Variational quantum nonlinear solver for nonlinear dynamics: Implementation and investigation

The variational quantum nonlinear solver (VQNLS), a quantum-enhanced computational method for nonlinear dynamical systems, converts solution tasks into global spatiotemporal optimization via parameterized ansatz circuits, thereby eliminating the need for iterative quantum state tomography. Herein, we present an extended implementation scheme of the VQNLS, which is compatible with most numerical discretization schemes and boundary conditions. The method is validated using a nonlinear ordinary differential equation (ODE) and the Burgers' equation using quantum virtual machines and superconducting quantum computers. Experimental results show that (1) the VQNLS can effectively solve nonlinear equations. On an ideal simulator, the ODE's solution achieves a relative error of only 0.17% when the loss function converges to 1×10−6. For the Burgers' equation, VQNLS captures key solution features with an average relative error below 5% on quantum virtual machines, while the average relative error increases to 7.91% on superconducting quantum computers. (2) VQNLS is highly sensitive to the loss function. (3) An appropriate additional loss function mitigates this issue by improving training efficiency and suppressing errors. At a fixed loss threshold of 1×10−3, it reduces training steps from 100 to 56 and cuts the maximum relative error from 15.3% to 4.1%. (4) Furthermore, under the same loss threshold, VQNLS's accuracy improves with the computational scale, with the average relative error decreasing monotonically in a power-law manner. Despite inherent limitations, including high error sensitivity and the challenge of selecting high-performance ansatz circuits, we consider that VQNLS provides a promising framework for engineering-oriented approximate simulations of nonlinear systems.