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YMSC-BIMSA Quantum Information Seminar
Quantum Eigenvalue Estimation for Non-Normal Matrices: Recent Developments
Quantum Eigenvalue Estimation for Non-Normal Matrices: Recent Developments
Organizers
Speaker
Yukun Zhang
Time
Friday, November 14, 2025 4:00 PM - 5:30 PM
Venue
Shuangqing-B627
Online
Zoom 230 432 7880
(BIMSA)
Abstract
Non-Hermitian operators appear across physics, engineering and probability — from 𝑃𝑇-symmetric quantum systems and open-system Liouvillians to nonreversible Markov chains — yet efficient numerical and quantum methods for their spectra lag far behind the Hermitian case. In this talk I present two complementary advances that bring powerful quantum techniques to bear on genuinely non-Hermitian and non-normal eigenproblems.
First, based on our recent work Phys. Rev. Lett. 135, 140601 (2025), I will introduce a quantum algorithm that isolates eigenvalues near any chosen line in the complex plane by combining a fuzzy quantum eigenvalue detector with a divide-and-conquer isolation strategy. The approach generalizes spectral-targeting methods beyond ground states and spectral gaps, and yields provable exponential speedups for broad classes of non-Hermitian matrices relevant to physics and stochastic processes. Applications include detecting spontaneous 𝑃𝑇-symmetry breaking, estimating Liouvillian gaps, and spectral analysis of classical Markov dynamics.
Second, based on our next recent work arXiv:2510.19651, I will present a new family of hybrid quantum-classical algorithms that synthesize eigenvalue signals via tailored quantum simulation protocols and recover them using advanced classical signal-processing. When supplied with purified input states, these methods achieve Heisenberg-limited precision, extending guided-local Hamiltonian ideas into the non-Hermitian regime and offering asymptotically optimal dependence on accuracy estimation for non-normal matrices.
Together these results significantly broaden the algorithmic toolkit for non-Hermitian linear algebra and point to concrete quantum advantages for physically motivated problems. Finally, I will discuss some possible future directions.
First, based on our recent work Phys. Rev. Lett. 135, 140601 (2025), I will introduce a quantum algorithm that isolates eigenvalues near any chosen line in the complex plane by combining a fuzzy quantum eigenvalue detector with a divide-and-conquer isolation strategy. The approach generalizes spectral-targeting methods beyond ground states and spectral gaps, and yields provable exponential speedups for broad classes of non-Hermitian matrices relevant to physics and stochastic processes. Applications include detecting spontaneous 𝑃𝑇-symmetry breaking, estimating Liouvillian gaps, and spectral analysis of classical Markov dynamics.
Second, based on our next recent work arXiv:2510.19651, I will present a new family of hybrid quantum-classical algorithms that synthesize eigenvalue signals via tailored quantum simulation protocols and recover them using advanced classical signal-processing. When supplied with purified input states, these methods achieve Heisenberg-limited precision, extending guided-local Hamiltonian ideas into the non-Hermitian regime and offering asymptotically optimal dependence on accuracy estimation for non-normal matrices.
Together these results significantly broaden the algorithmic toolkit for non-Hermitian linear algebra and point to concrete quantum advantages for physically motivated problems. Finally, I will discuss some possible future directions.
Speaker Intro
Yukun Zhang, Ph.D. student (admitted 2022) at the Center on Frontiers of Computing Studies, Peking University. His research interests include quantum algorithm design, quantum complexity theory, and the study of quantum advantage.