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. O ~ ( N / ϵ ) 9 9 In this work O ~ ignores terms that are polylogarithmic in N or 1 / ϵ . black-box queries to an oracle encoding the matrix, where N is the matrix dimension and ɛ is the desired precision. In contrast, the best classical algorithm for the same task requires Ω ( N ) 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.