Quantum computing of the nonlinear Schrödinger equation via measurement-assisted potential reconstruction

The nonlinear Schrödinger equation (NLSE) is a fundamental model that describes diverse complex phenomena in nature. However, simulating the NLSE on a quantum computer is inherently challenging due to the presence of the nonlinear term. We propose a hybrid quantum–classical framework for simulating the NLSE based on the split-step Fourier method. During the linear propagation step, we apply the kinetic evolution operator to generate an intermediate quantum state. The Hadamard test is then used to measure the Fourier components of low-wavenumber modes, from which an approximate nonlinear potential is reconstructed by classical post-processing. The phase transformation corresponding to the reconstructed potential is then implemented as a diagonal phase operation. To validate the proposed method, we numerically simulate the evolution of a Gaussian wave packet, a soliton wave, snake instability, and the wake flow past a cylinder. The simulation results show that the proposed method captures the dominant wave-function profiles, phase structures, and characteristic nonlinear dynamics. This work provides an implementation-level study and an error characterization to assess accuracy-cost trade-offs in hybrid quantum–classical simulations of nonlinear dispersive wave dynamics.