Variational Quantum Singular Value Decomposition

Singular value decomposition is central to many problems in both 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 quantum 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. We further conduct numerical simulations of the algorithm for singular-value decomposition of random matrices as well as its applications in image compression of handwritten digits. Finally, we discuss the applications of our algorithm in systems of linear equations, least squares estimation, and recommendation systems.