# CCRy gate (Controlled-Controlled-RY)

**CCRy** applies Y rotation by $\theta$ to the target if both controls are $|1\rangle$. Parametric doubly-controlled rotation for variational algorithms.

Action: $|c_1 c_2 t\rangle \to |c_1 c_2 (R_Y(\theta)|t\rangle)\rangle$ where rotation is applied only when both controls are $|1\rangle$.

$$\text{CCRy}(\theta) = \begin{pmatrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & \cos(\theta/2) & -\sin(\theta/2) & 0 & 0 \\ 0 & 0 & 0 & 0 & \sin(\theta/2) & \cos(\theta/2) & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & \cos(\theta/2) & -\sin(\theta/2) \\ 0 & 0 & 0 & 0 & 0 & 0 & \sin(\theta/2) & \cos(\theta/2) \end{pmatrix}$$

## Properties

- **Parametric**: rotation angle $\theta$ is tunable
- **Asymmetric**: control and target qubits are distinct
- **Real-valued**: unlike CCRx which has imaginary entries
- **Exponential form**: $\text{CCRy}(\theta) = e^{-i\theta Y/2}$ on target when controls are $|11\rangle$

## Uses

- **Variational quantum algorithms**: QAOA and VQE ansätze with doubly-controlled rotations
- **Conditional state preparation**: rotate target state based on control qubits
- **Quantum simulation**: tunable Y interactions for Heisenberg-like models
- **Parameterized circuits**: hybrid classical-quantum optimization

## Decomposition

CCRy can be decomposed using CX and single-qubit rotations:

$$\text{CCRy}(\theta) = (I \otimes I \otimes R_Y(\theta/2)) \text{CNOT}_{12} \text{CNOT}_{02} (I \otimes I \otimes R_Y(\theta/2))$$

More efficient decompositions exist using fewer gates.

## Implementation

- **Superconducting qubits**: decomposed into CX chains and single-qubit rotations
- **Trapped ions**: controlled rotations via laser interactions
- **Photonic**: challenging; typically decomposed

## Relations

- [[quantum-gate-ccrx|CCRx]], [[quantum-gate-ccry|CCRy]], [[quantum-gate-ccrz|CCRz]]: doubly-controlled rotations around three axes
- [[quantum-gate-ry|RY gate]]: single-qubit version; target rotation
- [[quantum-gate-rotation|Rotation gates]]: general parametric rotation category
- [[quantum-gate-three-qubit|Three-qubit gates]]: general category