quantum gate directory
Phase Shift Gate
Symbol
$P(\phi)$
Description:
The Phase Shift gate applies a rotation to the phase of the quantum state, leaving the $|0\rangle$ component unchanged while applying a phase shift $e^{i\phi}$ to the $|1\rangle$ component.
| Notation(s) | $P(\phi)$, $R_z(\phi)$ |
| Matrix | $\begin{bmatrix}1 & 0 \\ 0 & e^{i\phi} \end{bmatrix}$ |
| Arity | 1 |
| Group | $\mathsf{U}(2)$ |
Special Cases
The Phase Shift gate corresponds to other well-known gates for specific values of $\phi$:
| $\phi$ | Equivalent Gate | Matrix |
|---|---|---|
| $0$ | Identity ($I$) | $\begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}$ |
| $\pi/2$ | Phase ($S$) | $\begin{bmatrix}1 & 0 \\ 0 & i\end{bmatrix}$ |
| $\pi$ | Pauli-Z ($Z$) | $\begin{bmatrix}1 & 0 \\ 0 & -1\end{bmatrix}$ |
| $3\pi/2$ | $S^\dagger$ | $\begin{bmatrix}1 & 0 \\ 0 & -i\end{bmatrix}$ |
Back to home