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.