Quantum-inspired algorithms for solving low-rank linear equation systems with logarithmic dependence on the dimension
We present two efficient classical analogues of the quantum matrix inversion algorithm [16] for low-rank matrices. Inspired by recent work of Tang [27], assuming length-square sampling access to input data, we implement the pseudoinverse of a low-rank matrix allowing us to sample from the solution t...
| Main Authors: | , , , , , |
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| Other Authors: | |
| Format: | Article |
| Language: | English |
| Published: |
2022
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| Online Access: | https://hdl.handle.net/1721.1/138872 |
| _version_ | 1811083619530178560 |
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| author | Chia, NH Gilyén, A Lin, HH Lloyd, S Tang, E Wang, C |
| author2 | Massachusetts Institute of Technology. Department of Mechanical Engineering |
| author_facet | Massachusetts Institute of Technology. Department of Mechanical Engineering Chia, NH Gilyén, A Lin, HH Lloyd, S Tang, E Wang, C |
| author_sort | Chia, NH |
| collection | MIT |
| description | We present two efficient classical analogues of the quantum matrix inversion algorithm [16] for low-rank matrices. Inspired by recent work of Tang [27], assuming length-square sampling access to input data, we implement the pseudoinverse of a low-rank matrix allowing us to sample from the solution to the problem Ax = b using fast sampling techniques. We construct implicit descriptions of the pseudo-inverse by finding approximate singular value decomposition of A via subsampling, then inverting the singular values. In principle, our approaches can also be used to apply any desired “smooth” function to the singular values. Since many quantum algorithms can be expressed as a singular value transformation problem [15], our results indicate that more low-rank quantum algorithms can be effectively “dequantised” into classical length-square sampling algorithms. |
| first_indexed | 2024-09-23T12:35:57Z |
| format | Article |
| id | mit-1721.1/138872 |
| institution | Massachusetts Institute of Technology |
| language | English |
| last_indexed | 2024-09-23T12:35:57Z |
| publishDate | 2022 |
| record_format | dspace |
| spelling | mit-1721.1/1388722023-04-07T20:45:59Z Quantum-inspired algorithms for solving low-rank linear equation systems with logarithmic dependence on the dimension Chia, NH Gilyén, A Lin, HH Lloyd, S Tang, E Wang, C Massachusetts Institute of Technology. Department of Mechanical Engineering Massachusetts Institute of Technology. Department of Physics We present two efficient classical analogues of the quantum matrix inversion algorithm [16] for low-rank matrices. Inspired by recent work of Tang [27], assuming length-square sampling access to input data, we implement the pseudoinverse of a low-rank matrix allowing us to sample from the solution to the problem Ax = b using fast sampling techniques. We construct implicit descriptions of the pseudo-inverse by finding approximate singular value decomposition of A via subsampling, then inverting the singular values. In principle, our approaches can also be used to apply any desired “smooth” function to the singular values. Since many quantum algorithms can be expressed as a singular value transformation problem [15], our results indicate that more low-rank quantum algorithms can be effectively “dequantised” into classical length-square sampling algorithms. 2022-01-11T15:24:12Z 2022-01-11T15:24:12Z 2020-12-01 2022-01-11T15:17:55Z Article http://purl.org/eprint/type/ConferencePaper https://hdl.handle.net/1721.1/138872 Chia, NH, Gilyén, A, Lin, HH, Lloyd, S, Tang, E et al. 2020. "Quantum-inspired algorithms for solving low-rank linear equation systems with logarithmic dependence on the dimension." Leibniz International Proceedings in Informatics, LIPIcs, 181. en 10.4230/LIPIcs.ISAAC.2020.47 Leibniz International Proceedings in Informatics, LIPIcs Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. application/pdf DROPS |
| spellingShingle | Chia, NH Gilyén, A Lin, HH Lloyd, S Tang, E Wang, C Quantum-inspired algorithms for solving low-rank linear equation systems with logarithmic dependence on the dimension |
| title | Quantum-inspired algorithms for solving low-rank linear equation systems with logarithmic dependence on the dimension |
| title_full | Quantum-inspired algorithms for solving low-rank linear equation systems with logarithmic dependence on the dimension |
| title_fullStr | Quantum-inspired algorithms for solving low-rank linear equation systems with logarithmic dependence on the dimension |
| title_full_unstemmed | Quantum-inspired algorithms for solving low-rank linear equation systems with logarithmic dependence on the dimension |
| title_short | Quantum-inspired algorithms for solving low-rank linear equation systems with logarithmic dependence on the dimension |
| title_sort | quantum inspired algorithms for solving low rank linear equation systems with logarithmic dependence on the dimension |
| url | https://hdl.handle.net/1721.1/138872 |
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