quantum gate directory
Controlled-Z
Symbol
$\mathrm{C}Z$
Description:
Applies a $-1$ phase to $|11\rangle$.
Alternate notations:
- $\Lambda(Z)$
- $\text{controlled-}Z$
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.
Groups
This gate is contained in the following groups:
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