Factorization of large semiprimes remains one of the most challenging problems for classical computers. Shor’s algorithm offers a quantum approach that reduces computational complexity, but its practical application is currently limited by hardware constraints. Meanwhile, as a provisional approach, quantum annealing (QA) has been explored through formulations of the quadratic unconstrained binary optimization (QUBO) problem. Among existing methods, the blockwise partial-product approach effectively reduced the QUBO variable count but was limited to semiprimes up to 21 bits. To extend factorization to larger semiprimes, this paper addresses key engineering challenges in constructing efficient QUBO formulations for prime factorization. We propose five techniques to reduce variable counts and improve scalability with current QA hardware: (1) dividing the problem into subproblems with partial constraints; (2) applying analytical reductions near the LSB; (3) applying analytical reductions near the MSB; (4) exploiting special patterns in semiprimes, with odd bit widths and long MSB-side zero sequences; and (5) balancing variable usage across both sides of the subproblem. Integrated into a QUBO converter, these methods enable stable factorization of semiprimes up to 47-bits within 20 seconds and can extend to special 2049-bit instances with 1001 consecutive MSB-side zeros.

Conventional quantum error correction (QEC) de-coders 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.