Quantum Simulation of a Class of Highly-Oscillatory Transport Equations via Schrödingerisation (Journal of Computational Mathematics, 2025)
In this paper, we present quantum algorithms for a class of highly-oscillatory transport equations, which arise in semi-classical computation of surface hopping problems andother related non-adiabatic quantum dynamics, based on the Born-Oppenheimer approximation. Our method relies on the classical nonlinear geometric optics method, and therecently developed Schrödingerisation approach for quantum simulation of partial differential equations. The Schrödingerisation technique can transform any linear ordinary andpartial differential equations into Hamiltonian systems evolving under unitary dynamics,via a warped phase transformation that maps these equations to one higher dimension.We study possible paths for better recoveries of the solution to the original problem byshifting the bad eigenvalues in the Schrödingerized system. Our method ensures the uniform error estimates independent of the wave length, thus allowing numerical accuracy,in maximum norm, even without numerically resolving the physical oscillations. Variousnumerical experiments are performed to demonstrate the validity of this approach.
Keywords: Quantum simulation; Schrödingerisation; Highly oscillatory transport PDEs; Nonlinear geometric optics method.