Quantum-Accelerated Solution of Nonlinear Equations from Variational Principles

TL;DR

This paper introduces a quantum algorithm for solving nonlinear equilibrium equations based on variational principles, enabling scalable and memory-efficient quantum acceleration for complex physical systems.

Contribution

It presents a novel quantum algorithm that linearizes nonlinear equilibrium problems for fault-tolerant quantum computers, extending quantum advantages to nonlinear systems.

Findings

Successfully predicts nonlinear deformation in mechanical systems

Offers linear scaling of complexity independent of system size

Provides significant memory savings over classical methods

Abstract

The variational principle serves as a fundamental framework for describing equilibrium states of physical systems via the minimization or extremization of an energy-like functional. While quantum algorithms have demonstrated promising advances in efficiently solving linear problems rooted in this principle, extending these techniques to nonlinear equilibrium equations--ubiquitous in structural mechanics, fluid dynamics, and electromagnetism--remains an outstanding challenge. Here, we introduce a novel algorithm tailored for fault-tolerant quantum computers (FTQCs) that directly addresses nonlinear equilibrium conditions governed by the variational principle. Our approach recasts the static nonlinear problem as a time-evolution process, enabling an effective linearization amenable to quantum acceleration. This construction permits quantum acceleration of nonlinear equilibrium computations on FTQCs. Compared to classical solvers, our method offers significant memory savings without compromising accuracy, with computational complexity scaling linearly in simulation time and independent of system size. We validate the algorithm through accurate prediction of nonlinear deformation in springs and truss systems, demonstrating its potential for scalable quantum acceleration of nonlinear physical phenomena. This work paves the way toward leveraging fault-tolerant quantum computing for complex nonlinear systems across physics and engineering disciplines.