Exponential quantum advantages for practical non-Hermitian eigenproblems
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866916985909018624 |
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| author | Zhang, Xiao-Ming Zhang, Yukun He, Wenhao Yuan, Xiao |
| author_facet | Zhang, Xiao-Ming Zhang, Yukun He, Wenhao Yuan, Xiao |
| contents | Non-Hermitian physics has emerged as a rich field of study, with applications ranging from $PT$-symmetry breaking and skin effects to non-Hermitian topological phase transitions. Yet most studies remain restricted to small-scale or classically tractable systems. While quantum computing has shown strong performance in Hermitian eigenproblems, its extension to the non-Hermitian regime remains largely unexplored. Here, we develop a quantum algorithm to address general non-Hermitian eigenvalue problems, specifically targeting eigenvalues near a given line in the complex plane -- thereby generalizing previous results on ground state energy and spectral gap estimation for Hermitian matrices. Our method combines a fuzzy quantum eigenvalue detector with a divide-and-conquer strategy to efficiently isolate relevant eigenvalues. This yields a provable exponential quantum speedup for non-Hermitian eigenproblems. Furthermore, we discuss the broad applications in detecting spontaneous $PT$-symmetry breaking, estimating Liouvillian gaps, and analyzing classical Markov processes. These results highlight the potential of quantum algorithms in tackling challenging problems across quantum physics and beyond. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_12091 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Exponential quantum advantages for practical non-Hermitian eigenproblems Zhang, Xiao-Ming Zhang, Yukun He, Wenhao Yuan, Xiao Quantum Physics Mesoscale and Nanoscale Physics Data Structures and Algorithms Numerical Analysis Computational Physics Non-Hermitian physics has emerged as a rich field of study, with applications ranging from $PT$-symmetry breaking and skin effects to non-Hermitian topological phase transitions. Yet most studies remain restricted to small-scale or classically tractable systems. While quantum computing has shown strong performance in Hermitian eigenproblems, its extension to the non-Hermitian regime remains largely unexplored. Here, we develop a quantum algorithm to address general non-Hermitian eigenvalue problems, specifically targeting eigenvalues near a given line in the complex plane -- thereby generalizing previous results on ground state energy and spectral gap estimation for Hermitian matrices. Our method combines a fuzzy quantum eigenvalue detector with a divide-and-conquer strategy to efficiently isolate relevant eigenvalues. This yields a provable exponential quantum speedup for non-Hermitian eigenproblems. Furthermore, we discuss the broad applications in detecting spontaneous $PT$-symmetry breaking, estimating Liouvillian gaps, and analyzing classical Markov processes. These results highlight the potential of quantum algorithms in tackling challenging problems across quantum physics and beyond. |
| title | Exponential quantum advantages for practical non-Hermitian eigenproblems |
| topic | Quantum Physics Mesoscale and Nanoscale Physics Data Structures and Algorithms Numerical Analysis Computational Physics |
| url | https://arxiv.org/abs/2401.12091 |