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 $e^{i\varphi b}$.

A two-qubit parity phase gate multiplies odd-parity computational vectors by $e^{i\theta }$ and fixes even-parity ones. With $R_{\theta }=\mathrm{diag}(1,e^{i\theta })$,The first controlled-NOT gate computes $x\oplus y$ in the second wire, the phase gate multiplies by $e^{i\theta (x\oplus y)}$, and the last controlled-NOT gate restores the original bits. This compute-phase-uncompute construction leaves the data unchanged except for the desired phases. The more general Pauli-string phase by parity computation uses the same mechanism. In terms of the Pauli Z gate, $G_{\theta }=e^{i\theta /2}\exp [-i\theta (Z\otimes Z)/2]$; the scalar factor ensures equality as a matrix, rather than only up to global phase.

The phase gate with angle $\pi /4$. It rotates the Bloch vector by $\pi /4$ around $z$, and $T^2=S$. It is not a Clifford gate: it takes the positive $X$ eigenstate to a state with vector $(1/\sqrt{2},1/\sqrt{2},0)$, outside the stabilizer octahedron.

The phase gate with angle $\pi /2$. It rotates the Bloch vector by $\pi /2$ around $z$ and is a Clifford gate.

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.