Variational Quantum Singular Value Decomposition

Xin Wang1, Zhixin Song1, and Youle Wang1,2

1Institute for Quantum Computing, Baidu Research, Beijing 100193, China
2Centre for Quantum Software and Information, University of Technology Sydney, NSW 2007, Australia

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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.

Singular Value Decomposition (SVD) is central to many engineering and scientific fields. Its goal is to is find the decomposition of any given information matrix $M$ (can be non-hermitian) into three blocks M = UDV. SVD has been shown to support a wide range of real-world applications, including the recommendation system and image compression. However, it can not be naturally extended to work on near-term quantum devices. Our work formulates the task of SVD as an optimization problem and introduces a hybrid quantum-classical algorithm that can be implemented on near-term quantum computers. In particular, utilizing the variational principle and fundamental properties of SVD, we design a novel loss function that guarantees the validity of the VQSVD algorithm. Then we could train two quantum neural networks to evaluate the loss function on quantum devices and update the parameters using the classical optimization methods. When the matrix is a Hamiltonian, our approach can be applied to explore the properties of associated ground-state and excited states. It generalizes the conventional methods of Hamiltonian diagonalization to a non-hermitian regime, extending the matrix decomposition capabilities on near-term quantum computers. We also show extended applications of VQSVD in recommendation systems and polar decomposition. This VQSVD algorithm may further shed light on near-term quantum applications in quantum chemistry, quantum machine learning, and quantum optimization.

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