Quantum-Accelerated Solution of Nonlinear Equations from Variational Principles

Abstract

Nonlinear equilibrium problems derived from variational principles arise throughout physics and engineering, including structural mechanics, fluid dynamics, and electromagnetism. While fault-tolerant quantum algorithms have shown promising advantages for linear systems and linear dynamical simulations, extending quantum acceleration to nonlinear equilibrium problems remains a major challenge. Here we introduce a quantum algorithmic framework for nonlinear equilibrium analysis based on gradient-flow linearization. The key idea is to reformulate equilibrium conditions as nonlinear gradient-flow dynamics and transform the resulting evolution into a linear dynamical system using exact linearization techniques such as Carleman and Pivot Switching Carleman (PSC) linearization. This construction enables nonlinear equilibrium and energy-minimization problems to be addressed using quantum algorithms for linear dynamical simulation. We demonstrate the framework for nonlinear elasticity problems ranging from single nonlinear springs and chain-spring systems to two-dimensional truss structures. The resulting truncated linearized dynamics accurately reproduce nonlinear equilibrium states, while PSC linearization substantially improves stability in regimes where conventional Carleman linearization becomes unreliable. More broadly, our work establishes a connection between variational principles, nonlinear energy minimization, exact linearization, and quantum dynamical simulation. This perspective opens a route toward quantum algorithms for nonlinear physical systems beyond the scope of existing linear-system-based approaches.