Quantum singular value decomposition of non-sparse low-rank matrices

TL;DR

This paper introduces a quantum algorithm for efficiently computing the singular value decomposition of non-sparse, low-rank matrices, including non-Hermitian and non-square cases, with exponential speedup over classical methods.

Contribution

It presents a novel quantum method for exponentiating and decomposing non-sparse low-rank matrices, extending to non-Hermitian and non-square matrices, and discusses related optimization problems.

Findings

Quantum SVD can be performed exponentially faster than classical algorithms.

Method works for non-sparse, indefinite, low-rank matrices.

Extension to non-Hermitian and non-square matrices achieved.

Abstract

In this work, we present a method to exponentiate non-sparse indefinite low-rank matrices on a quantum computer. Given an operation for accessing the elements of the matrix, our method allows singular values and associated singular vectors to be found quantum mechanically in a time exponentially faster in the dimension of the matrix than known classical algorithms. The method extends to non-Hermitian and non-square matrices via embedding matrices. In the context of the generic singular value decomposition of a matrix, we discuss the Procrustes problem of finding a closest isometry to a given matrix.