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.
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.
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.
Abstract Quantum error correction is a technique that enhances a system’s ability to combat noise by encoding logical information into additional quantum bits, which plays a key role in building practical quantum computers. The XZZX surface code, with only one stabilizer generator on each face, demonstrates significant application potential under biased noise. However, the existing minimum weight perfect matching (MWPM) algorithm has high computational complexity and lacks flexibility in large-scale systems. Therefore, this paper proposes a decoding method that combines graph neural networks (GNN) with multi-classifiers, the syndrome is transformed into an undirected graph, and the features are aggregated by convolutional layers, providing a more efficient and accurate decoding strategy. In the experiments, we evaluated the performance of the XZZX code under different biased noise conditions (bias = 1, 20, 200) and different code distances (d = 3, 5, 7, 9, 11). The experimental results show that under low bias noise (bias = 1), the GNN decoder achieves a threshold of 0.18386, an improvement of approximately 19.12% compared to the MWPM decoder. Under high bias noise (bias = 200), the GNN decoder reaches a threshold of 0.40542, improving by approximately 20.76%, overcoming the limitations of the conventional decoder. They demonstrate that the GNN decoding method exhibits superior performance and has broad application potential in the error correction of XZZX code.
Neural decoders for quantum error correction have achieved high accuracy on surface code memory experiments, but practical fault-tolerant computation requires decoders that transfer across devices, noise conditions, and logical operations. We introduce a modular graph-recurrent decoder designed for this setting. Built from reusable local syndrome encoders, it is pretrained in simulation and fine-tuned to hardware using a common pipeline, with only lightweight spatial aggregation adapted between experiments. The decoder performs competitively on conventional memory benchmarks, outperforming minimum-weight perfect matching (MWPM) at distances 3 and 5 on an IBM Nighthawk-generation processor and closely approaching AlphaQubit on Google's below-threshold hardware data. We then test the framework in two settings beyond memory decoding on the same IBM device. First, in an X-type stability experiment, we continuously tune the effective measurement error rate using a variable pre-readout rotation and train a single decoder jointly across the resulting noise range. It matches or outperforms MWPM across the sweep without requiring a separately calibrated decoder for each setting. Second, we decode a ZZ-merge lattice-surgery experiment between two distance-3 surface-code patches, improving the decoded correlation between the merge outcome and the final ZZ parity relative to MWPM on real hardware. Together, these experiments demonstrate neural decoding that is portable not only across hardware, but also across qualitatively different surface code observables and logical operations, using models trainable on single-GPU academic resources.
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 computers can solve selected problems more efficiently than classical computers, but current devices are limited by high physical error rates. Quantum Error Correction (QEC) mitigates this by encoding many physical qubits into a logical qubit, with the surface code among the most widely studied approaches. Since syndrome measurements are produced continuously, the decoder must process them fast enough to avoid becoming a system bottleneck, making real-time decoding essential for fault-tolerant quantum computing. While most state-of-the-art real-time decoders rely on Minimum-Weight Perfect Matching (MWPM), we instead use a high-accuracy Graph Neural Network (GNN) that trades higher computational cost for lower logical error rates. To make this GNN practical for real-time decoding, we apply algorithm-hardware co-design. We reduce complexity through hardware-guided pruning and retraining, producing two hardware-friendly models that reduce parameter count by 3.1× and 6.5×. These target, respectively, an average decoding latency of one syndrome cycle and a worst-case latency within one syndrome cycle. We further reduce hardware cost using input-graph filtering and post-training quantization. Based on these optimized models, we propose an FPGA-based architecture for low-latency inference and real-time decoding. Evaluated on surface codes up to distance 7 under circuit-level noise at physical error rate p=10−3, our decoder outperforms MWPM in decoding accuracy for both average-latency and max-latency settings. It reduces logical error rate by 40% at 1μs average latency, and by 13% under a strict 1μs deadline.
Quantum error correction (QEC) is crucial for achieving fault-tolerant quantum computing by addressing errors in noisy qubits. The surface code, a leading QEC framework, traditionally employs the Minimum Weight Perfect Matching (MWPM) decoder. We present a step-by-step methodology that formulates surface code decoding as a supervised machine learning classification problem within a two-level framework, where low-level modules address physical qubit errors and highlevel modules correct logical errors. We implement Feedforward Neural Networks (FFNN) and Graph Neural Networks (GNN), which are individually employed to perform low-level physical qubit error correction and high-level logical error correction for rotated surface codes of distancesd=3,d=5, andd=7. Under depolarizing noise, both FFNN and GNN achieve lower logical error rates than MWPM, with GNN yielding up to 28% improvement atd=7. GNN achieves pseudothresholds of 0.1084, 0.1311, and 0.1441 ford=3,d=5, andd=7, surpassing MWPM's 0.0914, 0.1024, and 0.1126. Crossover points further indicate GNN's robustness at higher noise levels (e.g., 0.1632 vs. 0.1291 for MWPM). By learning error distributions in a two-level setting, both FFNN and GNN outperform MWPM, demonstrating their potential as scalable, high-performance decoders for near-term quantum devices.