Past exam of the computer science course of the University of Oxford / 2023 / Quantum information / 2 / b

Initialize one qubit in $|0⟩$, then apply a Hadamard gate, a T gate, and a Hadamard gate, in that temporal order. Defining the Pauli X eigenstates $|\pm ⟩=(|0⟩\pm |1⟩)/\sqrt{2}$, the successive quantum states are

Initialize three qubits in $|000⟩$. Apply a Hadamard gate on the first, then controlled-NOT gates $1\to 2$ and $1\to 3$:Apply the T gate twice to the first qubit. Because $T^2=\mathrm{diag}(1,i)$, the resulting GHZ state with the specified relative quantum phase is

Initialize two qubits in $|00⟩$. A Hadamard gate on the first followed by a controlled-NOT gate $1\to 2$ prepares the Bell state $|{\varphi }_{+}⟩$. The permitted T gates and Hadamard gates implement the needed Pauli gates exactly:Apply $T_1^4$, followed by $H_2,T_2,T_2,T_2,T_2,H_2$. These are a Pauli Z gate on the first qubit and a Pauli X gate on the second, givingEvery operation is in the allowed gate set; no quantum measurement or probabilistic branch is needed.