Variational Quantum Singular Value Decomposition

TL;DR

This paper introduces a variational quantum algorithm for singular value decomposition that leverages quantum neural networks and the Ky Fan Theorem, enabling matrix decomposition on near-term quantum devices for various applications.

Contribution

It presents a novel variational quantum algorithm for SVD using a new loss function and quantum neural networks, expanding quantum matrix decomposition capabilities beyond Hermitian data.

Findings

Numerical simulations demonstrate effective SVD on random matrices.

Application in image compression of handwritten digits shows practical utility.

Discussion of potential in recommendation systems and polar decomposition.

Abstract

Singular value decomposition is central to many problems in engineering and scientific fields. Several quantum algorithms have been proposed to determine the singular values and their associated singular vectors of a given matrix. Although these algorithms are promising, the required quantum subroutines and resources are too costly on near-term quantum devices. In this work, we propose a variational quantum algorithm for singular value decomposition (VQSVD). By exploiting the variational principles for singular values and the Ky Fan Theorem, we design a novel loss function such that two quantum neural networks (or parameterized quantum circuits) could be trained to learn the singular vectors and output the corresponding singular values. Furthermore, we conduct numerical simulations of VQSVD for random matrices as well as its applications in image compression of handwritten digits. Finally, we discuss the applications of our algorithm in recommendation systems and polar decomposition. Our work explores new avenues for quantum information processing beyond the conventional protocols that only works for Hermitian data, and reveals the capability of matrix decomposition on near-term quantum devices.