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Chinese Physics B

Quantum toric code decoding method based on syndrome-preliminary error fusion module and ResNet architecture

Naihua Ji, Ping-Li 平俐 Song 宋, Wei Wang, Hui-Qian 汇倩 Sun 孙, Hongyang Ma

Published Oct 28, 2025• 0 citations
Publisher

Abstract

Abstract Quantum error correction technology is based on the principle of redundant encoding, encoding logical quantum information into multiple physical qubits to provide important support for the stable operation of quantum computers. To address the issues of low decoding accuracy and limited feature extraction in quantum error correction, this paper proposes a toric code decoder based on a syndrome-preliminary error fusion module (SPEFM) and a ResNet architecture. This decoder takes full advantage of the correlations between X and Z errors. In the SPEFM, the syndrome and preliminary error predictions are deeply fused, while a unidirectional Swin transformer architecture is incorporated to extract global error features from the syndrome data, significantly improving both decoding accuracy and computational efficiency. In addition, this paper further extracts local error features from the fused features using the deep residual structure of ResNet, enhancing the decoder’s ability to capture quantum error patterns. Experimental results show that the decoder is applicable to different code distances ( d = 4, 6, 8, 10) under the depolarizing noise model. Its bit error rate is lower than that of the minimum weight perfect matching (MWPM) algorithm, and its logical error rate is lower than both the MWPM algorithm and the ResNet18 decoder. Furthermore, the decoding threshold is increased to 0.163, representing a 3.82% improvement over the MWPM algorithm threshold of 0.157.

DOI
10.1088/1674-1056/ae1818

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Abstract Quantum error correction technology is based on the principle of redundant encoding, encoding logical quantum information into multiple physical qubits to provide important support for the stable operation of quantum computers. To address the issues of low decoding accuracy and limited feature extraction in quantum error correction, this paper proposes a toric code decoder based on a syndrome-preliminary error fusion module (SPEFM) and a ResNet architecture. This decoder takes full advantage of the correlations between X and Z errors. In the SPEFM, the syndrome and preliminary error predictions are deeply fused, while a unidirectional Swin transformer architecture is incorporated to extract global error features from the syndrome data, significantly improving both decoding accuracy and computational efficiency. In addition, this paper further extracts local error features from the fused features using the deep residual structure of ResNet, enhancing the decoder’s ability to capture quantum error patterns. Experimental results show that the decoder is applicable to different code distances ( d = 4, 6, 8, 10) under the depolarizing noise model. Its bit error rate is lower than that of the minimum weight perfect matching (MWPM) algorithm, and its logical error rate is lower than both the MWPM algorithm and the ResNet18 decoder. Furthermore, the decoding threshold is increased to 0.163, representing a 3.82% improvement over the MWPM algorithm threshold of 0.157.
Realizing the full potential of quantum computation requires Quantum Error Correction (QEC). QEC reduces error rates by encoding logical information across redundant physical qubits, enabling errors to be detected and corrected. A common decoder used for this task is Minimum Weight Perfect Matching (MWPM) a graph-based algorithm that relies on edge weights to identify the most likely error chains. In this work, we propose a data-driven decoder named Neural Minimum Weight Perfect Matching (NMWPM). Our decoder utilizes a hybrid architecture that integrates Graph Neural Networks (GNNs) to extract local syndrome features and Transformers to capture long-range global dependencies, which are then used to predict dynamic edge weights for the MWPM decoder. To facilitate training through the non-differentiable MWPM algorithm, we formulate a novel proxy loss function that enables end-to-end optimization. Our findings on the toric code under depolarizing noise demonstrate thresholds of 17.9% and 10.95%, nearing the 18.9% and 11.0% maximum likelihood bounds, highlighting the advantage of hybrid decoders that combine the predictive capabilities of neural networks with the algorithmic structure of classical matching.
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 platforms based on arrays of tweezer-confined neutral atoms have recently emerged as a competitive modality thanks to a direct path toward high qubit count, rapidly advancing operation fidelities, and their ability to execute circuits with arbitrary qubit connectivity. These features will enable the use of efficient error correction schemes with high encoding-rates, time-efficient decoding, and resource-efficient architectures based on transversal gates. With these goals in mind, recent state of the art neutral atom demonstrations focus on the transition from the use of physical qubits to error-corrected logical qubits, but to date there has been no demonstration of repeated error correction scalable to arbitrary depth. Here, we demonstrate many cycles of syndrome extraction in a toric quantum error correcting code, using mid-circuit measurement and replacement of lost qubits, including reloading of a qubit reservoir for indefinite coherent operation. We characterize the logical error rate after up to 90 cycles, showing that logical information can be preserved through multiple rounds of qubit reloading. Comparing two distances of the code up to 8 rounds of syndrome extraction shows a lower absolute logical error rate for the larger distance code.
In the NISQ era, one of the most important bottlenecks for the realization of universal quantum computation is error correction. Stabiliser code is the most recognizable type of quantum error correction code. A scalable efficient decoder is most desired for the application of the quantum error correction codes. In this work, we propose a self-attention U-Net quantum decoder (SU-NetQD) for toric code, which outperforms the minimum weight perfect matching decoder, especially in the circuit level noise environments. Specifically, with our SU-NetQD, we achieve lower logical error rates compared with MWPM and discover an increased trend of code threshold as the increase of noise bias. We obtain a high threshold of 0.231 for the extremely biased noise environment. The combination of low-level decoder and high-level decoder is the key innovation for the high accuracy of our decoder. With transfer learning mechanics, our decoder is scalable for cases with different code distances. Our decoder provides a practical tool for quantum noise analysis and promotes the practicality of quantum error correction codes and quantum computing.
In quantum fault-tolerant computing, fast decoding of encoded bits is related to the running speed of quantum computers. In this paper, we propose a decoding agent based on reinforcement learning for correcting rotated surface codes in quantum error correction. The intelligent agent is trained using Deep Q-Network (DQN) and uses Convolutional Neural Network (CNN) to extract features of error syndrome and calculate the corresponding Q values for X, Y, and Z error correction operations. In the environment of depolarizing noise, the trained model achieves a fault-tolerant threshold that is superior to the Minimum Weight Perfect Matching (MWPM) and CNN decoder, providing a feasible framework for surface code error correction in practical applications.
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.