ag.algebraic geometry - algorithm for calculating the Chow groups of a variety over a finite field

I am not an expert, but let me point out that computing $CH^0(X)$ (which is freely generated by the irreducible components) is already quite hard. Algorithms do exist in this case, see page 206 of "Ideals, varieties and algorithms" by Cox, Little, O'Shea for references. I know of no way to compute the class groups (which can be identified with $CH^1(X)$ for smooth $X$) in general, but I will be very interested in what other people have to say about this.

Of course, in special situations, more is known. For example, the total Chow group of quadric hypersurfaces (at least up to tensoring with $mathbb Q$).