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arXiv:2606.08053

Numerical solution of the nonlinear Dirac equation by a splitting variational quantum algorithm

Qian Zuo, Ying He, Xiaofei Zhao

quant-phmath.NASubmitted Jun 6, 2026• Announced Jun 9, 2026• 0 citations
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Abstract

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.

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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.
A quantum algorithm that solves the time-dependent Dirac equation on a digital quantum computer is developed and analyzed. The time evolution is performed by an operator splitting decomposition technique that allows for a mapping of the Dirac operator to a quantum walk supplemented by unitary rotation steps in spinor space. Every step of the splitting method is decomposed into sets of quantum gates. It is demonstrated that the algorithm has an exponential speedup over the implementation of the same numerical scheme on a classical computer, as long as certain conditions are satisfied. Finally, an explicit decomposition of this algorithm into elementary gates from a universal set is carried out to determine the resource requirements. It is shown that a proof-of-principle calculation may be possible with actual quantum technologies.
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.
A quantum algorithm that solves the time-dependent Dirac equation on a digital quantum computer is developed and analyzed. The time evolution is performed by an operator-splitting decomposition technique that allows for a mapping of the Dirac operator to a quantum walk supplemented by unitary rotation steps in spinor space. Every step of the splitting method is decomposed into sets of quantum gates. It is demonstrated that the algorithm has an exponential speed-up over the implementation of the same numerical scheme on a classical computer, as long as certain conditions are satisfied. Finally, an explicit decomposition of this algorithm into elementary gates from a universal set is carried out to determine the resource requirements. It is shown that a proof-of-principle calculation may be possible with actual quantum technologies.
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.
Quantum algorithms for nonlinear partial differential equations remain challenging because nonlinear dynamics are not directly amenable to unitary quantum simulation. Building on the level-set formulation, we construct a quantum algorithm and provide an explicit gate-level implementation for solving scalar conservation laws. The nonlinear equation is first lifted to a linear Liouville equation, discretized by finite differences, and then embedded into a unitary evolution through Schrödingerisation. We further develop quantum procedures for estimating relevant observables from the evolved state. Error bounds and gate-complexity estimates are established for the complete algorithm. The resulting complexity comparison demonstrates a quantum advantage for observable estimation in sufficiently high spatial dimensions, under standard assumptions on state preparation and oracle access. Finally, numerical experiments validate the accuracy, multidimensional applicability, and predicted scaling of the proposed method.