Quantum Calculation for Two-Stream Instability and Advection Test of Vlasov-Maxwell Equations: Numerical Evaluation of Hamiltonian Simulation

The Vlasov-Maxwell equations provide kinetic simulations of collisionless plasmas, but numerically solving them on classical computers is often impractical. This is due to the computational resource constraints imposed by the time evolution in the 6-dimensional phase space, which requires broad spatial and temporal scales. In this study, we develop a quantum-classical hybrid Vlasov-Maxwell solver. Specifically, the Vlasov solver implements the Hamiltonian simulation based on Quantum Singular Value Transformation (QSVT), coupled with a classical Maxwell solver. We perform numerical simulation of a 1D advection test and a 1D1V two-stream instability test on the Qiskit-Aer-GPU quantum circuit emulator with an A100 GPU. The computational complexity of our quantum algorithm can potentially be reduced from the classical $O(N^6T^2)$ to $O(\text{poly}(\log(N),N,T))$ for the $N$ grid system and simulation time $T$. Furthermore, the numerical analysis reveals that our quantum algorithm is robust under larger time steps compared with classical algorithms with the constraint of Courant-Friedrichs-Lewy (CFL) condition.