Quantized tensor networks for solving the Vlasov-Maxwell equations

While the Vlasov-Maxwell equations provide an \textit{ab-initio} description of collisionless plasmas, solving them is often impractical due to high computational costs. In this work, we implement a semi-implicit Vlasov-Maxwell solver utilizing the quantized tensor network (QTN) framework. This framework allows one to efficiently represent and manipulate low-rank approximations of high-dimensional data sets. As a result, the cost of the solver scales polynomially with parameter $D$ (the so-called bond dimension), which is directly related to the error associated with the low-rank approximation. By increasing $D$, convergence to the dynamics that the solver would obtain without any low-rank approximation is guaranteed. We find that for the 2D3V test problems considered here, a modest $D=64$ appears to be sufficient for capturing the expected physics, despite the simulations using a total of $2^{36}$ grid points and thus requiring $D=2^{18}$ for exact calculations. Additionally, we utilize a QTN time evolution scheme based on the Dirac-Frenkel variational principle, which allows us to use larger time steps than that prescribed by the Courant-Friedrichs-Lewy (CFL) constraint. As such, the QTN format appears to be a promising means of approximately solving the Vlasov-Maxwell equations with significantly reduced cost.