  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.
