Christopher Cardullo

We study the fully generalized Grover's algorithm to find the optimal phase changes for each step of the iteration to maximize gain in probability of observation of the target, and when phase matching is required. We find that classical Grover's algorithm and phase matching remains to be optimal till the target probability gets close 1. If phase matching is enforced, we identify the threshold probability based on the size of the data set where the optimal phase change differs from pi. If phase matching is not enforced, we provide an optimization formula to identify the two phase changes based on the current amplitude vector and the size of the set. To analyze this formula, we approach it from a numerical and analytical perspective, with the analytical perspective focusing on special cases that simplify the optimization and allow for general statements about its behavior. Finally we identify a relationship between the complexity of the amplitude vector and the optimal phase change for both the enforced and non-enforced phase matching assumptions.