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 $e^{i\varphi b}$.

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 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.