rotations 1 qubit 1 parameter
$z$ Rotation
Rotation about the $z$ axis of the Bloch sphere.
The $R_z$ gate rotates a qubit by angle $\theta$ around the $z$ axis, adding a relative phase between $|0\rangle$ and $|1\rangle$.
$$ R_z(\theta) = \exp\left(-i\frac{\theta}{2}Z\right) = \begin{bmatrix} \mathrm{e}^{-i\frac{\theta}{2}} & 0 \\ 0 & \mathrm{e}^{i\frac{\theta}{2}} \end{bmatrix} $$
Properties
- Often implemented virtually as a frame update in many platforms.
- Combined with $R_x$ or $R_y$ for universal single-qubit control.
- Equivalent to a phase shift up to a global phase.
The unitary, drawn
Every entry of the matrix as a disc. The area is the transition probability $|U_{jk}|^2$; the colour and the hand are the phase. Rows are outputs $\langle j|$, columns inputs $|k\rangle$, in the computational basis. Each parameter has a slider; play runs through the family along the path of the Quirk link ($\theta = 2\pi t$).
Phase
Disc area $= |U_{jk}|^2$. Hover an entry for its value, or take the JSON.
SDK support
How to call this gate in each framework. Hover a ⓘ for a note; the support matrix compares every gate at once.
| SDK | Name |
|---|---|
| Qiskit | qiskit.circuit.library.RZGate |
| PennyLane | pennylane.RZ |
| Cirq | cirq.Rz |
| Q# | Std.Intrinsic.Rz |
| PyQuil | pyquil.gates.RZ |
| Braket | braket.circuits.gates.Rz |
| BQSKit | bqskit.ir.gates.RZGate |
| Qibo | qibo.gates.RZ |
| pytket | pytket.circuit.OpType.Rz ⓘ |
| Stim | — |
| OpenQASM | stdgates.inc: rz |