On the Success Probability of Quantum Order Finding

Abstract

We prove a lower bound on the probability of Shor’s order-finding algorithm successfully recovering the order r in a single run. The bound implies that by performing two limited searches in the classical post-processing part of the algorithm, a high success probability can be guaranteed, for any r, without re-running the quantum part or increasing the exponent length compared to Shor. Asymptotically, in the limit as r tends to infinity, the probability of successfully recovering r in a single run tends to one. Already for moderate r, a high success probability exceeding e.g. 1−10−4 can be guaranteed. As corollaries, we prove analogous results for the probability of completely factoring any integer N in a single run of the order-finding algorithm.