Is there a way to decompose a quantum circuit into a Clifford subcircuit and a non-Clifford subcircuit?

Question

Is there any way of decomposing a arbitrary quantum circuit into two sub circuits, first of them being composed only of Clifford gates and the second one only of non-Clifford gates?

Thank you!

Answer 1

You can make it even stronger than that. Any circuit can be approximated using the Clifford+T gate set. Then you can replace every T gate with gate teleportation through a T state. So any circuit can be decomposed into a Clifford part and a trivial non-Clifford part that does nothing except prepare many copies of the state $T |+\rangle$.