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

We present two eﬃcient classical analogues of the quantum matrix inversion algorithm [16] for low-rank matrices. Inspired by recent work of Tang [27], 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 [15], our results indicate that more low-rank quantum algorithms can be eﬀectively “dequantised” into classical length-square sampling algorithms. Acknowledgements 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.