Quantum Power Method by a Superposition of Time-Evolved States

Abstract

We propose a quantum-classical hybrid algorithm of the power method, here dubbed as quantum power method, to evaluate ${\hat{\mathcal{H}}}^n|\psi ⟩$ with quantum computers, where $n$ is a nonnegative integer, $\hat{\mathcal{H}}$ is a time-independent Hamiltonian of interest, and $|\psi ⟩$ is a quantum state. We show that the number of gates required for approximating ${\hat{\mathcal{H}}}^n$ 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 ${\hat{\mathcal{H}}}^n$ 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.

arXiv comment42 pages, 22 figures, 2 tables

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