Quantum-Inspired Classical Algorithms for Singular Value Transformation

TL;DR

This paper develops classical algorithms inspired by quantum computing techniques to perform singular value transformations on matrices, achieving efficiency comparable to recent quantum algorithms.

Contribution

It introduces quantum-inspired classical algorithms for a broad class of matrix transformations based on singular value decomposition, extending prior quantum dequantization methods.

Findings

Classical algorithms with polynomial complexity similar to quantum algorithms

Efficient approximation of matrix transformations via singular value decomposition

Broad applicability to low-rank matrix problems

Abstract

A recent breakthrough by Tang (STOC 2019) showed how to "dequantize" the quantum algorithm for recommendation systems by Kerenidis and Prakash (ITCS 2017). The resulting algorithm, classical but "quantum-inspired", efficiently computes a low-rank approximation of the users' preference matrix. Subsequent works have shown how to construct efficient quantum-inspired algorithms for approximating the pseudo-inverse of a low-rank matrix as well, which can be used to (approximately) solve low-rank linear systems of equations. In the present paper, we pursue this line of research and develop quantum-inspired algorithms for a large class of matrix transformations that are defined via the singular value decomposition of the matrix. In particular, we obtain classical algorithms with complexity polynomially related (in most parameters) to the complexity of the best quantum algorithms for singular value transformation recently developed by Chakraborty, Gily\'{e}n and Jeffery (ICALP 2019) and Gily\'{e}n, Su, Low and Wiebe (STOC19).