USDs: A universal stabilizer decoder framework using symmetry

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.

💡 Research Summary

This paper addresses a fundamental obstacle in applying deep‑learning techniques to quantum error‑correction decoding: the non‑uniqueness (label degeneracy) between measured syndromes and the underlying error patterns that serve as ground‑truth labels. Building on earlier work that resolved this issue for the toric code by exploiting the symmetry of its parity‑check structure, the authors propose a universal framework—USDs (Universal Stabilizer Decoders)—that can be applied to any stabilizer code.

The core idea is to define a continuous function f that maps an error vector E (2n binary components representing X‑ and Z‑errors on n physical qubits) to the syndrome values of the n − 1 stabilizer generators. Each component of f is expressed as f_i(E)= (1 − cos(π v_i))/2, where v_i is a linear combination of E with the binary incidence matrices S_X and S_Z that describe which qubits each stabilizer acts on. This function reproduces the exact syndrome when evaluated on binary error vectors, yet it is continuous over the compact domain