A quantum-inspired method for solving the Vlasov-Poisson equations

TL;DR

This paper investigates the use of quantum-inspired matrix product state methods to efficiently solve 1D Vlasov-Poisson equations, capturing key plasma dynamics with significant data compression.

Contribution

It demonstrates the practicality of MPS methods for plasma simulations and compares different mappings to optimize representation, paving the way for higher-dimensional applications.

Findings

MPS methods accurately reproduce damping and growth rates.

Significant compression of the solution is achieved.

Insights into MPS representation aid future extensions.

Abstract

Kinetic simulations of collisionless (or weakly collisional) plasmas using the Vlasov equation are often infeasible due to high resolution requirements and the exponential scaling of computational cost with respect to dimension. Recently, it has been proposed that matrix product state (MPS) methods, a quantum-inspired but classical algorithm, can be used to solve partial differential equations with exponential speed-up, provided that the solution can be compressed and efficiently represented as an MPS within some tolerable error threshold. In this work, we explore the practicality of MPS methods for solving the Vlasov-Poisson equations in 1D1V, and find that important features of linear and nonlinear dynamics, such as damping or growth rates and saturation amplitudes, can be captured while compressing the solution significantly. Furthermore, by comparing the performance of different mappings of the distribution functions onto the MPS, we develop an intuition of the MPS representation and its behavior in the context of solving the Vlasov-Poisson equations, which will be useful for extending these methods to higher dimensional problems.