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We describe a quantum algorithm for finding the smallest eigenvalue of aHermitian matrix. This algorithm combines Quantum Phase Estimation and QuantumAmplitude Estimation to achieve a quadratic speedup with respect to the bestclassical algorithm in terms of matrix dimensionality, i.e.,$\widetilde{\mathcal{O}}(\sqrt{N}/\epsilon)$ black-box queries to an oracleencoding the matrix, where $N$ is the matrix dimension and $\epsilon$ is thedesired precision. In contrast, the best classical algorithm for the same taskrequires $\Omega(N)\text{polylog}(1/\epsilon)$ queries. In addition, thisalgorithm allows the user to select any constant success probability. We alsoprovide a similar algorithm with the same runtime that allows us to prepare aquantum state lying mostly in the matrix's low-energy subspace. We implementsimulations of both algorithms and demonstrate their application to problems inquantum chemistry and materials science.

A square-root speedup for finding the smallest eigenvalue