Quantum algorithm for the nonlinear Schrödinger equation via the Lax-pair scattering

The nonlinear Schr\"odinger equation (NLSE) governs a broad class of wave phenomena, including deep-water waves, quantum turbulence, and solitons. The multiscale spatiotemporal coupling inherent in these systems imposes severe computational bottlenecks on classical high-fidelity numerical simulations. While quantum computing offers the potential for exponential speedup, its unitary dynamics pose a fundamental challenge to solve the NLSE. We propose a quantum framework based on the Lax-pair scattering for solving the 1D NLSE. Specifically, the physical field is first mapped into the spectral space via a quantum direct scattering circuit. Following a decoupled linear time evolution, the physical solution is reconstructed through an inverse scattering transform utilizing the quantum singular value transformation. Since the temporal evolution is performed analytically in the scattering domain, the framework bypasses iterative time stepping, rendering it highly advantageous for long-time simulations. To demonstrate the accuracy and noise resilience of this approach, we simulate a Gaussian wave packet under quantum noise, two-soliton collisions, breather dynamics, and modulational instability on a quantum emulator.