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arXiv · 2401.12091

Exponential quantum advantages for practical non-Hermitian eigenproblems

Xiao-Ming Zhang·Yukun Zhang·Wenhao He·Xiao Yuan
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Abstract

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.

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Xiao-Ming Zhang, Yukun Zhang, Wenhao He, Xiao Yuan. 2024-01-22. Exponential quantum advantages for practical non-Hermitian eigenproblems. https://doi.org/10.1103/3n8f-k8pl

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