In quantum computing we are often interested in cases where group of special unitary operators, G, for some d-dimensional system gives either the whole group SU(d) exactly or even just an approximation provided by a dense cover of SU(d).
A group of finite order, such as the Clifford group for a d-dimensional system C(d), will not give a dense cover. A group of infinite order will not give a dense cover if the group is Abelian. However, my rough intuition is that an infinite number of gates and basis changing operations of the Clifford group should suffice to provide a dense cover.
Formally, my question is:
I have a group G that is a subgroup of SU(d). G has infinite order and C(d) is a subgroup of G. Do all such G provide a dense cover of SU(d).
Note that I am particular interested in the case when d>2.
I take the Clifford group to be as defined here: 