  The Vlasov equation is a nonlinear partial differential equation that
provides a first-principles description of the dynamics of plasmas. Its linear
limit is routinely used in plasma physics to investigate plasma oscillations
and stability. In this work, we present a quantum algorithm that simulates the
linearized Vlasov equation with and without collisions, in the one-dimensional,
electrostatic limit. Rather than solving this equation in its native spatial
and velocity phase-space, we adopt an efficient representation in the dual
space yielded by a Fourier-Hermite expansion. The Fourier-Hermite
representation is exponentially more compact, thus yielding a classical
algorithm that can match the performance of a previously proposed quantum
algorithm for this problem. This representation results in a system of linear
ordinary differential equations which can be solved with well-developed quantum
algorithms: Hamiltonian simulation in the collisionless case, and quantum ODE
solvers in the collisional case. In particular, we demonstrate that a quadratic
speedup in system size is attainable.
