# Controlled-Phase Gate

**Controlled-Phase** (or CP gate) applies a phase to the $|11\rangle$ state, parameterized by angle $\theta$.

Matrix:

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

Special case: $\text{CP}(\pi) = \text{CZ}$.

## Relation to Other Gates

- **Controlled-RZ**: $\text{CP}(\theta) = I \otimes I + (I \otimes I - |00\rangle\langle 00| - |01\rangle\langle 01| - |10\rangle\langle 10|) \cdot R_Z(\theta)$
- **Parametric CZ**: CP is the generalization of CZ with tunable phase

## Properties

- **Parameterized**: vary $\theta$ to change gate strength
- **Diagonal**: only applies phases
- **Symmetric**: control and target interchangeable
- **Composition**: $\text{CP}(\alpha) \text{CP}(\beta) = \text{CP}(\alpha + \beta)$ on same qubits

## Uses

- **Tunable coupling**: variable phase enables dynamic control in parametric systems
- **Ansatz design**: parameterized two-qubit gates allow variational circuits to adapt interaction
- **Quantum simulation**: CP gates appear in Hamiltonians with tunable coupling

## Relation to XX/YY/ZZ

- **Related but distinct**: XX($\theta$), YY($\theta$), ZZ($\theta$) are parametric but not identical to CP
- **Physical origin**: correlated rotations more natural for certain interactions

## Implementation

- **Superconducting qubits**: parametric pumping or flux-dependent coupling; fidelity depends on phase control precision
- **Trapped ions**: engineered laser pulses with tunable Rabi frequency
- **Photonic**: programmable optical gates