  We introduce a new statistical and variational approach to the phase
estimation algorithm (PEA). Unlike the traditional and iterative PEAs which
return only an eigenphase estimate, the proposed method can determine any
unknown eigenstate-eigenphase pair from a given unitary matrix utilizing a
simplified version 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 to estimate the
proximity of the input state and input phase to the nearest
eigenstate-eigenphase pair and approaching this pair via a variational process
on the input state and phase. This method may search over the entire
computational space, or can efficiently search for eigenphases (eigenstates)
within some specified 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 Q platform
and on a local computer.

Quantum Physics