Quantum algorithms and lower bounds for convex optimization

We study to what extent quantum algorithms can speed up solving convex optimization problems. Following the classical literature we assume access to a convex set via various oracles, and we examine the efficiency of reductions between the different oracles. In particular, we show how a separation oracle can be implemented using O(1) quantum queries to a membership oracle, which is an exponential quantum speed-up over the Ω(n) membership queries that are needed classically. We show that a quantum computer can very efficiently compute an approximate subgradient of a convex Lipschitz function. Combining this with a simplification of recent classical work of Lee, Sidford, and Vempala gives our efficient separation oracle. This in turn implies, via a known algorithm, that O(n) quantum queries to a membership oracle suffice to implement an optimization oracle (the best known classical upper bound on the number of membership queries is quadratic). We also prove several lower bounds: Ω( √ n) quantum separation (or membership) queries are needed for optimization if the algorithm knows an interior point of the convex set, and Ω(n) quantum separation queries are needed if it does not. * We use [n] := {1, 2, . . . , n}. For p ≥ 1, ε ≥ 0, and a set C ⊆ R n we let B p (C, ε) = {x ∈ R n : ∃y ∈ C such that ||x − y|| p ≤ ε} be the set of points of distance at most ε from C in the p -norm. When C = {x} is a singleton set we abuse notation and write B p (x, ε). We overload notation by setting B p (C, −ε) = {x ∈ R n : B p (x, ε) ⊆ C}.Whenever p is omitted it is assumed that p = 2.

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“…Acknowledgments. We thank the authors of [CCLW18] for sending us a draft of their work. AG thanks Márió Szegedy for insightful discussions.…”

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confidence: 99%

Convex optimization using quantum oracles

Apeldoorn

Gilyén

Gribling

et al. 2020

Quantum

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“…Recently van Apeldoorn et al [vAGGdW20] and Chakrabarti et al [CCLW20] developed quantum algorithms for general black-box convex optimization, where one optimizes over a general convex set K, and the access to K is via membership and/or separation oracles. Since we work in a model where we are given access directly to the constraints defining the problem, our results are incomparable to theirs.…”

Section: Subsequent Workmentioning

confidence: 99%

Quantum SDP-Solvers: Better upper and lower bounds

Apeldoorn

Gilyén

Gribling

et al. 2020

Quantum

161

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Brandão and Svore \cite{brandao2016QSDPSpeedup} recently gave quantum algorithms for approximately solving semidefinite programs, which in some regimes are faster than the best-possible classical algorithms in terms of the dimension n of the problem and the number m of constraints, but worse in terms of various other parameters. In this paper we improve their algorithms in several ways, getting better dependence on those other parameters. To this end we develop new techniques for quantum algorithms, for instance a general way to efficiently implement smooth functions of sparse Hamiltonians, and a generalized minimum-finding procedure.We also show limits on this approach to quantum SDP-solvers, for instance for combinatorial optimization problems that have a lot of symmetry. Finally, we prove some general lower bounds showing that in the worst case, the complexity of every quantum LP-solver (and hence also SDP-solver) has to scale linearly with mn when m≈n, which is the same as classical.

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“…In this section, we show how to apply the importance sampling technique to reduce the cost of estimating the loss function in Eq. (12). For convenience, we restate Eq.…”

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confidence: 99%

“…Quantum computing is believed to deliver new technology to speed up computation, and it already promises speedups for integer factoring [4] and database search [5] in theory. Enormous efforts have been made in exploring the possibility of using quantum resources to speed up other important tasks, including linear system solvers [6][7][8][9][10], convex optimizations [11][12][13][14], and machine learning [15][16][17]. Quantum algorithms for SVD have been proposed in [18,19], which leads to applications in solving linear systems of equations [9] and developing quantum recommendation systems [18].…”

Section: Introductionmentioning

confidence: 99%

Variational Quantum Singular Value Decomposition

Wang

Song

Wang

2021

Quantum

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Singular value decomposition is central to many problems in engineering and scientific fields. Several quantum algorithms have been proposed to determine the singular values and their associated singular vectors of a given matrix. Although these algorithms are promising, the required quantum subroutines and resources are too costly on near-term quantum devices. In this work, we propose a variational quantum algorithm for singular value decomposition (VQSVD). By exploiting the variational principles for singular values and the Ky Fan Theorem, we design a novel loss function such that two quantum neural networks (or parameterized quantum circuits) could be trained to learn the singular vectors and output the corresponding singular values. Furthermore, we conduct numerical simulations of VQSVD for random matrices as well as its applications in image compression of handwritten digits. Finally, we discuss the applications of our algorithm in recommendation systems and polar decomposition. Our work explores new avenues for quantum information processing beyond the conventional protocols that only works for Hermitian data, and reveals the capability of matrix decomposition on near-term quantum devices.

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Quantum algorithms and lower bounds for convex optimization

Cited by 35 publications

References 35 publications

Convex optimization using quantum oracles

Convex optimization using quantum oracles

Quantum SDP-Solvers: Better upper and lower bounds

Variational Quantum Singular Value Decomposition

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