# Quantum Algorithm for the Vlasov Equation

## TL;DR

This paper introduces a quantum algorithm for simulating the Vlasov-Maxwell system in plasma physics, promising exponential speedups over classical methods for high-dimensional problems, with discussions on extensions and current limitations.

## Contribution

The paper develops a quantum algorithm for the Vlasov-Maxwell system, specifically for electrostatic Landau damping, demonstrating potential exponential speedup in high-dimensional plasma simulations.

## Key findings

- Quantum algorithm scales as polylogarithm of velocity grid points
- Classical costs scale linearly with grid points and time
- Quantum approach faces measurement error challenges

## Abstract

The Vlasov-Maxwell system of equations, which describes classical plasma physics, is extremely challenging to solve, even by numerical simulation on powerful computers. By linearizing and assuming a Maxwellian background distribution function, we convert the Vlasov-Maxwell system into a Hamiltonian simulation problem. Then for the limiting case of electrostatic Landau damping, we design and verify a quantum algorithm, appropriate for a future error-corrected universal quantum computer. While the classical simulation has costs that scale as $\mathcal{O}(N_v t)$ for a velocity grid with $N_v$ grid points and simulation time $t$, our quantum algorithm scales as $\mathcal{O}(\text{polylog}(N_v) t/\delta)$ where $\delta$ is the measurement error, and weaker scalings have been dropped. Extensions, including electromagnetics and higher dimensions, are discussed. A quantum computer could efficiently handle a high-resolution, six-dimensional phase-space grid, but the $1/\delta$ cost factor to extract an accurate result remains a difficulty. This paper provides insight into the possibility of someday achieving efficient plasma simulation on a quantum computer.