quantum gate directory
Phase Shift
Symbol
$P(\phi)$
Description:
Applies a relative phase $e^{i\phi}$ to the $|1\rangle$ component.
Alternate notations:
- $U_1(\phi)$
The phase shift gate is diagonal and leaves $|0\rangle$ unchanged while multiplying $|1\rangle$ by $e^{i\phi}$.
$$ P(\phi) = \begin{bmatrix} 1 & 0 \\ 0 & \mathrm{e}^{i\phi} \end{bmatrix} $$
Properties
- Equivalent to $R_z(\phi)$ up to a global phase.
- Clifford when $\phi$ is an integer multiple of $\pi/2$.
Special Cases
| $\phi$ | Equivalent Gate | Matrix |
|---|---|---|
| $0$ | Identity | $\begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}$ |
| $\pi / 4$ | $T$ | $\begin{bmatrix}1 & 0 \\ 0 & \mathrm{e}^{i\pi/8}\end{bmatrix}$ |
| $\pi/2$ | $S$ | $\begin{bmatrix}1 & 0 \\ 0 & i\end{bmatrix}$ |
| $\pi$ | $Z$ | $\begin{bmatrix}1 & 0 \\ 0 & -1\end{bmatrix}$ |
| $3\pi/2$ | $S^\dagger$ | $\begin{bmatrix}1 & 0 \\ 0 & -i\end{bmatrix}$ |
Groups
This gate is contained in the following groups:
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