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The Hadamard gate conjugates the Pauli X gate into the Pauli Z gate:When the control qubit is $|0⟩$, all three circuits act as the identity on the target. When it is $|1⟩$, the right-hand circuit acts on the target as $HXH=Z$. It therefore agrees with the Controlled-Z gate on every computational basis state, provingConjugating both sides by $I\otimes H$ and using $H^2=I$ immediately gives the reverse identity

A unitary operation on a fixed number of qubits used as a component of a quantum circuit. Examples include the Hadamard gate and Controlled-Z gate. Measurements are described separately by quantum measurement; a gate is a reversible operation on its full input state.