A Novel Quantum Algorithm for Solving Non-Linear Differential Equations with Potential Exponential Speedup

Abstract

Non-linear differential equations constitute the mathematical foundation of complex physical, biological, and engineering systems, yet classical numerical solvers often suffer from prohibitive computational costs as system dimensionality increases. Quantum computation offers a promising pathway for accelerating such calculations, although existing quantum algorithms primarily address linear differential models and fail to generalize efficiently to non-linear regimes. This study aims to develop and evaluate a novel quantum algorithm designed specifically to approximate solutions to non-linear differential equations with a potential exponential speedup over classical methods. The proposed approach integrates a variational quantum ansatz with non-linear Hamiltonian embedding and amplitude encoding to capture non-linearity within a tractable quantum framework. Simulations were conducted on noisy intermediate-scale quantum (NISQ) models and idealized quantum circuits to benchmark accuracy, convergence behavior, and computational scaling. The results indicate that the algorithm achieves stable convergence across representative non-linear systems while demonstrating a significant reduction in computational complexity relative to classical solvers, particularly for high-dimensional models. The study concludes that the proposed algorithm represents a viable direction for quantum-enhanced numerical analysis and may serve as a foundation for future quantum solvers targeting complex dynamical systems.