controlled 2 qubits 1 parameter
Controlled Phase
Applies a phase $\mathrm{e}^{i\phi}$ to $|11\rangle$ and leaves other basis states unchanged.
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.
The unitary, drawn
Every entry of the matrix as a disc. The area is the transition probability $|U_{jk}|^2$; the colour and the hand are the phase. Rows are outputs $\langle j|$, columns inputs $|k\rangle$, in the computational basis. Each parameter has a slider; play runs through the family along the path of the Quirk link ($\phi = 2\pi t$).
Phase
Disc area $= |U_{jk}|^2$. Hover an entry for its value, or take the JSON.
Weyl chamber
Where this gate sits among all two-qubit gates, up to single-qubit rotations before and after.
Coordinates:
not entangling entangling perfect entangler
$(a, b, c)$ means the gate equals $e^{i(a\,XX + b\,YY + c\,ZZ)}$ up to single-qubit gates, reduced to the chamber $\pi/4 \ge a \ge b \ge |c|$ (the convention of the canonical gate, Qiskit and Cirq). The $a = \pi/4$ face is a mirror: $(\pi/4, b, c)$ and $(\pi/4, b, -c)$ are the same gate. Drag the figure to rotate it, double-click to reset.
SDK support
How to call this gate in each framework. Hover a ⓘ for a note; the support matrix compares every gate at once.
| SDK | Name |
|---|---|
| Qiskit | qiskit.circuit.library.CPhaseGate |
| PennyLane | pennylane.ControlledPhaseShift |
| Cirq | cirq.CZPowGate ⓘ |
| Q# | — ⓘ |
| PyQuil | pyquil.gates.CPHASE |
| Braket | braket.circuits.gates.CPhaseShift ⓘ |
| BQSKit | bqskit.ir.gates.CPGate |
| Qibo | qibo.gates.CU1 ⓘ |
| pytket | pytket.circuit.OpType.CU1 ⓘ |
| Stim | — |
| OpenQASM | stdgates.inc: cp |