A Quantum Speed-Up for Approximating the Top Eigenvectors of a Matrix

Finding a good approximation of the top eigenvector of a given d d matrix A is a basic and important computational problem, with many applications. We give two different quantum algorithms that, given query access to the entries of a Hermitian matrix A and assuming a constant eigenvalue gap, output a classical description of a good approximation of the top eigenvector: one algorithm with time complexity O (d^1. 75) and one with time complexity d^1. 5+o (1) (the first algorithm has a slightly better dependence on the ₂-error of the approximating vector than the second, and uses different techniques of independent interest). Both of our quantum algorithms provide a polynomial speed-up over the best-possible classical algorithm, which needs (d²) queries to entries of A, and hence (d²) time. We extend this to a quantum algorithm that outputs a classical description of the subspace spanned by the top-q eigenvectors in time qd^1. 5+o (1). We also prove a nearly-optimal lower bound of (d^1. 5) on the quantum query complexity of approximating the top eigenvector. Our quantum algorithms run a version of the classical power method that is robust to certain benign kinds of errors, where we implement each matrix-vector multiplication with small and well-behaved error on a quantum computer, in different ways for the two algorithms. Our first algorithm estimates the matrix-vector product one entry at a time, using a new ``Gaussian phase estimation'' procedure. Our second algorithm uses block-encoding techniques to compute the matrix-vector product as a quantum state, from which we obtain a classical description by a new time-efficient unbiased pure-state tomography procedure.