Quantum Signal Processing and Quantum Singular Value Transformation on $U(N)$
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866915901737009152 |
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| author | Lu, Xi Liu, Yuan Lin, Hongwei |
| author_facet | Lu, Xi Liu, Yuan Lin, Hongwei |
| contents | Quantum signal processing and quantum singular value transformation are powerful tools to implement polynomial transformations of block-encoded matrices on quantum computers, and has achieved asymptotically optimal complexity in many prominent quantum algorithms. We propose a framework of quantum signal processing and quantum singular value transformation on $U(N)$, which realizes multiple polynomials simultaneously from a block-encoded input, as a generalization of those on $U(2)$ in the original frameworks. We provide a comprehensive characterization of achievable polynomial matrices and give recursive algorithms to construct the quantum circuits that realize desired polynomial transformations. As three example applications, we propose a framework to realize bi-variate polynomial functions, demonstrate $N$-interval decision achieving $O(d)$ query complexity with a $\log_2 N$ improvement over iterative $U(2)$-QSP requiring $O(d\log_2 N)$ queries, and present a quantum amplitude estimation algorithm achieving the Heisenberg limit without adaptive measurements. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_01439 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Quantum Signal Processing and Quantum Singular Value Transformation on $U(N)$ Lu, Xi Liu, Yuan Lin, Hongwei Quantum Physics Quantum signal processing and quantum singular value transformation are powerful tools to implement polynomial transformations of block-encoded matrices on quantum computers, and has achieved asymptotically optimal complexity in many prominent quantum algorithms. We propose a framework of quantum signal processing and quantum singular value transformation on $U(N)$, which realizes multiple polynomials simultaneously from a block-encoded input, as a generalization of those on $U(2)$ in the original frameworks. We provide a comprehensive characterization of achievable polynomial matrices and give recursive algorithms to construct the quantum circuits that realize desired polynomial transformations. As three example applications, we propose a framework to realize bi-variate polynomial functions, demonstrate $N$-interval decision achieving $O(d)$ query complexity with a $\log_2 N$ improvement over iterative $U(2)$-QSP requiring $O(d\log_2 N)$ queries, and present a quantum amplitude estimation algorithm achieving the Heisenberg limit without adaptive measurements. |
| title | Quantum Signal Processing and Quantum Singular Value Transformation on $U(N)$ |
| topic | Quantum Physics |
| url | https://arxiv.org/abs/2408.01439 |