correction code. Our decoder outperforms state-of-the-art algorithmic decoders on real-world data from Google's Sycamore quantum processor for distance 3 and 5 surface codes. On distances up to 11, the decoder maintains its advantage on simulated data with realistic noise including cross-talk, leakage, and analog readout signals, and sustains its accuracy far beyond the 25 cycles it was trained on. Our work illustrates the ability of machine learning to go beyond human-designed algorithms by learning from data directly, highlighting machine learning as a strong contender for decoding in quantum computers.

Neural decoders for quantum error correction (QEC) rely on neural networks to classify syndromes extracted from error correction codes and find appropriate recovery operators to protect logical information against errors. Despite the good performance of neural decoders, important practical requirements remain to be achieved, such as minimizing the decoding time to meet typical rates of syndrome generation in repeated error correction schemes, and ensuring the scalability of the decoding approach as the code distance increases. Designing a dedicated integrated circuit to perform the decoding task in co-integration with a quantum processor appears necessary to reach these decoding time and scalability requirements, as routing signals in and out of a cryogenic environment to be processed externally leads to unnecessary delays and an eventual wiring bottleneck. In this work, we report the design and performance analysis of a neural decoder inference accelerator based on an in-memory computing (IMC) architecture, where crossbar arrays of resistive memory devices are employed to both store the synaptic weights of the decoder neural network and perform analog matrix-vector multiplications during inference. In proof-of-concept numerical experiments supported by experimental measurements, we investigate the impact of TiO$_\textrm{x}$-based memristive devices' non-idealities on decoding accuracy. Hardware-aware training methods are developed to mitigate the loss in accuracy, allowing the memristive neural decoders to achieve a pseudo-threshold of $9.23\times 10^{-4}$ for the distance-three surface code, whereas the equivalent digital neural decoder achieves a pseudo-threshold of $1.01\times 10^{-3}$. This work provides a pathway to scalable, fast, and low-power cryogenic IMC hardware for integrated QEC.

We propose a general framework for decoding quantum error-correcting codes with generative modeling. The model utilizes autoregressive neural networks, specifically Transformers, to learn the joint probability of logical operators and syndromes. This training is in an unsupervised way, without the need for labeled training data, and is thus referred to as pre-training. After the pre-training, the model can efficiently compute the likelihood of logical operators for any given syndrome, using maximum likelihood decoding. It can directly generate the most-likely logical operators with computational complexity $\mathcal O(2k)$ in the number of logical qubits $k$, which is significantly better than the conventional maximum likelihood decoding algorithms that require $\mathcal O(4^k)$ computation. Based on the pre-trained model, we further propose refinement to achieve more accurately the likelihood of logical operators for a given syndrome by directly sampling the stabilizer operators. We perform numerical experiments on stabilizer codes with small code distances, using both depolarizing error models and error models with correlated noise. The results show that our approach provides significantly better decoding accuracy than the minimum weight perfect matching and belief-propagation-based algorithms. Our framework is general and can be applied to any error model and quantum codes with different topologies such as surface codes and quantum LDPC codes. Furthermore, it leverages the parallelization capabilities of GPUs, enabling simultaneous decoding of a large number of syndromes. Our approach sheds light on the efficient and accurate decoding of quantum error-correcting codes using generative artificial intelligence and modern computational power.