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

• URL: http://arxiv.org/abs/2604.14079v1
• Date: Wed, 15 Apr 2026 16:50:41 GMT
• Title: Hybrid quantum-classical algorithms for complex nonlinear partial differential equations with Ginzburg-Landau potential and vortex motion laws
• Abstract summary: We propose quantum algorithms for complex-valued nonlinear partial differential equations in the strongly nonlinear regime.<n>Examples include the complex-valued nonlinear Schrdinger equation, as well as nonlinear heat wave equations with Ginzburg-Landau-type nonlinearity.<n>We show that the same principle extends to dissipative Ginzburg--Landau vortex dynamics and to vortex filaments in three-dimensional superconductivity.
• Score: 35.75479170931658
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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 Schrödinger 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 Schrödinger equation, we also combine quantum BPX preconditioning with Schrödingerization 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.

Related papers

• Quantum simulation of the Liouville equation in classical mechanics with discontinuous potential via Schrödingerization [42.22857110018992]
  We develop quantum simulation algorithms for the Liouville equation of classical mechanics with discontinuous potential.<n>We combine Hamiltonian-preserving schemes by Jin and Wen in Commun. Math. Sci. 3(3), 285-315 (2005) with the Schrdingerization method, which embeds the resulting nonunitary semi-discrete dynamics into a unitary Schrdinger type system.
  arXiv  Detail & Related papers  (2026-06-13T02:49:05Z)
• Quantum algorithms for stochastic nonlinear differential equations [2.0065923589074735]
  We present a quantum subroutine for a broad class of $N$-dimensional differential equations with dissipation and quadratic drift.<n>The norm-preserving drifts applies to strongly nonlinear systems with all-to-all interactions.<n>Our method approximates expectation values of low-order correlation functions with rigorous error bounds.
  arXiv  Detail & Related papers  (2026-06-06T21:45:53Z)
• Adaptive-Growth Randomized Neural Networks for Level-Set Computation of Multivalued Nonlinear First-Order PDEs with Hyperbolic Characteristics [38.23142730599331]
  This paper proposes an Adaptive-Growth Randomized Neural Network (AG-RaNN) method for computing multivalued solutions of nonlinear first-order PDEs with hyperbolic characteristics.<n>Such solutions arise in geometric optics, seismic waves, semiclassical limit of quantum dynamics and high frequency limit of linear waves.
  arXiv  Detail & Related papers  (2026-03-01T13:16:25Z)
• Quantum algorithms for viscosity solutions to nonlinear Hamilton-Jacobi equations based on an entropy penalisation method [40.18480271945435]
  We present a framework for efficient extraction of the viscosity solutions of nonlinear Hamilton-Jacobi equations with convex Hamiltonians.<n>These viscosity solutions play a central role in areas such as front propagation, mean-field games, optimal control, machine learning, and a direct application to the forced Burgers' equation.<n>We provide quantum algorithms, both analog and digital, for extracting pointwise values, gradients, minima, and function evaluations at the minimiser of the viscosity solution.
  arXiv  Detail & Related papers  (2025-12-08T14:19:32Z)
• Simulating Time Dependent and Nonlinear Classical Oscillators through Nonlinear Schrödingerization [0.5729426778193399]
  We present quantum algorithms for simulating the dynamics of a broad class of classical oscillator systems.<n>Our work extends the applicability of quantum algorithms to simulate the dynamics of non-conservative and nonlinear classical systems.
  arXiv  Detail & Related papers  (2025-05-22T17:48:42Z)
• Tensor-Programmable Quantum Circuits for Solving Differential Equations [0.0]
  We present a quantum solver for partial differential equations based on a flexible matrix product operator representation.<n>The capabilities of the framework are demonstrated for linear and non-linear partial differential equations.
  arXiv  Detail & Related papers  (2025-02-06T18:23:38Z)
• Quantum Simulation of Nonlinear Dynamical Systems Using Repeated Measurement [42.896772730859645]
  We present a quantum algorithm based on repeated measurement to solve initial-value problems for nonlinear ordinary differential equations. We apply this approach to the classic logistic and Lorenz systems in both integrable and chaotic regimes.
  arXiv  Detail & Related papers  (2024-10-04T18:06:12Z)
• Quantum Algorithms for Nonlinear Dynamics: Revisiting Carleman Linearization with No Dissipative Conditions [0.7373617024876725]
  We explore the embedding of nonlinear dynamical systems into linear ordinary differential equations (ODEs) via the Carleman linearization method.<n>Our analysis extends these findings by exploring error bounds beyond the traditional dissipative condition.<n>We prove how this resonance condition leads to a linear convergence with respect to the truncation level $N$ in Carleman linearization.
  arXiv  Detail & Related papers  (2024-05-21T12:09:34Z)
• Dynamical chaos in nonlinear Schr\"odinger models with subquadratic power nonlinearity [137.6408511310322]
  We deal with a class of nonlinear Schr"odinger lattices with random potential and subquadratic power nonlinearity. We show that the spreading process is subdiffusive and has complex microscopic organization. The limit of quadratic power nonlinearity is also discussed and shown to result in a delocalization border.
  arXiv  Detail & Related papers  (2023-01-20T16:45:36Z)
• Decimation technique for open quantum systems: a case study with driven-dissipative bosonic chains [62.997667081978825]
  Unavoidable coupling of quantum systems to external degrees of freedom leads to dissipative (non-unitary) dynamics. We introduce a method to deal with these systems based on the calculation of (dissipative) lattice Green's function. We illustrate the power of this method with several examples of driven-dissipative bosonic chains of increasing complexity.
  arXiv  Detail & Related papers  (2022-02-15T19:00:09Z)
• Designing Kerr Interactions for Quantum Information Processing via Counterrotating Terms of Asymmetric Josephson-Junction Loops [68.8204255655161]
  static cavity nonlinearities typically limit the performance of bosonic quantum error-correcting codes. Treating the nonlinearity as a perturbation, we derive effective Hamiltonians using the Schrieffer-Wolff transformation. Results show that a cubic interaction allows to increase the effective rates of both linear and nonlinear operations.
  arXiv  Detail & Related papers  (2021-07-14T15:11:05Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.