The Clifford group of quantum operators is generated by the quantum operations:
- Hadamard, and
- Phase ($= |0\rangle\langle0| + i |1\rangle\langle1|$).
A circuit composed only of these gates can be simulated efficiently on a classical computer. However, if I understand correctly, not all classical algorithms can be implemented efficiently using Clifford group operations, at least as far as we know.
Is there a construction to implement, even inefficiently or approximately, a classical algorithm using Clifford group operations? For instance, how do you implement a Toffoli gate using Clifford group gates, if it's possible?