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.

Featured image: The workflow of VQSVD

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Popular summary

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.

► BibTeX data

@article{Wang2021variationalquantum, doi = {10.22331/q-2021-06-29-483}, url = {https://doi.org/10.22331/q-2021-06-29-483}, title = {Variational {Q}uantum {S}ingular {V}alue {D}ecomposition}, author = {Wang, Xin and Song, Zhixin and Wang, Youle}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{“{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {5}, pages = {483}, month = jun, year = {2021}<br /> }

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Cited by

[1] Kishor Bharti, Alba Cervera-Lierta, Thi Ha Kyaw, Tobias Haug, Sumner Alperin-Lea, Abhinav Anand, Matthias Degroote, Hermanni Heimonen, Jakob S. Kottmann, Tim Menke, Wai-Keong Mok, Sukin Sim, Leong-Chuan Kwek, and Alán Aspuru-Guzik, “Noisy intermediate-scale quantum (NISQ) algorithms”, arXiv:2101.08448.

[2] Ranyiliu Chen, Zhixin Song, Xuanqiang Zhao, and Xin Wang, “Variational Quantum Algorithms for Trace Distance and Fidelity Estimation”, arXiv:2012.05768.

[3] Chenfeng Cao and Xin Wang, “Noise-Assisted Quantum Autoencoder”, Physical Review Applied 15 5, 054012 (2021).

[4] Jinfeng Zeng, Chenfeng Cao, Chao Zhang, Pengxiang Xu, and Bei Zeng, “A variational quantum algorithm for Hamiltonian diagonalization”, arXiv:2008.09854.

[5] Ajinkya Borle, Vincent E. Elfving, and Samuel J. Lomonaco, “Quantum Approximate Optimization for Hard Problems in Linear Algebra”, arXiv:2006.15438.

[6] Jinfeng Zeng, Zipeng Wu, Chenfeng Cao, Chao Zhang, Shiyao Hou, Pengxiang Xu, and Bei Zeng, “Simulating noisy variational quantum eigensolver with local noise models”, arXiv:2010.14821.

[7] Fanxu Meng, “Quantum Algorithm for DOA Estimation in Hybrid Massive MIMO”, arXiv:2102.03963.

[8] Guangxi Li, Zhixin Song, and Xin Wang, “VSQL: Variational Shadow Quantum Learning for Classification”, arXiv:2012.08288.

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This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.

Generative Data Intelligence

Variational Quantum Singular Value Decomposition

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