Enhancing the Practical Reliability of Shor's Quantum Algorithm via Generalized Period Decomposition: Theory and Large-Scale Empirical Validation

This work presents a generalized period decomposition approach, significantly improving the practical reliability of Shor's quantum factoring algorithm. Although Shor's algorithm theoretically enables polynomial-time integer factorization, its real-world performance heavily depends on stringent conditions related to the period obtained via quantum phase estimation. Our generalized decomposition method relaxes these conditions by systematically exploiting arbitrary divisors of the obtained period, effectively broadening the applicability of each quantum execution. Extensive classical simulations were performed to empirically validate our approach, involving over one million test cases across integers ranging from 2 to 8 digits. The proposed method achieved near-perfect success rates, exceeding 99.998% for 7-digit numbers and 99.999% for 8-digit numbers, significantly surpassing traditional and recently improved variants of Shor's algorithm. Crucially, this improvement is achieved without compromising the algorithm's polynomial-time complexity and integrates seamlessly with existing quantum computational frameworks. Moreover, our method enhances the efficiency of quantum resource usage by minimizing unnecessary repetitions, making it particularly relevant for quantum cryptanalysis with noisy intermediate-scale quantum (NISQ) devices. This study thus provides both theoretical advancements and substantial practical benefits, contributing meaningfully to the field of quantum algorithm research and the broader field of quantum information processing.