# CCP gate (Controlled-Controlled-Phase)

**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 $|111\rangle \to e^{i\theta}|111\rangle$.

## Properties

- **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

## Uses

- **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

## Decomposition

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.

## Implementation

- **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

## Relations

- [[quantum-gate-ccz|CCZ]]: special case when $\theta = \pi$; non-parametric doubly-controlled phase
- [[quantum-gate-cphase|Controlled-Phase]]: two-qubit version; CCP adds second control
- [[quantum-gate-phase|Phase gates]]: single-qubit version (parametric $R_Z$)
- [[quantum-gate-three-qubit|Three-qubit gates]]: general category