Abstract Quantum computing has the potential to solve complex problems that are inefficiently handled by classical computation. However, the high sensitivity of qubits to environmental interference and the high error rates in current quantum devices exceed the error correction thresholds required for effective algorithm execution. Therefore, quantum error correction technology is crucial to achieving reliable quantum computing. In this work, we study a topological surface code with a two-dimensional lattice structure that protects quantum information by introducing redundancy across multiple qubits and using syndrome qubits to detect and correct errors. However, errors can occur not only in data qubits but also in syndrome qubits, and different types of errors may generate the same syndromes, complicating the decoding task and creating a need for more efficient decoding methods. To address this challenge, we used a transformer decoder based on an attention mechanism. By mapping the surface code lattice, the decoder performs a self-attention process on all input syndromes, thereby obtaining a global receptive field. The performance of the decoder was evaluated under a phenomenological error model. Numerical results demonstrate that the decoder achieved a decoding accuracy of 93.8%. Additionally, we obtained decoding thresholds of 5% and 6.05% at maximum code distances of 7 and 9, respectively. These results indicate that the decoder used demonstrates a certain capability in correcting noise errors in surface codes.
Abstract Quantum computing has the potential to solve complex problems that are inefficiently handled by classical computation. However, the high sensitivity of qubits to environmental interference and the high error rates in current quantum devices exceed the error correction thresholds required for effective algorithm execution. Therefore, quantum error correction technology is crucial to achieving reliable quantum computing. In this work, we study a topological surface code with a two-dimensional lattice structure that protects quantum information by introducing redundancy across multiple qubits and using syndrome qubits to detect and correct errors. However, errors can occur not only in data qubits but also in syndrome qubits, and different types of errors may generate the same syndromes, complicating the decoding task and creating a need for more efficient decoding methods. To address this challenge, we used a transformer decoder based on an attention mechanism. By mapping the surface code lattice, the decoder performs a self-attention process on all input syndromes, thereby obtaining a global receptive field. The performance of the decoder was evaluated under a phenomenological error model. Numerical results demonstrate that the decoder achieved a decoding accuracy of 93.8%. Additionally, we obtained decoding thresholds of 5% and 6.05% at maximum code distances of 7 and 9, respectively. These results indicate that the decoder used demonstrates a certain capability in correcting noise errors in surface codes.
Abstract To address qubits' high environmental sensitivity and reduce the significant error rates in current quantum devices, quantum error correction stands as one of the most dependable approaches. The topological surface code, renowned for its unique qubit lattice structure, is widely considered a pivotal tool for enabling fault-tolerant quantum computation. Through redundancy introduced across multiple qubits, the surface code safeguards quantum information and identifies errors via state changes captured by syndrome qubits. However, simultaneous errors in data and syndrome qubits substantially escalate decoding complexity. Quantum Generative Adversarial Networks (QGANs) have emerged as promising deep learning frameworks, effectively harnessing quantum advantages for practical tasks such as image processing and data optimization. Consequently, a topological code trainer for quantum-classical hybrid GANs is proposed as an auxiliary model to enhance error correction in machine learning-based decoders, demonstrating significantly improved training accuracy compared to the traditional Minimum Weight Perfect Matching (MWPM) algorithm, which achieves an accuracy of 65 . Numerical experiments reveal that the decoder achieves a fidelity threshold of P=0.1978, substantially surpassing the traditional algorithm's threshold of P=0.1024. To enhance decoding efficiency, a Transformer decoder is integrated, incorporating syndrome error outputs trained via QGANs into its framework. By leveraging its self-attention mechanism, the Transformer effectively captures long-range qubit dependencies at a global scale, enabling high-fidelity error correction over larger dimensions. Numerical validation of the surface code error threshold demonstrates an 8.5 threshold with a correction success rate exceeding 94, whereas the local MWPM decoder achieves only 55 and fails to support large-scale computation at a 4 threshold.
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.
Abstract Quantum error-correcting codes are essential for fault-tolerant quantum computing, as they effectively detect and correct noise-induced errors by distributing information across multiple physical qubits. The subsystem surface code with three-qubit check operators demonstrates significant application potential due to its simplified measurement operations and low logical error rates. However, the existing minimum-weight perfect matching (MWPM) algorithm exhibits high computational complexity and lacks flexibility in large-scale systems. Therefore, this paper proposes a decoder based on a graph attention network (GAT), representing error syndromes as undirected graphs with edge weights, and employing a multi-head attention mechanism to efficiently aggregate node features and enable parallel computation. Compared to MWPM, the GAT decoder exhibits linear growth in computational complexity, adapts to different quantum code structures, and demonstrates stronger robustness under high physical error rates. The experimental results demonstrate that the proposed decoder achieves an overall accuracy of 89.95% under various small code lattice sizes (L = 2,3,4,5), with the logical error rate threshold increasing to 0.0078, representing an improvement of approximately 13.04% compared to the MWPM decoder. This result significantly outperforms traditional methods, showcasing superior performance under small code lattice sizes and providing a more efficient decoding solution for large-scale quantum error correction.
To perform reliable information processing in quantum computers, quantum error-correction (QEC) codes are essential for the detection and correction of errors in the qubits. Among QEC codes, topological QEC codes are designed to interact between the neighboring qubits, which is a promising property for easing the implementation requirements. In addition, the locality to the qubits provides unusual tolerance to local errors. Recently, various decoding algorithms based on machine learning have been proposed to improve the decoding performance and latency of QEC codes. In this work, we propose a new decoding algorithm for surface codes, i.e., a type of topological codes, by using convolutional neural networks (CNN) tailored for the topological lattice structure of the surface codes. In particular, the proposed algorithm takes advantage of the syndrome pattern which is represented as a part of a rectangular lattice given to the CNN as its input. The remaining part of the rectangular lattice is filled with a carefully selected incoherent value for better logical error-rate performance. In addition, we introduce how to optimize the hyper-parameters in the CNN, according to the lattice structure of a given surface code. This reduces the overall decoding complexity and makes the CNN-based decoder computationally more suitable for implementation. The numerical results show that the proposed decoding algorithm effectively improves the decoding performance in terms of logical error-rate as compared to the existing algorithms, on various quantum error models.
Quantum error correction, which utilizes logical qubits that are encoded as redundant multiple physical qubits to find and correct errors in physical qubits, is indispensable for practical quantum computing. Surface code is considered to be a promising encoding method with a high error threshold that is defined by stabilizer generators. However, previous methods have suffered from the problem that the decoder acquires solely the error probability distribution because of the non-uniqueness of correct prediction obtained from the input. To circumvent this problem, we propose a technique to reoptimize the decoder model by approximating syndrome measurements with a continuous function that is mathematically interpolated by neural network. We evaluated the improvement in accuracy of a multilayer perceptron based decoder for code distances of 5 and 7 as well as for decoders based on convolutional and recurrent neural networks and transformers for a code distance of 5. In all cases, the reoptimized decoder gave better accuracy than the original models, demonstrating the universal effectiveness of the proposed method that is independent of code distance or network architecture. These results suggest that re-framing the problem of surface code decoding into a regression problem that can be tackled by deep learning is a useful strategy.
Quantum error correction is essential to achieve fault-tolerant quantum computation in the presence of noisy qubits. Among the most promising approaches to quantum error correction is the surface code, thanks to a scalable two-dimensional architecture, only nearest-neighbor interactions, and a high error threshold. Decoding the surface code, i.e. finding the most likely error chain given a syndrome measurement outcome is a computationally complex task. Traditional decoders rely on classical algorithms, which, especially for larger systems, can be slow and may not always converge to the optimal solution. This thesis presents a novel approach to decoding the surface code using graph neural networks. By mapping the syndrome measurements to a graph and performing graph classification, we find that the graph neural networks can predict the most likely error configuration with high accuracy. Our results show that the GNN-based decoder outperforms the classic minimum weight perfect matching (MWPM) decoder in terms of accuracy. With a phenomenological noise model with depolarizing noise and perfect syndrome measurements, our networks beat MWPM up to code-size 15 across all relevant error rates. Furthermore, the GNN is capable of surpassing MWPM under circuit-level noise up to code size 7. We also show that training the network on repetition code data from a recent experiment [Google Quantum AI, Nature 614, 676 (2023)] produces per-step error rates comparable to those achieved with a matching decoder specifically adapted to the error rates of the physical qubits. This indicates that graph neural network decoders are capable of learning the underlying error distribution on the qubits. Our findings advance the field of quantum error correction and provide a promising new direction for the development of efficient decoding algorithms.