quantum gate directory
Phased Rotation
Symbol
$R(\theta, \phi)$
Qubits: $1$ Parameters: $2$
Description:
Rotation by $\theta$ about the equatorial Bloch-sphere axis at azimuthal angle $\phi$.
Alternate notations:
- $R(\theta, \phi)$
- $R_\phi(\theta)$
- $\mathrm{PhasedX}$
SDK Support
| 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 | — |
The phased rotation generalizes $R_x$ and $R_y$ to an arbitrary axis in the equatorial ($xy$) plane of the Bloch sphere:
$$ R(\theta, \phi) = \exp\left(-i \frac{\theta}{2} (\cos\phi \, X + \sin\phi \, Y)\right) = \begin{bmatrix} \cos\frac{\theta}{2} & -i\mathrm{e}^{-i\phi}\sin\frac{\theta}{2} \\ -i\mathrm{e}^{i\phi}\sin\frac{\theta}{2} & \cos\frac{\theta}{2} \end{bmatrix} $$
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.
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