Exercise 5
1. Clifford Gates and Paulis
(a) For n qubits, there are 4^n possible tensor products of Paulis (one of which is the $n$-qubit identity). Show that (up to a phase) these can be expressed as a product of 2n $n$-qubit Paulis.
(b) If U is a Clifford gate, the following property holds
U P U^\dagger \sim P' \,\,\,\,\, \forall P,
where P and P' are Paulis and \sim denotes equality up to a factor of \pm 1 or \pm i. If this relation holds for the 2n Pauli generators of part (a), show that it also holds for all $n$-qubit Paulis.
2. Single-Qubit Clifford Gates
(a) Show that the Paulis are Cliffords themselves.
(b) Show that H, S and S^\dagger are Clifford gates.
(c) Show that T=S^{1/2} is not a Clifford gate.
3. Two-Qubit Clifford Gates
For more than one qubit, Clifford gates map between tensor products of Pauli operators.
For two qubits
U \,( P \otimes Q )\, U^\dagger \sim P' \otimes Q' \,\,\,\,\, \forall P,Q
where P, Q, P' and Q' are all Paulis and \sim denotes equality up to a factor of \pm 1 or \pm i.
(a) Show that the controlled-NOT is a Clifford gate.
(b) Show that the controlled-Hadamard is not a Clifford gate.
4. Three-Qubit Clifford Gates
(a) Provide an example of a three-qubit Clifford gate, and show that it is indeed a Clifford. This should be a truly three qubit gate, and therefore not one that can be expressed purely as a tensor product of single- and two-qubit gates.
(b) Show that the Toffoli gate is not Clifford.
