Quantum simulation of ODEs, PDEs and related problems. Part II: Nonlinear problems and applications

In the second part of the lectures on quantum simulation for partial differential equations, we will focus on nonlinear partial differential equations and applications of Schrodingerisation.

1. For (nonlinear) Hamilton-Jacobi equation and scalar nonlinear hyperbolic equations, we use the level set method to map them exactly to phase space linear transport PDEs so they can be implemented with quantum algorithms. We gain quantum advantage for various physical and numerical parameters.
2. We show how to apply Schrodingerisation to a system of (nonlinear) ordinary differential equations.
3. For (both artificial and physical) boundary value and interface problems of linear PDEs, we show how one can Schrodingerise them so they become suitable for quantum simulations.
4. We give an example of Schrodingerisation for Maxwell’s equations.