A square-root speedup for finding the smallest eigenvalue

Abstract

We describe a quantum algorithm for finding the smallest eigenvalue of a Hermitian matrix. This algorithm combines Quantum Phase Estimation and Quantum Amplitude Estimation to achieve a quadratic speedup with respect to the best classical algorithm in terms of matrix dimensionality, i.e., $\tilde{\mathcal{O}}(\sqrt{N}/ϵ)$ black-box queries to an oracle encoding the matrix, where $N$ is the matrix dimension and $ϵ$ is the desired precision. In contrast, the best classical algorithm for the same task requires $\Omega (N)\text{polylog}(1/ϵ)$ queries. In addition, this algorithm allows the user to select any constant success probability. We also provide a similar algorithm with the same runtime that allows us to prepare a quantum state lying mostly in the matrix's low-energy subspace. We implement simulations of both algorithms and demonstrate their application to problems in quantum chemistry and materials science.