Quantum Power Method by a Superposition of Time-Evolved States

We propose a quantum-classical hybrid algorithm of the power method, here dubbed as the quantum power method, to evaluate (H) over cap (n)vertical bar psi > with quantum computers, where n is a non-negative integer, (H) over cap is a time-independent Hamiltonian of interest, and vertical bar psi > is a quantum state. We show that the number of gates required for approximating (H) over cap (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 (H) over cap (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.