For a 1+1 dimensional Vlasov model with Krook collisions, a Carleman linearization quantum algorithm has polynomially worse complexity than classical finite difference methods and requires unphysically large collision rates to provably converge.
Alekseenko and Craig Euler
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Solving the Nonlinear Vlasov Equation on a Quantum Computer
For a 1+1 dimensional Vlasov model with Krook collisions, a Carleman linearization quantum algorithm has polynomially worse complexity than classical finite difference methods and requires unphysically large collision rates to provably converge.