Heisenberg-Limited Quantum Eigenvalue Estimation for Non-normal Matrices

• URL: http://arxiv.org/abs/2510.19651v1
• Date: Wed, 22 Oct 2025 14:55:44 GMT
• Title: Heisenberg-Limited Quantum Eigenvalue Estimation for Non-normal Matrices
• Authors: Yukun Zhang, Yusen Wu, Xiao Yuan,
• Abstract summary: Estimating the eigenvalues of non-normal matrices is a foundational problem with far-reaching implications.<n>Here we introduce a new class of quantum algorithms that directly address this challenge.<n>Our work lays the foundation for a rigorous and scalable quantum computing approach to one of the most demanding problems in linear algebra.
• Score: 5.733109475878588
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Estimating the eigenvalues of non-normal matrices is a foundational problem with far-reaching implications, from modeling non-Hermitian quantum systems to analyzing complex fluid dynamics. Yet, this task remains beyond the reach of standard quantum algorithms, which are predominantly tailored for Hermitian matrices. Here we introduce a new class of quantum algorithms that directly address this challenge. The central idea is to construct eigenvalue signals through customized quantum simulation protocols and extract them using advanced classical signal-processing techniques, thereby enabling accurate and efficient eigenvalue estimation for general non-normal matrices. Crucially, when supplied with purified quantum state inputs, our algorithms attain Heisenberg-limited precision--achieving optimal performance. These results extend the powerful guided local Hamiltonian framework into the non-Hermitian regime, significantly broadening the frontier of quantum computational advantage. Our work lays the foundation for a rigorous and scalable quantum computing approach to one of the most demanding problems in linear algebra.

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