Exploring quantum algorithms for solving the full Vlasov equation

Given the progress of quantum algorithms in solving linear partial differential equations, there are potential quantum algorithms for solving nonlinear partial differential equations with reduced computational cost compared to classical solvers. However, it is unclear what a general framework for such an algorithm would actually be. We consider a quantum algorithm for solving the Vlasov-Poisson equation in 1D1V that takes the form of an iterative hybrid quantum-classical algorithm in which each iteration involves classical information being encoded with a parameterized ansatz and evolved quantumly according to a time marching scheme. The resulting quantum encoded solution is then processed classically to obtain relevant classical information required for optimization and influencing the next step of the algorithm. In this talk, we will introduce the proposed algorithm and then discuss how to move towards fully near-term or fault-tolerant quantum algorithms.