Exponential quantum advantages for practical non-Hermitian eigenproblems

• URL: http://arxiv.org/abs/2401.12091v3
• Date: Fri, 03 Oct 2025 07:45:01 GMT
• Title: Exponential quantum advantages for practical non-Hermitian eigenproblems
• Authors: Xiao-Ming Zhang, Yukun Zhang, Wenhao He, Xiao Yuan,
• Abstract summary: We develop a quantum algorithm to address general non-Hermitian eigenvalue problems.<n>Our method combines a fuzzy quantum eigenvalue detector with a divide-and-conquer strategy to efficiently isolate relevant eigenvalues.<n>We discuss the broad applications in detecting spontaneous $PT$-symmetry breaking, estimating Liouvillian gaps, and analyzing classical Markov processes.
• Score: 9.59420259768862
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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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