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

Yanlin Chen, András Gilyén, Ronald de Wolf

TL;DR

Two different quantum algorithms are given that, given query access to the entries of a Hermitian matrix, output a classical description of a good approximation of the top eigenvector, and prove a nearly-optimal lower bound of $\tilde{\Omega}(d^{1.5})$ on the quantum query complexity of approximating the top eigenvector.

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 $Ω(d^2)$ 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 $\tildeΩ(d^{1.5})$ 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.

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

TL;DR

Two different quantum algorithms are given that, given query access to the entries of a Hermitian matrix, output a classical description of a good approximation of the top eigenvector, and prove a nearly-optimal lower bound of on the quantum query complexity of approximating the top eigenvector.

Abstract

Finding a good approximation of the top eigenvector of a given matrix 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 and assuming a constant eigenvalue gap, output a classical description of a good approximation of the top eigenvector: one algorithm with time complexity and one with time complexity (the first algorithm has a slightly better dependence on the -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 queries to entries of , and hence time. We extend this to a quantum algorithm that outputs a classical description of the subspace spanned by the top- eigenvectors in time . We also prove a nearly-optimal lower bound of 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.
Paper Structure (37 sections, 50 theorems, 104 equations, 3 algorithms)

This paper contains 37 sections, 50 theorems, 104 equations, 3 algorithms.

Table of Contents

  1. Introduction
  2. The power method for approximating a top eigenvector
  3. Our results: quantum algorithms
  4. Quantum noisy power method.
  5. Finding the top-q eigenvectors.
  6. Our computational model.
  7. Our results: lower bounds
  8. Related work
  9. Future work
  10. Preliminaries
  11. Computational model and quantum algorithms
  12. Quantum subroutines
  13. KP-tree and state-preparation
  14. Block-encoding and Hamiltonian simulation
  15. Singular-value and singular-vector-perturbation bounds
  16. ...and 22 more sections

Key Result

Theorem 2.1

Let $a,\delta>0$, $U$ be a unitary that maps $\ket{0}\rightarrow \ket{\psi}$ and $R_\mathcal{A},R_{\ket{0}}$ be quantum circuits that reflect through subspaces $\mathcal{A}$ and (the span of) $\ket{0}$ respectively. Suppose $\|\Pi_{\mathcal{A}}\ket{\psi}\|\geq a$. There is a quantum algorithm that p

Theorems & Definitions (76)

  • Theorem 2.1: gilyen2018QSingValTransfyoder2014FixedPointSearch, fixed-point amplitude amplification
  • Theorem 2.2
  • Theorem 2.3: follows from Section 4 of brassard2002AmpAndEst, amplitude estimation
  • Definition 2.4: KP-tree
  • Theorem 2.5: Modified Theorem 2.12 of CdW21QLasso
  • Definition 2.6
  • Theorem 2.7: low2016HamSimQubitization
  • Theorem 2.8: low2018HamSimNearlyOptSpecNorm
  • Theorem 2.9: gilyen2018QSingValTransf
  • Theorem 2.10
  • ...and 66 more