rotations 1 qubit 2 parameters
Phased Rotation
Rotation by $\theta$ about the equatorial Bloch-sphere axis at azimuthal angle $\phi$.
The phased rotation generalizes $R_x$ and $R_y$ to an arbitrary axis in the equatorial ($xy$) plane of the Bloch sphere:
$$ \begin{align*} R(\theta, \phi) & = \exp\left(-i \frac{\theta}{2} (\cos\phi \, X + \sin\phi \, Y)\right) \\ & = \begin{bmatrix} \cos\frac{\theta}{2} & -i\mathrm{e}^{-i\phi}\sin\frac{\theta}{2} \\ -i\mathrm{e}^{i\phi}\sin\frac{\theta}{2} & \cos\frac{\theta}{2} \end{bmatrix} \end{align*} $$
Special values
| Parameters | Gate |
|---|---|
| $\phi = 0$ | $R_x(\theta)$ |
| $\phi = \pi/2$ | $R_y(\theta)$ |
| $\theta = \pi$ | GPi$(\phi)$ up to global phase |
| $\theta = \pi/2$ | GPi2$(\phi)$ |
Properties
- Sandwich form: $R(\theta, \phi) = R_z(\phi) \, R_x(\theta) \, R_z(-\phi)$ — the phase $\phi$ just rotates the frame in which the $x$ rotation happens.
- Covers half of the Bloch sphere's rotation axes with one pulse shape, which is why it is the native single-qubit gate on ion traps (as GPi/GPi2) and on Google hardware (as PhasedX).
Usage
- On hardware with virtual $z$ rotations, any single-qubit unitary becomes two phased rotations: the compiler tracks $\phi$ offsets in software and only the equatorial pulses are physically played.
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 ($\theta = 2\pi t,\ \phi = 4\pi t$).
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.RGate |
| PennyLane | — ⓘ |
| Cirq | cirq.PhasedXPowGate ⓘ |
| Q# | — ⓘ |
| PyQuil | — |
| Braket | braket.circuits.gates.PRx ⓘ |
| BQSKit | bqskit.ir.gates.U1qGate ⓘ |
| Qibo | qibo.gates.PRX ⓘ |
| pytket | pytket.circuit.OpType.PhasedX ⓘ |
| Stim | — |
| OpenQASM | — |