  Quantum computers can be used to simulate nonlinear non-Hamiltonian classical
dynamics on phase space by using the generalized Koopman-von Neumann
formulation of classical mechanics. The Koopman-von Neumann formulation implies
that the conservation of the probability distribution function on phase space,
as expressed by the Liouville equation, can be recast as an equivalent
Schr\"odinger equation on Hilbert space with a Hermitian Hamiltonian operator
and a unitary propagator. This Schr\"odinger equation is linear in the momenta
because it derives from a constrained Hamiltonian system with twice the
classical phase space dimension. A quantum computer with finite resources can
be used to simulate a finite-dimensional approximation of this unitary
evolution operator. Quantum simulation of classical dynamics is exponentially
more efficient than a deterministic Eulerian discretization of the Liouville
equation if the Koopman-von Neumann Hamiltonian is sparse. Utilizing quantum
walk techniques for state preparation and amplitude estimation for the
calculation of observables leads to a quadratic improvement over classical
probabilistic Monte Carlo algorithms.
