License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ISAAC.2020.47
URN: urn:nbn:de:0030-drops-133916
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2020/13391/
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Chia, Nai-Hui ; Gilyén, András ; Lin, Han-Hsuan ; Lloyd, Seth ; Tang, Ewin ; Wang, Chunhao

Quantum-Inspired Algorithms for Solving Low-Rank Linear Equation Systems with Logarithmic Dependence on the Dimension

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Abstract

We present two efficient classical analogues of the quantum matrix inversion algorithm [Harrow et al., 2009] for low-rank matrices. Inspired by recent work of Tang [Tang, 2019], assuming length-square sampling access to input data, we implement the pseudoinverse of a low-rank matrix allowing us to sample from the solution to the problem Ax = b using fast sampling techniques. We construct implicit descriptions of the pseudo-inverse by finding approximate singular value decomposition of A via subsampling, then inverting the singular values. In principle, our approaches can also be used to apply any desired "smooth" function to the singular values. Since many quantum algorithms can be expressed as a singular value transformation problem [András Gilyén et al., 2019], our results indicate that more low-rank quantum algorithms can be effectively "dequantised" into classical length-square sampling algorithms.

BibTeX - Entry

@InProceedings{chia_et_al:LIPIcs:2020:13391,
  author =	{Nai-Hui Chia and Andr{\'a}s Gily{\'e}n and Han-Hsuan Lin and Seth Lloyd and Ewin Tang and Chunhao Wang},
  title =	{{Quantum-Inspired Algorithms for Solving Low-Rank Linear Equation Systems with Logarithmic Dependence on the Dimension}},
  booktitle =	{31st International Symposium on Algorithms and Computation (ISAAC 2020)},
  pages =	{47:1--47:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-173-3},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{181},
  editor =	{Yixin Cao and Siu-Wing Cheng and Minming Li},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2020/13391},
  URN =		{urn:nbn:de:0030-drops-133916},
  doi =		{10.4230/LIPIcs.ISAAC.2020.47},
  annote =	{Keywords: sublinear algorithms, quantum-inspired, regression, importance sampling, quantum machine learning}
}

Keywords: sublinear algorithms, quantum-inspired, regression, importance sampling, quantum machine learning
Collection: 31st International Symposium on Algorithms and Computation (ISAAC 2020)
Issue Date: 2020
Date of publication: 04.12.2020

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