Approximating uniform quantum channels

Let G be a finite subgroup of unitary matrices acting on the space of $N$-qubits. We associate with G a uniform quantum channel QU from the space on $N$-qubits to itself. We give a quantum algorithm to approximate this channel by considering a set of generators on G. Under suitable assumptions this approximation is BPQ. We then apply this approximation to study the orbit equivalence of two density matrices under the action of G. We show that for some special cases of G and two pure states the orbit equivalence in BPQ, if a specific quantum observation can be implemented efficiently. We discuss the application of our problem to the graph isomorphism problem.