Quantum simulation of the nonlinear Schrödinger equation via measurement-induced potential reconstruction

TL;DR

This paper tackles the challenge of simulating the nonlinear Schrödinger equation (NLSE) on quantum hardware by developing a hybrid quantum–classical split-step spectral framework. It preserves the classical split-step structure, using a quantum Fourier transform for linear propagation and a measurement-assisted, spectrally filtered nonlinear update via the Hadamard test to reconstruct the nonlinear potential with reduced measurement costs. The authors demonstrate that, with appropriate spectral truncation and renormalization, the quantum scheme reproduces key features of the classical NLSE dynamics for 1D solitons, 2D Gaussian wave packets, and 2D wake flows past obstacles, while offering potential scalability advantages in high dimensions when the spectral support is compact. The work provides a concrete methodology and complexity estimates for balancing accuracy and measurement overhead in quantum–classical NLSE simulations, laying the groundwork for extending to more complex nonlinear wave phenomena.

Abstract

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. 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.

Figures (8)

• Figure 1: Schematic of the first three QSSFM time-step iterations. Blue lines denote the classical computer (CC). The operators $U_k$ and $U_p$ implement linear and nonlinear propagation, respectively. The operators enclosed in the dashed box represent computational processes that can be canceled. The measurement block extracts selected Fourier modes of the wave function. Following a Fourier transform on the CC, the approximate nonlinear potential is utilized to reconstruct the nonlinear operator $U_p$ for the subsequent quantum evolution step.
• Figure 2: Workflow of the filtered-QSSFM. The blue and pink regions represent the quantum and classical computation components, respectively. Rectangular boxes indicate computational steps, while the diamond-shaped box checks whether the current time $t$ equals the final time $t'$. Normalization of the values following the inverse Fourier transform enhances the simulation accuracy.
• Figure 3: Quantum circuit of the filtered-QSSFM incorporating the Hadamard test. Here, $H$ and $S$ denote the Hadamard gate and the phase gate, respectively, and $U$ is defined as $U = U_{\ell}^\dagger U_{\psi}$. The two Hadamard tests, implemented with and without the phase gate, correspond to measuring the real and imaginary parts of the retained mode.
• Figure 4: Soliton wave evolution obtained via the filtered-QSSFM with $n=8$ and varying retained qubits $m$. The amplitude spectra of the initial state are illustrated for (a) $m=4$ and (c) $m=3$. In these panels, blue circles denote the retained modes, while gray dashed lines represent the original modes. The corresponding time evolution in physical space over $t \in [0,5]$ is displayed for (b) $m=4$ and (d) $m=3$. Solid lines represent the filtered-QSSFM results, whereas the shaded regions indicate the ground-truth solution with the same grid resolution.
• Figure 5: Soliton wave evolution obtained by the filtered-QSSFM with retained qubits $m=4$ and different total qubits: (a) $n=5$, (b) $n=8$, and (c) $n=11$. Solid lines represent the filtered-QSSFM results, whereas the shaded regions indicate the ground-truth solution with the same grid resolution.