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

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

$$\text{CCRz}(\theta) = \begin{pmatrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & e^{-i\theta/2} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & e^{i\theta/2} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & e^{-i\theta/2} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & e^{i\theta/2} \end{pmatrix}$$

• Parametric: rotation angle $\theta$ is tunable
• Asymmetric: control and target qubits are distinct
• Diagonal: only applies phases, like RZ
• Exponential form: $\text{CCRz}(\theta) = e^{-i\theta Z/2}$ on target when controls are $|11\rangle$

• Variational quantum algorithms: QAOA and VQE ansätze with doubly-controlled rotations
• Conditional phase accumulation: rotate target phase based on control qubits
• Quantum simulation: tunable Z interactions (Ising-like) for Hamiltonians
• Parameterized circuits: hybrid classical-quantum optimization

CCRz can be decomposed using CX and single-qubit phase rotations:

$$\text{CCRz}(\theta) = (I \otimes I \otimes R_Z(\theta/2)) \text{CNOT}_{12} (I \otimes I \otimes R_Z(\theta/2)) \text{CNOT}_{12}$$

More efficient decompositions exist; often combined with CCP or other parametric gates.

• Superconducting qubits: typically virtual (reference frame adjustment); very low cost if decomposition is free
• Trapped ions: controlled phase via detuned laser interactions
• Photonic: challenging; usually decomposed

• CCRx, CCRy, CCRz: doubly-controlled rotations around three axes
• RZ gate: single-qubit version; target rotation
• CCP: related parametric gate; CCP applies global phase instead of rotation
• Rotation gates: general parametric rotation category
• Three-qubit gates: general category