Abstract: This disclosure relates to a method for estimating a solution to a problem represented by a Hamiltonian on a quantum computer having a quantum state. A classical computer determines a trial state and evolves the quantum state of the quantum computer based on the trial state. The computer then determines estimates for expectation values with respect to the trial state of powers of the Hamiltonian based on measurements from the quantum computer and calculates an estimate of the solution based on the estimates for the expectation values of powers of the Hamiltonian for the trial state. The computer then repeated updates the trial state and repeats the measuring steps to iteratively improve the estimate of the solution.