# RY Gate (Rotation around Y)

**RY** rotates a qubit around the y-axis by angle $\theta$.

Matrix:

$$R_Y(\theta) = \begin{pmatrix} \cos(\theta/2) & -\sin(\theta/2) \\ \sin(\theta/2) & \cos(\theta/2) \end{pmatrix}$$

Action: $R_Y(\theta)|\psi\rangle$ rotates $|\psi\rangle$ by $\theta$ radians around y-axis on Bloch sphere.

Special cases:

- $R_Y(0) = I$
- $R_Y(\pi/2) = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 & -1 \\ 1 & 1 \end{pmatrix}$
- $R_Y(\pi) = -i Y$ (equivalent to Pauli Y up to global phase)

## Properties

- **Periodicity**: $R_Y(\theta + 2\pi) = R_Y(\theta)$
- **Inverse**: $R_Y(\theta)^\dagger = R_Y(-\theta)$
- **Composition**: $R_Y(\alpha) R_Y(\beta) = R_Y(\alpha + \beta)$
- **Real entries**: RY matrix is real (unlike RX and RZ which have complex entries)

## Uses

- **Ansatz design**: common in variational quantum algorithms; often preferred over RX due to real matrix
- **State preparation**: combined with other rotations, can prepare arbitrary single-qubit states
- **Bloch sphere**: y-axis rotation

## Decomposition

- **Exponential form**: $R_Y(\theta) = e^{-i\theta Y/2}$
- **Relation to X and Z**: $R_Y(\theta) = \text{RX}(\pi/2) \text{RZ}(\theta) \text{RX}(-\pi/2)$

## Implementation

- **Superconducting qubits**: microwave pulse with specific phase; gate time ~20–40 ns
- **Trapped ions**: laser pulse at angle between x and z transitions
- **Photonic**: rotation via wave plates or optical elements
- **Fidelity**: 99–99.9%