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arXiv:2405.14765

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

Yanlin Chen, András Gilyén, Ronald de Wolf

Published May 23, 2024• 0 citations
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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.

DOI
10.48550/arxiv.2405.14765

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Finding a good approximation of the top eigenvector of a given d×dd\times dd×d matrix AAA 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 AAA and assuming a constant eigenvalue gap, output a classical description of a good approximation of the top eigenvector: one algorithm with time complexity O~(d1.75)\mathcal{\tilde{O}}(d^{1.75})O~(d1.75) and one with time complexity d1.5+o(1)d^{1.5+o(1)}d1.5+o(1) (the first algorithm has a slightly better dependence on the ℓ2\ell_2ℓ2​-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(d2)Ω(d^2)Ω(d2) queries to entries of AAA, and hence Ω(d2)Ω(d^2)Ω(d2) time. We extend this to a quantum algorithm that outputs a classical description of the subspace spanned by the top-qqq eigenvectors in time qd1.5+o(1)qd^{1.5+o(1)}qd1.5+o(1). We also prove a nearly-optimal lower bound of Ω~(d1.5)\tildeΩ(d^{1.5})Ω~(d1.5) 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.
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.
In the quantum linear systems problem (QLSP), we are given query access to a ddd-sparse N×NN\times NN×N matrix AAA with condition number κκκ, and the ability to prepare a quantum state proportional to a vector b⃗\vec bb. The goal is to output an εεε-approximation to the quantum state proportional to the solution x⃗\vec xx of Ax⃗=b⃗A\vec{x}=\vec{b}Ax=b. Following a long line of work, the best previously known quantum algorithms for the QLSP had query complexities O(κdlog⁡(1/ε))O(κd\log(1/ε))O(κdlog(1/ε)) and κd(κd/ε)o(1)κ\sqrt d(κd/ε)^{o(1)}κd​(κd/ε)o(1), while the best known lower bounds were Ω(κlog⁡(1/ε))Ω(κ\log(1/ε))Ω(κlog(1/ε)) and Ω(κd)Ω(κ\sqrt d)Ω(κd​). We improve these bounds and show that the complexity of the QLSP is Θ(κdlog⁡(1/ε))Θ(κ\sqrt d\log(1/ε))Θ(κd​log(1/ε)). We also resolve an open problem of Berry and Childs by showing that any N×NN\times NN×N unitary can be implemented with bounded error using O(N)O(\sqrt N)O(N​) queries to its matrix entries.
Matrix scaling is a simple to state, yet widely applicable linear-algebraic problem: the goal is to scale the rows and columns of a given non-negative matrix such that the rescaled matrix has prescribed row and column sums. Motivated by recent results on first-order quantum algorithms for matrix scaling, we investigate the possibilities for quantum speedups for classical second-order algorithms, which comprise the state-of-the-art in the classical setting. We first show that there can be essentially no quantum speedup in terms of the input size in the high-precision regime: any quantum algorithm that solves the matrix scaling problem for n×nn \times nn×n matrices with at most mmm non-zero entries and with ℓ2\ell_2ℓ2​-error ε=Θ~(1/m)\varepsilon=\widetildeΘ(1/m)ε=Θ(1/m) must make Ω~(m)\widetildeΩ(m)Ω(m) queries to the matrix, even when the success probability is exponentially small in nnn. Additionally, we show that for ε∈[1/n,1/2]\varepsilon\in[1/n,1/2]ε∈[1/n,1/2], any quantum algorithm capable of producing ε100\frac{\varepsilon}{100}100ε​-ℓ1\ell_1ℓ1​-approximations of the row-sum vector of a (dense) normalized matrix uses Ω(n/ε)Ω(n/\varepsilon)Ω(n/ε) queries, and that there exists a constant ε0>0\varepsilon_0>0ε0​>0 for which this problem takes Ω(n1.5)Ω(n^{1.5})Ω(n1.5) queries. To complement these results we give improved quantum algorithms in the low-precision regime: with quantum graph sparsification and amplitude estimation, a box-constrained Newton method can be sped up in the large-ε\varepsilonε regime, and outperforms previous quantum algorithms. For entrywise-positive matrices, we find an ε\varepsilonε-ℓ1\ell_1ℓ1​-scaling in time O~(n1.5/ε2)\widetilde O(n^{1.5}/\varepsilon^2)O(n1.5/ε2), whereas the best previously known bounds were O~(n2polylog(1/ε))\widetilde O(n^2\mathrm{polylog}(1/\varepsilon))O(n2polylog(1/ε)) (classical) and O~(n1.5/ε3)\widetilde O(n^{1.5}/\varepsilon^3)O(n1.5/ε3) (quantum).
We prove lower bounds on the time and space required for quantum computers to solve a wide variety of problems involving matrices, many of which have only been analyzed classically in prior work. Using a novel way of applying recording query methods we show that for many linear algebra problems—including matrix-vector product, matrix inversion, matrix multiplication and powering—existing classical time-space tradeoffs also apply to quantum algorithms with at most a constant factor loss. For example, for almost all fixed matrices A, including the discrete Fourier transform (DFT) matrix, we prove that quantum circuits with at most T input queries and S qubits of memory require T=Ω(n2/S) to compute matrix-vector product Ax for x ∈ {0,1}n. We similarly prove that matrix multiplication for n× n binary matrices requires T=Ω(n3 / √S). Because many of our lower bounds are matched by deterministic algorithms with the same time and space complexity, our results show that quantum computers cannot provide any asymptotic advantage for these problems at any space bound. We also improve the previous quantum time-space tradeoff lower bounds for n× n Boolean (i.e. AND-OR) matrix multiplication from T=Ω(n2.5/S1/2) to T=Ω(n2.5/S1/4) which has optimal exponents for the powerful query algorithms to which it applies. Our method also yields improved lower bounds for classical algorithms.
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Along with the development of quantum technology, finding useful applications of quantum computers has been a central pursuit. Despite various quantum algorithms have been developed, many of them often require strong input assumptions, which is hardware demanding. In particular, recent advances on dequantization have revealed that the quantum advantage is more of a mere artifact of strong input assumptions. In this work, we propose a variant of these algorithms, leveraging both classical and quantum resources. Provided the classical knowledge (the entries) of the matrix/vector of interest, a classical procedure is used to pre-process this information. Then they are fed into a quantum circuit which is shown to be a block encoding of the matrix of interest. From this block-encoding, we show how to use it to tackle a wide range of problems, including principal component analysis, linear equation solving, Hamiltonian simulation, preparing ground state, and data fitting. We also analyze our protocol, showing that both the classical and quantum procedure can achieve logarithmic complexity in the input dimension, thus implying its potential for near term realization. We then discuss several implications and corollaries of our result. First,, our results suggest there are certain matrices/Hamiltonians where our method can provide exponential improvement compared to the existing ones with respect to the sparsity. Regarding dense linear systems, our method achieves exponential speed-up with respect to the inverse of error tolerance, compared to the best previously known quantum algorithm for dense systems. Last, and most importantly, regarding quantum data fitting, we show how the output of our quantum algorithms can be leveraged to predict unseen data. Thus, it provides an end-to-end application, which has been an open aspect of the previous quantum data fitting algorithm.