Bounded depth probability distributions
There are some relatively recent papers by Emanuele Viola et al., which deal with the complexity of sampling distributions. They focus on restricted model of computations, like bounded depth decision trees or bounded depth circuits.

Unfortunately they don't discuss reversible gates. On the contrary there is often loss in the output length. Nevertheless these papers may be a good starting point.

Bounded-Depth Circuits Cannot Sample Good Codes

The complexity of Distributions