Phase gate ($P(\varphi )$)

The single-qubit phase gate is $P(\varphi )=\mathrm{diag}(1,e^{i\varphi })$. On a computational-basis bit $b$, it contributes the phase $e^{i\varphi b}$.

For $R(\theta )=\mathrm{diag}(1,e^{i\theta })$, the difference vanishes on $|0⟩$ and multiplies $|1⟩$ by $e^{i\theta }(e^{i\delta }-1)$. Its operator norm is therefore $|e^{i\delta }-1|=2|\sin (\delta /2)|$. Choosing the nearest angle on the grid $2\pi m/2^n$, with circular distance, gives $|\delta |\le \pi /2^n$ and a uniform bound of this size on the output-state norm error. This bound concerns state vectors, not an asserted equality with measurement-probability error.

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.

The magic state $|A_{\theta }⟩=(|0⟩+e^{i\theta }|1⟩)/\sqrt{2}$ and a short controlled-NOT gate circuit can apply either $P(\theta )$ or $P(-\theta )$ to an input, with a measured bit identifying the branch.