Hybrid quantum-classical algorithms for complex nonlinear partial differential equations with Ginzburg-Landau potential and vortex motion laws

TL;DR

This paper introduces hybrid quantum-classical algorithms for nonlinear PDEs with vortex dynamics, achieving exponential speedup in spatial problem size and extending to 3D superconductivity models.

Contribution

The paper develops novel hybrid algorithms leveraging asymptotic PDE properties, combining classical vortex dynamics with quantum elliptic solvers for improved efficiency.

Findings

Exponential speedup in spatial problem size for 2D nonlinear Schrödinger equation.

Effective quantum algorithms for vortex dynamics in 2D and 3D superconductivity.

Numerical results validate the PDE reduction and algorithm effectiveness.

Abstract

We propose quantum algorithms for complex-valued nonlinear partial differential equations in the strongly nonlinear regime, where the dynamics is governed by vortex cores, phase singularities, and nonlinear vortex interactions. Examples include the complex-valued nonlinear Schr\"odinger equation, as well as nonlinear heat and wave equations with Ginzburg--Landau-type nonlinearity. In the strongly nonlinear regime, the solutions to these equations are asymptotically governed by, in leading order, linear elliptic equations, coupled with low-dimensional vortex dynamics, where the vortex cores correspond to topological defects in superconductors. Our hybrid quantum-classical algorithms utilize this asymptotic property, in which the vortex dynamic is advanced classically while the boundary-value problem of linear elliptic equation is handled by quantum algorithms. For the two-dimensional nonlinear Schr\"odinger equation, we also combine quantum BPX preconditioning with Schr\"odingerization to estimate physically relevant observables in the small-output regime. This yields, already in two dimensions, an {\it exponential} improvement in the dependence on the spatial problem size, while the dependence on the target accuracy remains essentially linear up to polylogarithmic factors. We further show that the same principle extends to dissipative Ginzburg--Landau vortex dynamics and to vortex filaments in three-dimensional superconductivity. Numerical results support the validity of this PDE reduction and the effectiveness of the proposed approach.