Controlled phase gate ($CP(\varphi )$)

The controlled phase gate contributes $e^{i\varphi }$ exactly when both of its computational-basis input bits are one. It is diagonal and symmetric between its two qubits.

The controlled-Z gate is $CP(\pi )=\mathrm{diag}(1,1,1,-1)$. Conjugating the target of a controlled-NOT gate by Hadamard gates turns controlled-NOT into controlled-Z.