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)

• 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)

• 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

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

• 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%