Imaginarity as a resource in Shor's algorithm

Abstract

We investigate the sequential version of Shor's order-finding algorithm from the perspective of the imaginarity of quantum channels. By fixing an eigenbranch of the modular multiplication unitary and a target output path, the process is effectively decomposed into a real diagonal factor and a residual phase channel. We find that for the ideal residual phase channel, the three trace-norm induced imaginarity measures are all equal. On good output paths, a smaller residual imaginarity corresponds to a higher single-block success probability. By combining the branch-wise probabilities, we obtain the exact ideal success probability of the sequential order-finding protocol and derive lower and upper bounds in terms of residual phase-channel imaginarity. We further extend the analysis to noisy residual phase channels, and obtain the lower and upper bounds on the noisy success probability. These results clarify the role of imaginarity of channels in Shor's algorithm and provide a refined perspective on quantum-resource behavior in ideal and noisy sequential quantum algorithms.