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

• 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

• 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

• 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

• 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

• 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