Quantum error correction is indispensable to achieving reliable quantum computation. When quantum information is encoded redundantly, a larger Hilbert space is constructed using multiple physical qubits, and the computation is performed within a designated subspace. When applying deep learning to the decoding of quantum error-correcting codes, a key challenge arises from the non-uniqueness between the syndrome measurements provided to the decoder and the corresponding error patterns that constitute the ground-truth labels. Building upon prior work that addressed this issue for the toric code by re-optimizing the decoder with respect to the symmetry inherent in the parity-check structure, we generalize this approach to arbitrary stabilizer codes. In our experiments, we employed multilayer perceptrons to approximate continuous functions that complement the syndrome measurements of the Color code and the Golay code. Using these models, we performed decoder re-optimization for each code. For the Color code, we achieved an improvement of approximately 0.8% in decoding accuracy at a physical error rate of 5%, while for the Golay code the accuracy increased by about 0.1%. Furthermore, from the evaluation of the geometric and algebraic structures in the continuous function approximation for each code, we showed that the design of generalized continuous functions is advantageous for learning the geometric structure inherent in the code. Our results also indicate that approximations that faithfully reproduce the code structure can have a significant impact on the effectiveness of reoptimization. This study demonstrates that the re-optimization technique previously shown to be effective for the Toric code can be generalized to address the challenge of label degeneracy that arises when applying deep learning to the decoding of stabilizer codes.
Quantum error correction is indispensable to achieving reliable quantum computation. When quantum information is encoded redundantly, a larger Hilbert space is constructed using multiple physical qubits, and the computation is performed within a designated subspace. When applying deep learning to the decoding of quantum error-correcting codes, a key challenge arises from the non-uniqueness between the syndrome measurements provided to the decoder and the corresponding error patterns that constitute the ground-truth labels. Building upon prior work that addressed this issue for the toric code by re-optimizing the decoder with respect to the symmetry inherent in the parity-check structure, we generalize this approach to arbitrary stabilizer codes. In our experiments, we employed multilayer perceptrons to approximate continuous functions that complement the syndrome measurements of the Color code and the Golay code. Using these models, we performed decoder re-optimization for each code. For the Color code, we achieved an improvement of approximately 0.8% in decoding accuracy at a physical error rate of 5%, while for the Golay code the accuracy increased by about 0.1%. Furthermore, from the evaluation of the geometric and algebraic structures in the continuous function approximation for each code, we showed that the design of generalized continuous functions is advantageous for learning the geometric structure inherent in the code. Our results also indicate that approximations that faithfully reproduce the code structure can have a significant impact on the effectiveness of reoptimization. This study demonstrates that the re-optimization technique previously shown to be effective for the Toric code can be generalized to address the challenge of label degeneracy that arises when applying deep learning to the decoding of stabilizer codes.
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.
Scaling quantum computing to practical applications necessitates reliable quantum error correction. Although numerous correction codes have been proposed, the overall correction efficiency critically limited by the decode algorithms. We introduce GraphQEC, a code-agnostic decoder leveraging machine-learning on the graph structure of stabilizer codes with linear time complexity. GraphQEC demonstrates unprecedented accuracy and efficiency across all tested code families, including surface codes, color codes, and quantum low-density parity-check (QLDPC) codes. For instance, on a distance-12 QLDPC code, GraphQEC achieves a logical error rate of 9.55×10−5, an 18-fold improvement over the previous best specialized decoder's 1.74×10−3 under p=0.005 physical error rates, while maintaining 157μs/cycle decoding speed. Our approach represents the first universal solution for real-time quantum error correction across arbitrary stabilizer codes.
Quantum computers have the possibility of a much reduced calculation load compared with classical computers in specific problems. Quantum error correction (QEC) is vital for handling qubits, which are vulnerable to external noise. In QEC, actual errors are predicted from the results of syndrome measurements by stabilizer generators, in place of making direct measurements of the data qubits. Here, we propose Quantum Golay code Error Correction (QGEC), a QEC method using Golay code, which is an efficient coding method in classical information theory. We investigated our method's ability in decoding calculations with the Transformer. We evaluated the accuracy of the decoder in a code space defined by the generative polynomials with three different weights sets and three noise models with different correlations of bit-flip error and phase-flip error. Furthermore, under a noise model following a discrete uniform distribution, we compared the decoding performance of Transformer decoders with identical architectures trained respectively on Golay and toric codes. The results showed that the noise model with the smaller correlation gave better accuracy, while the weights of the generative polynomials had little effect on the accuracy of the decoder. In addition, they showed that Golay code requiring 23 data qubits and having a code distance of 7 achieved higher decoding accuracy than toric code which requiring 50 data qubits and having a code distance of 5. This suggests that implementing quantum error correction using a Transformer may enable the Golay code to realize fault-tolerant quantum computation more efficiently.
Quantum Error Correction (QEC) decoding faces a fundamental accuracy-efficiency tradeoff. Classical methods like Minimum Weight Perfect Matching (MWPM) exhibit variable performance across noise models and suffer from polynomial complexity, while tensor network decoders achieve high accuracy but at prohibitively high computational cost. Recent neural decoders reduce complexity but lack the accuracy needed to compete with computationally expensive classical methods. We introduce SAQ-Decoder, a unified framework combining transformer-based learning with constraint aware post-processing that achieves both near Maximum Likelihood (ML) accuracy and linear computational scalability with respect to the syndrome size. Our approach combines a dual-stream transformer architecture that processes syndromes and logical information with asymmetric attention patterns, and a novel differentiable logical loss that directly optimizes Logical Error Rates (LER) through smooth approximations over finite fields. SAQ-Decoder achieves near-optimal performance, with error thresholds of 10.99% (independent noise) and 18.6% (depolarizing noise) on toric codes that approach the ML bounds of 11.0% and 18.9% while outperforming existing neural and classical baselines in accuracy, complexity, and parameter efficiency. Our findings establish that learned decoders can simultaneously achieve competitive decoding accuracy and computational efficiency, addressing key requirements for practical fault-tolerant quantum computing systems.
Quantum computation promises to revolutionize information processing by exploiting superposition and entanglement, enabling applications such as quantum simulation, combinatorial optimization, and secure quantum communication. Beyond individual processors, the quantum internet aims to connect quantum nodes through entanglement distribution, enabling distributed computation. However, quantum states are fragile: environmental interactions, imperfect gate operations, and measurement errors can quickly degrade encoded information. Reliable quantum computation therefore requires carefully designed quantum error-correcting codes and fault-tolerant protocols. Unlike classical redundancy schemes, quantum error correction must preserve delicate phase relationships and relies on indirect error detection through syndrome extraction with ancillary qubits, enabling error identification without collapsing encoded states. Topological codes, in particular, emerged early as the preferred choice for many architectures due to their planar structure, local interactions, and natural support for fault-tolerant logical operations. Color codes, for instance, allow transversal implementation of all Clifford gates, whereas non-Clifford operations require magic-state injection. More recent constructions, including quantum low-density parity-check codes, offer higher encoding rates and reduced resource overhead, but require long-range qubit connectivity and more complex protocols. Despite experimental progress, several challenges remain. Many platforms exhibit biased noise, motivating codes that exploit channel asymmetry. Real-time decoding remains a critical bottleneck, as fault-tolerant computation requires low-latency processing of rapidly generated syndrome data. Fault-tolerant logical state initialization also remains resource-intensive and scales poorly with code distance. Addressing these challenges, this thesis develops a unified framework spanning theory and experiment. It provides precise performance analysis of quantum error-correcting codes in the low physical error-rate regime, introduces codes tailored to biased noise, develops fast low-complexity decoding algorithms for real-time operation, and presents modular and efficient protocols for fault-tolerant state preparation of CSS codes. The latter is demonstrated experimentally through the preparation of a logical zero state of the Golay code on the Quantinuum H2 trapped-ion quantum computer.
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.