Variational Quantum Singular Value Decomposition
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 subrout...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
Verein zur Förderung des Open Access Publizierens in den Quantenwissenschaften
2021-06-01
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| Series: | Quantum |
| Online Access: | https://quantum-journal.org/papers/q-2021-06-29-483/pdf/ |
| _version_ | 1818590929811931136 |
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| author | Xin Wang Zhixin Song Youle Wang |
| author_facet | Xin Wang Zhixin Song Youle Wang |
| author_sort | Xin Wang |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-12-16T10:04:21Z |
| format | Article |
| id | doaj.art-36e16ea074db45e3bbf04b6db8e1e428 |
| institution | Directory Open Access Journal |
| issn | 2521-327X |
| language | English |
| last_indexed | 2024-12-16T10:04:21Z |
| publishDate | 2021-06-01 |
| publisher | Verein zur Förderung des Open Access Publizierens in den Quantenwissenschaften |
| record_format | Article |
| series | Quantum |
| spelling | doaj.art-36e16ea074db45e3bbf04b6db8e1e4282022-12-21T22:35:42ZengVerein zur Förderung des Open Access Publizierens in den QuantenwissenschaftenQuantum2521-327X2021-06-01548310.22331/q-2021-06-29-48310.22331/q-2021-06-29-483Variational Quantum Singular Value DecompositionXin WangZhixin SongYoule WangSingular 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.https://quantum-journal.org/papers/q-2021-06-29-483/pdf/ |
| spellingShingle | Xin Wang Zhixin Song Youle Wang Variational Quantum Singular Value Decomposition Quantum |
| title | Variational Quantum Singular Value Decomposition |
| title_full | Variational Quantum Singular Value Decomposition |
| title_fullStr | Variational Quantum Singular Value Decomposition |
| title_full_unstemmed | Variational Quantum Singular Value Decomposition |
| title_short | Variational Quantum Singular Value Decomposition |
| title_sort | variational quantum singular value decomposition |
| url | https://quantum-journal.org/papers/q-2021-06-29-483/pdf/ |
| work_keys_str_mv | AT xinwang variationalquantumsingularvaluedecomposition AT zhixinsong variationalquantumsingularvaluedecomposition AT youlewang variationalquantumsingularvaluedecomposition |