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 $|1\rangle$.

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

• 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 $|11\rangle$

• 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

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.

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

• CCRx, CCRy, CCRz: doubly-controlled rotations around three axes
• RY gate: single-qubit version; target rotation
• Rotation gates: general parametric rotation category
• Three-qubit gates: general category