Clifford group quantum operations and classical computation
As pointed out in a comment above, if it were possible to coherently implement a Toffoli gate using Clifford group gates, then the Clifford group would be universal for quantum computation. It was noted in Section 5 of this paper that something even stronger is true: informally speaking, if there exists a class of quantum circuits which can be simulated efficiently classically, and which is universal for classical computation, then BQP=BPP. Thus we would expect simulable classes of quantum circuits not to be universal for classical computation.

Clifford group circuits themselves are particularly weak, and correspond to the complexity class Parity-L, as was shown here.