Variational Quantum Singular Value Decomposition
This work addresses the challenge of costly quantum subroutines for matrix decomposition in engineering and scientific fields, offering a practical approach for near-term quantum devices, though it appears incremental as it builds on existing variational principles.
The authors tackled the problem of performing singular value decomposition on near-term quantum devices by proposing a variational quantum algorithm that uses two quantum neural networks to learn singular vectors and output singular values, with numerical simulations showing applications in image compression of handwritten digits.
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.