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DOI: 10.4230/LIPIcs.ISAAC.2020.47
Quantum-Inspired Algorithms for Solving Low-Rank Linear Equation Systems with Logarithmic Dependence on the Dimension

Authors: Nai-Hui Chia, András Gilyén, Han-Hsuan Lin, Seth Lloyd, Ewin Tang, and Chunhao Wang

Published in: LIPIcs, Volume 181, 31st International Symposium on Algorithms and Computation (ISAAC 2020)

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.
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Nai-Hui Chia, András Gilyén, Han-Hsuan Lin, Seth Lloyd, Ewin Tang, and Chunhao Wang. Quantum-Inspired Algorithms for Solving Low-Rank Linear Equation Systems with Logarithmic Dependence on the Dimension. In 31st International Symposium on Algorithms and Computation (ISAAC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 181, pp. 47:1-47:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{chia_et_al:LIPIcs.ISAAC.2020.47,
  author =	{Chia, Nai-Hui and Gily\'{e}n, Andr\'{a}s and Lin, Han-Hsuan and Lloyd, Seth and Tang, Ewin and Wang, Chunhao},
  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 =	{Cao, Yixin and Cheng, Siu-Wing and Li, Minming},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2020.47},
  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}
}
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