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.
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 d-sparse N×N matrix A with condition number κ, and the ability to prepare a quantum state proportional to a vector b. The goal is to output an ε-approximation to the quantum state proportional to the solution x of Ax=b. Following a long line of work, the best previously known quantum algorithms for the QLSP had query complexities O(κdlog(1/ε)) and κd(κd/ε)o(1), while the best known lower bounds were Ω(κlog(1/ε)) and Ω(κd). We improve these bounds and show that the complexity of the QLSP is Θ(κdlog(1/ε)). We also resolve an open problem of Berry and Childs by showing that any N×N unitary can be implemented with bounded error using O(N) queries to its matrix entries.
Finding a good approximation of the top eigenvector of a given d×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 O~(d1.75) and one with time complexity d1.5+o(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(d2) queries to entries of A, and hence Ω(d2) time. We extend this to a quantum algorithm that outputs a classical description of the subspace spanned by the top-q eigenvectors in time qd1.5+o(1). We also prove a nearly-optimal lower bound of 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.
Matrix scaling and matrix balancing are two basic linear-algebraic problems with a wide variety of applications, such as approximating the permanent, and pre-conditioning linear systems to make them more numerically stable. We study the power and limitations of quantum algorithms for these problems. We provide quantum implementations of two classical (in both senses of the word) methods: Sinkhorn’s algorithm for matrix scaling and Osborne’s algorithm for matrix balancing. Using amplitude estimation as our main tool, our quantum implementations both run in time Õ(√{mn}/ε⁴) for scaling or balancing an n × n matrix (given by an oracle) with m non-zero entries to within 𝓁₁-error ε. Their classical analogs use time Õ(m/ε²), and every classical algorithm for scaling or balancing with small constant ε requires Ω(m) queries to the entries of the input matrix. We thus achieve a polynomial speed-up in terms of n, at the expense of a worse polynomial dependence on the obtained 𝓁₁-error ε. Even for constant ε these problems are already non-trivial (and relevant in applications). Along the way, we extend the classical analysis of Sinkhorn’s and Osborne’s algorithm to allow for errors in the computation of marginals. We also adapt an improved analysis of Sinkhorn’s algorithm for entrywise-positive matrices to the 𝓁₁-setting, obtaining an Õ(n^{1.5}/ε³)-time quantum algorithm for ε-𝓁₁-scaling. We also prove a lower bound, showing our quantum algorithm for matrix scaling is essentially optimal for constant ε: every quantum algorithm for matrix scaling that achieves a constant 𝓁₁-error w.r.t. uniform marginals needs Ω(√{mn}) queries.
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×n matrices with at most m non-zero entries and with ℓ2-error ε=Θ(1/m) must make Ω(m) queries to the matrix, even when the success probability is exponentially small in n. Additionally, we show that for ε∈[1/n,1/2], any quantum algorithm capable of producing 100ε-ℓ1-approximations of the row-sum vector of a (dense) normalized matrix uses Ω(n/ε) queries, and that there exists a constant ε0>0 for which this problem takes 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 regime, and outperforms previous quantum algorithms. For entrywise-positive matrices, we find an ε-ℓ1-scaling in time O(n1.5/ε2), whereas the best previously known bounds were O(n2polylog(1/ε)) (classical) and 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.
Approximating the k -th spectral gap Δ k = | λ k − λ k + 1 | and the corresponding midpoint μ k = λ k + λ k + 1 2 of an N × N Hermitian matrix with eigenvalues λ 1 ≥ λ 2 ≥ … ≥ λ N , is an important special case of the eigenproblem with numerous applications in science and engineering. In this work, we present a quantum algorithm which approximates these values up to additive error ϵ Δ k using a logarithmic number of qubits. Notably, in the QRAM model, its total complexity (queries and gates) is bounded by O ( N 2 ϵ 2 Δ k 2 p o l y l o g ( N , 1 Δ k , 1 ϵ , 1 δ ) ) , where ϵ , δ ∈ ( 0 , 1 ) are the accuracy and the failure probability, respectively. For large gaps Δ k , this provides a speed-up against the best-known complexities of classical algorithms, namely, O ( N ω p o l y l o g ( N , 1