quantum gate directory
Controlled Phase
Symbol
$\mathrm{C}P$
Description:
Applies a phase $\mathrm{e}^{i\phi}$ to $|11\rangle$ and leaves other basis states unchanged.
Alternate notations:
- $\mathrm{CPhase}(\phi)$
- $\Lambda(\mathrm{e}^{i\phi})$
SDK Support
| SDK | Name |
|---|---|
| Qiskit |
qiskit.circuit.library.CPhaseGate
|
| PennyLane |
pennylane.ControlledPhaseShift
|
| Cirq |
cirq.CZPowGate
ⓘ
|
| Q# | — |
| PyQuil | — |
| Braket | — |
| BQSKit | — |
Groups
This gate is contained in the following groups:
The controlled-phase gate is a two-qubit diagonal gate that adds a phase to the $|11\rangle$ component.
$$ \mathrm{C}P(\phi) = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & \mathrm{e}^{i\phi} \end{bmatrix} $$
Properties
- Diagonal and symmetric under qubit exchange.
- Clifford when $\phi$ is a multiple of $\pi/2$.
- Related to controlled-$Z$ at $\phi = \pi$.
Usage
- Phase-kickback in algorithms and controlled phase rotations.
- Building block for QFT and diagonal unitaries.
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