A square-root speedup for finding the smallest eigenvalue

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., \(\widetilde{\mathcal{O}}(\sqrt{N}/\epsilon)\) black-box queries to an oracle encoding the matrix, where \(N\) is the matrix dimension and \(\epsilon\) is the desired precision. In contrast, the best classical algorithm for the same task requires \(\Omega(N)\text{polylog}(1/\epsilon)\) 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.