Solving linear and nonlinear Caputo fractional differential equations with a quantum pseudo-spectral approach

Abstract

Linear and nonlinear Caputo time-fractional differential equations play a fundamental role in pure and applied mathematics as well as theoretical physics. This article develops a hybrid methodology that combines quantum computing paradigms with spectral methods to solve such equations, employing shifted fractional Chebyshev polynomials as basis functions. The simultaneous treatment of linear and nonlinear fractional equations requires careful selection of both basis functions and collocation points. This choice proves essential for avoiding the chain rule complication inherent in Caputo’s derivative formulation. Crucially, the chosen basis functions generate a triangular operational matrix, thereby improving both the accuracy and computational efficiency of the pseudo-spectral approach. Within our computational framework, the solution at the terminal time is encoded as a final quantum state. We demonstrate the method’s efficacy through numerical experiments and comparative analysis with existing approaches.