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

• URL: http://arxiv.org/abs/2405.14765v1
• Date: Thu, 23 May 2024 16:33:13 GMT
• Title: A Quantum Speed-Up for Approximating the Top Eigenvectors of a Matrix
• Authors: Yanlin Chen, András Gilyén, Ronald de Wolf,
• Abstract summary: Finding a good approximation of the top eigenvector of a given $dtimes d$ matrix $A$ is a basic and important computational problem. We give two different quantum algorithms that output a classical description of a good approximation of the top eigenvector.
• Score: 2.7050250604223693
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Finding a good approximation of the top eigenvector of a given $d\times d$ 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 $\mathcal{\tilde{O}}(d^{1.75})$ and one with time complexity $d^{1.5+o(1)}$ (the first algorithm has a slightly better dependence on the $\ell_2$-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 $\Omega(d^2)$ queries to entries of $A$, and hence $\Omega(d^2)$ 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 $\tilde{\Omega}(d^{1.5})$ 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.

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