Quantum simulation of a class of highly-oscillatory transport equations via Schr\"odingerisation

In this paper, we present quantum algorithms for a class of highly-oscillatory transport equations, which arise in semiclassical computation of surface hopping problems and other related non-adiabatic quantum dynamics, based on the Born-Oppenheimer approximation. Our method relies on the classical nonlinear geometric optics method, and the recently developed Schr\"odingerisation approach for quantum simulation of partial differential equations. The Schr\"odingerisation technique can transform any linear ordinary and partial 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 by shifting the bad eigenvalues in the Schr\"odingerized 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. Various numerical experiments are performed to demonstrate the validity of this approach.