  We present a super-polynomial improvement in the precision scaling of quantum
simulations for coupled classical-quantum systems in this paper. Such systems
are found, for example, in molecular dynamics simulations within the
Born-Oppenheimer approximation. By employing a framework based on the
Koopman-von Neumann formalism, we express the Liouville equation of motion as
unitary dynamics and utilize phase kickback from a dynamical quantum simulation
to calculate the quantum forces acting on classical particles. This approach
allows us to simulate the dynamics of these particles without the overheads
associated with measuring gradients and solving the equations of motion on a
classical computer, resulting in a super-polynomial advantage at the price of
increased space complexity. We demonstrate that these simulations can be
performed in both microcanonical and canonical ensembles, enabling the
estimation of thermodynamic properties from the prepared probability density.
