A Quantum Speed-Up for Approximating the Top Eigenvectors of a Matrix
Yanlin Chen, András Gilyén, Ronald de Wolf · Society for Industrial and Applied Mathematics eBooks · 2025
Finding a good approximation of the top eigenvector of a given d x 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 Õ (d 1.75 ) and one with time complexity d 1.5+0(1) (the first algorithm has a slightly better dependence on the ℓ2-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 Ω (d 2) queries to entries of A, and hence Ω(d 2) 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 on the quantum query complexity of approximating the top eigenvector.