quantum gate directory

Qubits: $1$ Parameters: $1$

SDK Support

SDK	Name
Qiskit	qiskit.circuit.library.PhaseGate
PennyLane	pennylane.PhaseShift
Cirq	cirq.ZPowGate ⓘ
Q#	Std.Intrinsic.R1 ⓘ
PyQuil	pyquil.gates.PHASE
Braket	braket.circuits.gates.PhaseShift
BQSKit	bqskit.ir.gates.U1Gate ⓘ
Qibo	qibo.gates.U1 ⓘ
pytket	pytket.circuit.OpType.U1 ⓘ
Stim	—
OpenQASM	stdgates.inc: p

Groups

This gate is contained in the following groups:

• diagonal
• number-preserving

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	$\begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}$
$\pi / 4$	$T$	$\begin{bmatrix}1 & 0 \\ 0 & \mathrm{e}^{i\pi/8}\end{bmatrix}$
$\pi/2$	$S$	$\begin{bmatrix}1 & 0 \\ 0 & i\end{bmatrix}$
$\pi$	$Z$	$\begin{bmatrix}1 & 0 \\ 0 & -1\end{bmatrix}$
$3\pi/2$	$S^\dagger$	$\begin{bmatrix}1 & 0 \\ 0 & -i\end{bmatrix}$