SOME: Symmetric One-Hot Matching Elector -- A Lightweight Microsecond Decoder for Quantum Error Correction

Abstract: Conventional quantum error correction (QEC) decoders such as Minimum-Weight Perfect Matching (MWPM) and Union-Find (UF) offer high thresholds and fast decoding, respectively, but both suffer from high topological complexity. In contrast, Ising model-based decoders reduce topological complexity but demand considerable decoding time. We propose the Symmetric One-Hot Matching Elector (SOME), a novel decoder that reformulates the QEC decoding task as a Quadratic Unconstrained Binary Optimization (QUBO) problem -- termed the One-Hot QUBO (OHQ). Each variable in the QUBO represents whether a given pair of flipped syndromes is matched, while the error probabilities between the pair are encoded as interaction coefficients (weight). Constraints ensure that each flipped syndrome is matched exactly once. Valid solutions of OHQ correspond to self-inverse permutation matrices, characterized by symmetric one-hot encoding. To solve the OHQ efficiently, SOME reformulates the decoding task as the construction of permutation matrices that minimize the total weight. It initializes each candidate matrix from one of the minimum-weight syndrome pairs, then iteratively appends additional pairs in ascending order of weight, and finally selects the permutation matrix with the lowest total energy. SOME achieves up to a 99.9x reduction in variable count and reduces decoding times from milliseconds to microseconds on a single-threaded commodity CPU. OHQ also maintains performance up to a 10.5% physical error rate, surpassing the highest known threshold of MWPM@.