  We consider perturbations of the semiclassical Schr{\"o}dinger equation on a
compact Riemannian surface with constant negative curvature and without
boundary. We show that, for scales of times which are logarithmic in the size
of the perturbation, the solutions associated to initial data in a small
spectral window become equidistributed in the semiclassical limit. As an
application of our method, we also derive some properties of the quantum
Loschmidt echo below and beyond the Ehrenfest time for initial data in a small
spectral window.
