Quantum computing of the nonlinear Schr\"odinger equation via measurement-induced potential reconstruction

Abstract

The nonlinear Schr\"odinger 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. Subsequently, the Hadamard test is employed to measure the Fourier components of low-wavenumber modes, enabling the efficient reconstruction of nonlinear potentials. The phase transformation corresponding to the reconstructed potential is then implemented via a quantum circuit using the phase kickback technique. To validate the efficacy of the proposed algorithm, we numerically simulate the evolution of a Gaussian wave packet, a soliton wave, and the wake flow past a cylinder. The simulation results demonstrate excellent agreement with the corresponding classical solutions. This work provide a concrete basis for analyzing accuracy-cost trade-offs in quantum-classical simulations of nonlinear dispersive wave dynamics.