Treatment of linear and nonlinear pantograph equations with a quantum pseudo-spectral method

Abstract We present a hybrid computational framework that integrates quantum representations by helping tensor products with classical spectral methods to solve pantograph equations, a prominent class of delay-differential equations with applications across applied mathematics. Both single and multiple delay cases are considered. Our approach establishes a Chebyshev-based spectral framework, leveraging careful selection of basis functions and collocation points. Nonlinearities are addressed via successive linearization, which iteratively reduces the problem to systems of linear equations solved with a quantum algorithm. The solution is then encoded as a quantum state, enabling efficient processing. The methodology is validated through numerical experiments, with comparative results confirming its efficiency. The numerical results presented in this work were obtained using classical computation. The quantum formalism serves as a blueprint for potential future implementation on quantum hardware.