Statistical Approach to Quantum Phase Estimation

We introduce a new statistical and variational approach to the phase estimation algorithm (PEA). Un-like the traditional and iterative PEAs which return only an eigenphase estimate, the proposed methodcan determine any unknown eigenstate-eigenphase pair from a given unitary matrix utilizing a simplifiedversion of the hardware intended for the Iterative PEA (IPEA). This is achieved by treating the probabilistic output of an IPEA-like circuit as an eigenstate-eigenphase proximity metric, using this metric toestimate the proximity of the input state and input phase to the nearest eigenstate-eigenphase pair andapproaching this pair via a variational process on the input state and phase. This method may searchover the entire computational space, or can efficiently search for eigenphases (eigenstates) within somespecified range (directions), allowing those with some prior knowledge of their system to search for particular solutions. We show the simulation results of the method with the Qiskit package on the IBM Qplatform and on a local computer.