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YMSC-BIMSA量子信息讨论班
Quantum Eigenvalue Estimation for Non-Normal Matrices: Recent Developments
Quantum Eigenvalue Estimation for Non-Normal Matrices: Recent Developments
演讲者
张宇鹍
时间
2025年11月14日 16:00 至 17:30
地点
Shuangqing-B627
线上
Zoom 230 432 7880
(BIMSA)
摘要
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.
演讲者介绍
张宇鹍，2022级北京大学前沿计算研究中心博士生，他的兴趣包括量子算法设计，量子复杂度论及量子优越性的研究。