 Solution

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.