Heisenberg-Limited Quantum Eigenvalue Estimation for Non-normal Matrices

TL;DR

This paper introduces quantum algorithms capable of estimating eigenvalues of non-normal matrices with Heisenberg-limited precision, expanding quantum computational capabilities beyond Hermitian matrices.

Contribution

The authors develop a novel quantum algorithm framework for non-normal matrices, achieving optimal precision and extending existing methods into the non-Hermitian domain.

Findings

Achieves Heisenberg-limited eigenvalue estimation for non-normal matrices.

Extends the guided local Hamiltonian framework to non-Hermitian systems.

Provides a scalable quantum approach to complex linear algebra problems.

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.