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, Chunhao Wang

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Nai-Hui Chia
  • Department of Computer Science, University of Texas at Austin, TX, USA
  • Joint Center for Quantum Information and Computer Science, University of Maryland, College Park, MD, USA
András Gilyén
  • QuSoft, CWI and University of Amsterdam, The Netherlands
Han-Hsuan Lin
  • Department of Computer Science, University of Texas at Austin, TX, USA
  • Department of Computer Science, National Tsing Hua University, Hsinchu, Taiwan
Seth Lloyd
  • MIT, Departments of Mechanical Engineering and Physics, Cambridge, MA, USA
  • Xanadu, Toronto, Canada
Ewin Tang
  • University of Washington, Seattle, WA, USA
Chunhao Wang
  • Department of Computer Science, University of Texas at Austin, TX, USA
  • Department of Computer Science and Engineering, Pennsylvania State University, Philadelphia, PA, USA


NHC, HHL, and CW thank Scott Aaronson for the valuable feedback on a draft of this paper. AG thanks Márió Szegedy for introduction to the problem and sharing insights, Ravi Kannan for helpful discussions and Ronald de Wolf for useful comments on the manuscript. We thank the anonymous reviewers for their valuable feedback on both submissions.

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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)


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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Sketching and sampling
  • sublinear algorithms
  • quantum-inspired
  • regression
  • importance sampling
  • quantum machine learning


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