Quantum Power Method by a Superposition of Time-Evolved States

TL;DR

This paper introduces a quantum-classical hybrid algorithm called the quantum power method, which efficiently approximates Hamiltonian powers using quantum computers, enabling improved ground-state energy estimation and other moment-based quantum algorithms.

Contribution

The paper presents a novel quantum power method that scales linearly with the power and qubits, and demonstrates its effectiveness in controlling errors and enhancing quantum eigenvalue algorithms.

Findings

Efficient approximation of Hamiltonian powers up to n=100.

Improved ground-state energy and fidelity estimation.

Potential applications in moment-based quantum methods.

Abstract

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.