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.

翻译：量子纠错是实现可靠量子计算不可或缺的技术。当量子信息被冗余编码时，通过多个物理量子比特构建了一个更大的希尔伯特空间，计算在指定的子空间内进行。将深度学习应用于量子纠错码的解码时，一个关键挑战源于提供给解码器的校验子测量值与构成真实标签的对应错误模式之间的非唯一性。基于先前针对环面码通过利用奇偶校验结构固有的对称性对解码器进行重新优化的研究工作，我们将此方法推广至任意稳定子码。在实验中，我们采用多层感知机来近似连续函数，以补充Color码和Golay码的校验子测量。利用这些模型，我们对每种编码进行了解码器重新优化。对于Color码，在物理错误率为5%时，解码准确率提升了约0.8%；而对于Golay码，准确率提高了约0.1%。此外，通过对每种编码在连续函数近似中的几何与代数结构进行评估，我们证明了广义连续函数的设计有利于学习编码固有的几何结构。我们的结果还表明，能够忠实复现编码结构的近似方法可对重新优化的效果产生显著影响。本研究表明，先前被证明对环面码有效的重新优化技术，可推广用于解决将深度学习应用于稳定子码解码时出现的标签简并性挑战。