We study both dissipative and nondissipative quantum transport in disc
rete Brownian rectifiers being driven by nonthermal noise that is unbi
ased on average. In the absence of dissipation the current is always z
ero and the ballistic diffusion changes into normal diffusion. The dis
sipative quantum dynamics exhibits current (with distinctive reversals
) as a result of the cooperative interplay between dissipative forces
and external fluctuations. Considering the nonlinear current response
to aperiodic, noisy forces we predict aperiodic quantum stochastic res
onance. The nonthermal fluctuations can considerably enhance, as well
as suppress, the thermal quantum diffusion.