research-article

Sampling-based Sublinear Low-rank Matrix Arithmetic Framework for Dequantizing Quantum Machine Learning

Authors:

András Pal Gilyén,

Chunhao WangAuthors Info & Claims

Journal of the ACM, Volume 69, Issue 5

Article No.: 33, Pages 1 - 72

https://doi.org/10.1145/3549524

Published: 27 October 2022 Publication History

Abstract

We present an algorithmic framework for quantum-inspired classical algorithms on close-to-low-rank matrices, generalizing the series of results started by Tang’s breakthrough quantum-inspired algorithm for recommendation systems [STOC’19]. Motivated by quantum linear algebra algorithms and the quantum singular value transformation (SVT) framework of Gilyén et al. [STOC’19], we develop classical algorithms for SVT that run in time independent of input dimension, under suitable quantum-inspired sampling assumptions. Our results give compelling evidence that in the corresponding QRAM data structure input model, quantum SVT does not yield exponential quantum speedups. Since the quantum SVT framework generalizes essentially all known techniques for quantum linear algebra, our results, combined with sampling lemmas from previous work, suffice to generalize all prior results about dequantizing quantum machine learning algorithms. In particular, our classical SVT framework recovers and often improves the dequantization results on recommendation systems, principal component analysis, supervised clustering, support vector machines, low-rank regression, and semidefinite program solving. We also give additional dequantization results on low-rank Hamiltonian simulation and discriminant analysis. Our improvements come from identifying the key feature of the quantum-inspired input model that is at the core of all prior quantum-inspired results: ℓ²-norm sampling can approximate matrix products in time independent of their dimension. We reduce all our main results to this fact, making our exposition concise, self-contained, and intuitive.

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Index Terms

Sampling-based Sublinear Low-rank Matrix Arithmetic Framework for Dequantizing Quantum Machine Learning
1. Theory of computation

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cover image Journal of the ACM

Journal of the ACM Volume 69, Issue 5

October 2022

420 pages

ISSN:0004-5411

EISSN:1557-735X

DOI:10.1145/3563903

Editor:
Venkatesan Guruswami
University of California, Berkeley, United States

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Publication History

Published: 27 October 2022

Online AM: 10 August 2022

Accepted: 07 June 2022

Revised: 20 May 2022

Received: 20 August 2020

Published in JACM Volume 69, Issue 5

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Scott Aaronson’s Vannevar Bush Faculty Fellowship from the US Department of Defense
Samsung Electronics Co., Ltd., for the project “The Computational Power of Sampling on Quantum Computers”
Institute for Quantum Information and Matter, an NSF Physics Frontiers Center
EU’s Horizon 2020 Marie Skłodowska-Curie program
IBM PhD Fellowship, QISE-NET Triplet Award
U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research, Quantum Algorithms Teams program
National Science Foundation Graduate Research Fellowship Program

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Cited By

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Zuo QWei YXiang H(2024)Quantum-inspired algorithm for truncated total least squares solutionJournal of Computational and Applied Mathematics10.1016/j.cam.2024.116042451(116042)Online publication date: Dec-2024
https://doi.org/10.1016/j.cam.2024.116042
Incudini MMartini FDi Pierro A(2023)Higher-Order Topological Kernels via Quantum Computation2023 IEEE International Conference on Quantum Computing and Engineering (QCE)10.1109/QCE57702.2023.00076(621-629)Online publication date: 17-Sep-2023
https://doi.org/10.1109/QCE57702.2023.00076
Kashif MAl-Kuwari S(2023)The unified effect of data encoding, ansatz expressibility and entanglement on the trainability of HQNNsInternational Journal of Parallel, Emergent and Distributed Systems10.1080/17445760.2023.223116338:5(362-400)Online publication date: 17-Jul-2023
https://doi.org/10.1080/17445760.2023.2231163

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