quantum gate directory
$y$ Rotation
Symbol
$R_y$
Description:
Rotation about the $y$ axis of the Bloch sphere.
Alternate notations:
- $R_y(\theta)$
- $\mathrm{Ry}(\theta)$
- $Y(\theta)$
- $Y_\theta$
SDK Support
| SDK | Name |
|---|---|
| Qiskit |
qiskit.circuit.library.RYGate
|
| PennyLane |
pennylane.RY
|
| Cirq |
cirq.Ry
|
| Q# | — |
| PyQuil |
pyquil.gates.RY
|
| Braket |
braket.circuits.gates.Ry
|
| BQSKit |
bqskit.ir.gates.RYGate
|
Groups
This gate is contained in the following groups:
The $R_y$ gate rotates a qubit by angle $\theta$ around the $y$ axis. It is the only purely real rotation matrix among $R_x$, $R_y$, and $R_z$.
$$ R_y(\theta) = \exp\left(-i\frac{\theta}{2}Y\right) = \begin{bmatrix} \cos\frac{\theta}{2} & -\sin\frac{\theta}{2} \\ \sin{\frac{\theta}{2}} & \phantom{+}\cos\frac{\theta}{2} \end{bmatrix} $$
Properties
- Unitary with generator $Y$: $R_y(\theta) = e^{-i\theta Y/2}$.
- Real-valued matrix; useful for preparing real-amplitude states.
- $R_y(\pi) = -iY$.
Usage
- State preparation and mixing amplitudes in variational circuits.
- Combined with $R_z$ for arbitrary single-qubit rotations.
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