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arXiv:2601.19184

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

Kaiwen Weng, Zhaoyuan Meng, Zixuan Yang, Guohui Hu

quant-phSubmitted Jan 27, 2026• Announced Jan 28, 2026• 0 citations
v1 · Jan 27, 2026v2 · Feb 14, 2026
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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.

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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.
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.
The nonlinear Schrödinger 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.
In this work, we propose an operator-splitting variational quantum algorithm, termed Dirac-sVQA, for simulating the nonlinear Dirac equation (NLDE). The main difficulty arises from the state-dependent nonlinear interaction, its time-discrete update depends explicitly on the intermediate spinor state and, in general, cannot be implemented as a fixed state-independent unitary circuit. To address this difficulty, we decompose the NLDE evolution into a structured linear Dirac substep and a nonlinear variational correction. The linear substep is implemented by a spinor-Fourier Dirac propagator on a joint position-spin register, preserving the spin-momentum coupling and mass-induced spin evolution of the Dirac operator. The nonlinear correction is reformulated as a measurement-based variational update through a small set of overlap, self-channel, and cross-channel observables. We provide the corresponding quantum circuits and derive measurement-aware resource and complexity estimates. Numerical experiments in several nonlinear regimes show that Dirac-sVQA accurately captures both the total density and the componentwise spinor dynamics, agrees well with classical Fourier pseudospectral splitting solutions, and exhibits stable error behavior over time. These results provide numerical evidence for the feasibility of operator-splitting variational quantum simulation for nonlinear relativistic wave equations.
The time-dependent one-dimensional nonlinear Schrödinger equation (NLSE) is solved numerically by a hybrid pseudospectral-variational quantum algorithm that connects a pseudospectral step for the Hamiltonian term with a variational step for the nonlinear term. The Hamiltonian term is treated as an integrating factor by forward and backward Fourier transforms, which are here carried out classically. This split allows us to avoid higher-order time integration schemes, to apply a first-order explicit time stepping for the remaining nonlinear NLSE term in a variational algorithm block, and thus to avoid numerical instabilities. We demonstrate that the analytical solution is reproduced with a small root mean square error for a long time interval over which a nonlinear soliton propagates significantly forward in space while keeping its shape. We analyze the accuracy and complexity of the quantum algorithm, the expressibility of the ansatz circuit and compare it with classical approaches. Furthermore, we investigate the influence of algorithm parameters on the accuracy of the results, including the temporal step width and the depth of the quantum circuit.
From fluid flow and transport to collective dynamics, numerical simulation of nonlinear partial differential equations underpins modern scientific computing. Extending this capability to quantum computers remains a longstanding challenge because nonlinear and non-Hermitian evolution is fundamentally incompatible with conventional Hamiltonian-based quantum simulation. Here we experimentally realize the time evolution of nonlinear fluid dynamics on a quantum processor using a hybrid variational framework for the viscous and inviscid Burgers equations. Our approach directly encodes the nonlinear dynamics into a variational optimization procedure, avoiding the enlarged linear embeddings and truncation overhead associated with Carleman linearization-based quantum algorithms. We further demonstrate convection-dominated dynamics corresponding to Reynolds numbers of order 10210^2102. We encode the governing evolution into parametrized quantum circuits and iteratively reconstruct the time-dependent field through quantum-classical optimization. By introducing a zero-noise extrapolation method without additional circuit-folding overhead, we accurately execute deep error-circuits with entangling-gate counts beyond those typical of Hadamard test circuits. We accurately reconstruct the time evolution across multiple timesteps despite hardware noise and finite device coherence. Our results constitute, to our knowledge, the first experimental realization of nonlinear time propagation on a quantum processor, extending quantum simulation beyond predominantly linear settings and establishing a route toward quantum computation for nonlinear continuum dynamics.
We introduce the Quantum Split-Step Fourier (QSSF) algorithm for nonlinear optical waveguides, a numerical framework that combines split-step propagation of the nonlinear Schrödinger equation with a commutator-preserving Bogoliubov evolution of Gaussian quantum fluctuations. The method propagates the classical mean field together with the Bogoliubov matrices UUU and VVV, from which reduced second moments, covariance matrices, symplectic eigenvalues, and entropic measures are constructed for arbitrary spectral windows. Applied to soliton-driven resonant radiation, QSSF shows that the selected radiation band acquires a steadily increasing von Neumann entropy and a corresponding loss of purity, quantifying its entanglement with the rest of the spectrum in the lossless Gaussian setting. The analysis also reveals a surprisingly pronounced low-dimensional structure: although the radiation occupies many Fourier bins, its reduced Gaussian state is dominated by only a few Williamson modes. QSSF therefore provides a practical information-theoretic diagnostic for quantum correlations in nonlinear frequency conversion, supercontinuum generation, and multimode squeezed-light formation in ultrafast waveguide platforms.