  The power method (or iteration) is a well-known classical technique that can
be used to find the dominant eigenpair of a matrix. Here, we present a
variational quantum circuit method for the power iteration, which can be used
to find the eigenpairs of unitary matrices and so their associated
Hamiltonians. We discuss how to apply the circuit to combinatorial optimization
problems formulated as a quadratic unconstrained binary optimization and
discuss its complexity. In addition, we run numerical simulations for random
problem instances with up to 21 parameters and observe that the method can
generate solutions to the optimization problems with only a few number of
iterations and the growth in the number of iterations is polynomial in the
number of parameters. Therefore, the circuit can be simulated on the near term
quantum computers with ease.
