  We propose a quantum-classical hybrid algorithm of the power method, here
dubbed as quantum power method, to evaluate $\hat{\cal H}^n |\psi\rangle$ with
quantum computers, where $n$ is a nonnegative integer, $\hat{\cal H}$ is a
time-independent Hamiltonian of interest, and $|\psi \rangle$ is a quantum
state. We show that the number of gates required for approximating $\hat{\cal
H}^n$ scales linearly in the power and the number of qubits, making it a
promising application for near term quantum computers. Using numerical
simulation, we show that the power method can control systematic errors in
approximating the Hamiltonian power ${\hat{\cal H}^n}$ for $n$ as large as 100.
As an application, we combine our method with a multireference
Krylov-subspace-diagonalization scheme to show how one can improve the
estimation of ground-state energies and the ground-state fidelities found using
a variational-quantum-eigensolver scheme. Finally, we outline other
applications of the quantum power method, including several moment-based
methods. We numerically demonstrate the connected-moment expansion for the
imaginary-time evolution and compare the results with the multireference
Krylov-subspace diagonalization.
