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

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

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

• Parametric: rotation angle $\theta$ is tunable
• Asymmetric: control and target qubits are distinct
• Exponential form: $\text{CCRx}(\theta) = e^{-i\theta X/2}$ 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 X interactions for Ising-like models
• Parameterized circuits: hybrid classical-quantum optimization

CCRx can be decomposed using CCx and single-qubit rotations:

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

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
• RX gate: single-qubit version; target rotation
• Rotation gates: general parametric rotation category
• Three-qubit gates: general category