A quantum nonlinear solver based on the asymptotic numerical method

TL;DR

This paper introduces the quantum asymptotic numerical method (qANM), a framework leveraging quantum computing to solve nonlinear problems efficiently through high-order perturbation techniques and quantum linear solvers.

Contribution

The work presents a novel quantum-based approach for nonlinear problem solving, combining Taylor series expansions with quantum algorithms and demonstrating practical validation on quantum hardware.

Findings

Numerical simulations confirm convergence of qANM on quantum simulators.

High-order ANM effectively captures solution paths in nonlinear problems.

Quantum hardware experiment achieves 98% accuracy despite noise.

Abstract

Quantum computing offers a promising avenue for advancing computational methods in science and engineering. In this work, we introduce the quantum asymptotic numerical method (qANM), a framework for solving nonlinear problems using quantum computing. Based on the principle of high-order perturbation techniques, the proposed method uses Taylor series expansions to transform complex nonlinear systems into sequences of linear equations. We integrate the method with the variational quantum linear solver and a quantum-enhanced Jacobi method. Numerical simulations on a quantum simulator validate the convergence of the method. In particular, the high-order ANM formulation demonstrates robustness in addressing nonlinear problems by effectively capturing the solution path through Taylor series expansions. Furthermore, a highlight of this work is a proof-of-principle experiment on a superconducting quantum processor. Despite the noise inherent in near-term quantum hardware, the experiment achieves 98% accuracy in tracking the nonlinear solution path. We believe this work provides a useful reference for applying quantum computing to nonlinear computational mechanics.