CCP applies phase $e^{i\theta}$ to the target if both controls are $|1\rangle$. Parametric doubly-controlled gate for phase estimation and variational algorithms.

Action: $|c_1 c_2 t\rangle \to e^{i c_1 c_2 \theta}|c_1 c_2 t\rangle$ where phase is applied only when both controls are $|1\rangle$.

$$\text{CCP}(\theta) = \begin{pmatrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & e^{i\theta} \end{pmatrix}$$

Basis state mapping: identity on all states except $|111\rangle$. Maps $|111\rangle \to e^{i\theta}|111\rangle$.

• Parametric: phase angle $\theta$ is tunable
• Symmetric: all three qubits play equivalent roles under qubit permutations
• Diagonal: only applies phases, preserving basis structure
• Special cases: $\theta = \pi$ gives CCZ; $\theta = 0$ gives identity

• Phase estimation: quantum algorithm for eigenvalue extraction
• Variational quantum algorithms: QAOA and VQE circuits with parametric phases
• Quantum simulation: simulating phase interactions in Hamiltonians
• Grover's algorithm: amplitude amplification with controlled phases

CCP can be decomposed using CCZ and single-qubit phase gates:

$$\text{CCP}(\theta) = (I \otimes I \otimes R_Z(\theta/2)) \text{CCZ}^{1/2} (I \otimes I \otimes R_Z(\theta/2))$$

Alternatively, decompose directly into CX/phase gate chains.

• Superconducting qubits: typically decomposed into CX chains with phase rotations; parametric implementation possible
• Trapped ions: controlled phase interactions via laser detuning
• Photonic: challenging; usually decomposed

• CCZ: special case when $\theta = \pi$; non-parametric doubly-controlled phase
• Controlled-Phase: two-qubit version; CCP adds second control
• Phase gates: single-qubit version (parametric $R_Z$)
• Three-qubit gates: general category