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

Finding a good approximation of the top eigenvector of a given d ˆd 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 Õpd 1.75 q and one with time complexity d 1.5op1q (the first algorithm has a slightly better dependence on the ℓ 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 Ωpd 2 q queries to entries of A, and hence Ωpd 2 q 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 1.5op1q . We also prove a nearly-optimal lower bound of Ωpd 1.5 q 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. This procedure uses an essentially optimal number O `d logpdqε 2 ˘of "conditional sample states"; if we have a statepreparation unitary available rather than just copies of the state, then this ε-dependence can be improved further quadratically. Our procedure comes with improved statistical properties and faster runtime compared to earlier pure-state tomography algorithms. We also develop an almost optimal time-efficient process-tomography algorithm for reflections around boundedrank subspaces, providing the basis for our top-eigensubspace estimation algorithm, and in turn providing a pure-state tomography algorithm that only requires a reflection about the state rather than a state preparation unitary as input.

˚QuSoft, CWI, the Netherlands.