Variational Quantum Singular Value 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.

23 pages, v3 accepted by Quantum

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• Simulating noisy variational quantum eigensolver with local noise models
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• Quantum gradient descent algorithms for nonequilibrium steady states and linear algebraic systems
• Dissipative variational quantum algorithms for Gibbs state preparation
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• Variational Quantum Algorithms for Trace Distance and Fidelity Estimation
• Near-term Efficient Quantum Algorithms for Entanglement Analysis
• Hardware-efficient entangled measurements for variational quantum algorithms
• Implementation and Empirical Evaluation of a Quantum Machine Learning Pipeline for Local Classification
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• Hybrid Quantum Singular Spectrum Decomposition for Time Series Analysis
• An Efficient Gradient Sensitive Alternate Framework for VQE with Variable Ansatz
• Quantum Algorithms for tensor-SVD
• On Modifying the Variational Quantum Singular Value Decomposition Algorithm
• Matrix Product State on a Quantum Computer