Solving the Nonlinear Vlasov Equation on a Quantum Computer

TL;DR

This paper maps the nonlinear Vlasov equation onto a quantum algorithm using Carleman linearization, analyzes its complexity, and discusses limitations due to dissipation requirements.

Contribution

It introduces a quantum algorithm for solving the nonlinear Vlasov equation and provides complexity bounds and convergence analysis.

Findings

Quantum algorithm's complexity is polynomially larger than classical counterparts.

High dissipation levels are necessary for convergence, limiting practical applications.

The approach is applicable to discretized (1+1)D plasma physics problems.

Abstract

We present a mapping of the nonlinear, electrostatic Vlasov equation with Krook-type collision operators, discretized on a (1+1) dimensional grid, onto a recent Carleman linearization-based quantum algorithm for solving ordinary differential equations (ODEs) with quadratic nonlinearities. We derive upper bounds for the query- and gate complexities of the quantum algorithm in the limit of large grid sizes. We conclude that these are polynomially larger than the time complexity of the corresponding classical algorithms. We find that this is mostly due to the dimension, sparsity and norm of the Carleman linearized evolution matrix. We show that the convergence criteria of the quantum algorithm places severe restrictions on potential applications. This is due to the high level of dissipation required for convergence, that far exceeds the physical dissipation effect provided by the Krook operator for typical plasma physics applications.