rotations 1 qubit 1 parameter
$y$ Rotation
Rotation about the $y$ axis of the Bloch sphere.
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.
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.RYGate |
| PennyLane | pennylane.RY |
| Cirq | cirq.Ry |
| Q# | Std.Intrinsic.Ry |
| PyQuil | pyquil.gates.RY |
| Braket | braket.circuits.gates.Ry |
| BQSKit | bqskit.ir.gates.RYGate |
| Qibo | qibo.gates.RY |
| pytket | pytket.circuit.OpType.Ry ⓘ |
| Stim | — |
| OpenQASM | stdgates.inc: ry |