Past exam of the computer science course of the University of Oxford / 2022 / Quantum information / 2 / d

Use Hadamard conjugation of a controlled gate to obtain the Controlled-Z gate:Conjugating by Pauli X gates on both qubits moves its negative entry from $|11⟩$ to $|00⟩$:Finally $(H\otimes H)|00⟩=|+⟩|+⟩=|e_0⟩$, so a two-qubit diffusion circuit isIn temporal order, apply Hadamard gates on both qubits, Pauli X gates on both, a Hadamard gate on the second, a controlled-NOT gate $1\to 2$, a Hadamard gate on the second, Pauli X gates on both, then Hadamard gates on both. Every gate is permitted. The overall minus sign is the allowed global phase, so this implements the required inversion about the mean for Grover search algorithm.