Exponential Quantum Advantages for Practical Non-Hermitian Eigenproblems

2024-01-22

Physical Review Letters

AI Summary

This work addresses non-Hermitian eigenvalue problems, specifically the localization of eigenvalues within a prescribed neighborhood of a line in the complex plane. Method: We propose the first general quantum algorithmic framework for this task, integrating quantum phase estimation, block-encoding techniques, and non-Hermitian Hamiltonian simulation—overcoming prior restrictions to Hermitian systems. Contributions/Results: (1) We achieve the first efficient quantum verification of PT-symmetry breaking, with a rigorous proof of exponential speedup; (2) we provide the first quantum algorithm for estimating the Liouvillian spectral gap, yielding provable exponential acceleration for determining relaxation times in open quantum systems; (3) the framework supports both unconstrained spectral search and precise eigenvalue localization within user-defined target regions. Our results extend quantum computation to critical domains including non-Hermitian physics, open quantum systems, and Markov processes, significantly enhancing computational efficiency.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Non-convex Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Economic ramifications for generative AI infrastructure and applications
• Security and Privacy: Large-scale security measurements

Abstract

While non-Hermitian physics has attracted considerable attention, current studies are limited to small or classically solvable systems. Quantum computing, as a powerful eigensolver, have predominantly been applied to Hermitian domain, leaving their potential for studying non-Hermitian problems largely unexplored. We extend the power of quantum computing to general non-Hermitian eigenproblems. Our approach works for finding eigenvalues without extra constrains, or eigenvalues closest to specified points or lines, thus extending results for ground energy and energy gap problems for Hermitian matrices. Our algorithms have broad applications, and as examples, we consider two central problems in non-Hermitian physics. Firstly, our approach is the first to offer an efficient quantum solution to the witness of spontaneous $PT$-symmetry breaking, and provide provable, exponential quantum advantage. Secondly, our approach enables the estimation of Liouvillian gap, which is crucial for characterizing relaxation times. Our general approach can also find applications in many other areas, such as the study of Markovian stochastic processes. These results underscore the significance of our quantum algorithms for addressing practical eigenproblems across various disciplines.

Problem

• Developing quantum algorithms for non-Hermitian eigenvalue problems
• Achieving exponential speedup over classical computational methods
• Addressing applications in PT-symmetry breaking and Liouvillian gaps

Innovation

• Quantum algorithm for non-Hermitian eigenvalue problems
• Fuzzy quantum eigenvalue detector with divide-and-conquer
• Exponential speedup for complex plane eigenvalue isolation