The controlled-$Z$ gate is a two-qubit diagonal entangling gate.
$$ \mathrm{C}Z = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & -1 \end{bmatrix} $$
Properties
- Clifford, Hermitian, and self-inverse.
- Related to CNOT by Hadamards on the target: $\mathrm{CNOT} = (I \otimes H)\,\mathrm{CZ}\,(I \otimes H)$.
- Symmetric under exchange of the two qubits.
Usage
- Native entangling gate on several hardware platforms.
- Controlled phase flips and stabilizer circuits.
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.
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.CZGate |
| PennyLane | pennylane.CZ |
| Cirq | cirq.CZ |
| Q# | Std.Canon.CZ |
| PyQuil | pyquil.gates.CZ |
| Braket | braket.circuits.gates.CZ |
| BQSKit | bqskit.ir.gates.CZGate |
| Qibo | qibo.gates.CZ |
| pytket | pytket.circuit.OpType.CZ |
| Stim | CZ |
| OpenQASM | stdgates.inc: cz |