Efficient Quantum Circuit for the Vlasov Equation with External Electromagnetic Fields

Quantum computers and quantum algorithms offer the possibility to develop new numerical schemes to solve partial differential equations (PDEs) with memory and time advantages over classical methods. Many physics phenomena are challenging to simulate even on classical computers, especially plasma phenomena which evolve under a kinetic equation in a six dimensional phase space, implying a need of large memory and long-time simulations. In this context, an efficient quantum algorithm is derived for the six-dimensional Vlasov equation with external spatio-temporal-dependent electromagnetic fields. This quantum scheme is based on a mapping of the initial PDE into a Schrödinger form and the use of Walsh functions in the implementation of the external electromagnetic field and in the initialization step. This work is a crucial step in the development of numerical tools for an efficient quantum algorithm to solve the Maxwell-Vlasov system of equations on digital quantum computers.