  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.
