quantum gate directory
Controlled-CZ
Symbol
$\mathrm{CC}Z$
Description:
Applies a $-1$ phase to $|111\rangle$ and leaves all other basis states unchanged.
Alternate notations:
- $C^2Z$
- $\Lambda^2(Z)$
SDK Support
| SDK | Name |
|---|---|
| Qiskit |
qiskit.circuit.library.CCZGate
|
| PennyLane |
pennylane.CCZ
|
| Cirq |
cirq.CCZ
|
| Q# | — |
| PyQuil | — |
| Braket | — |
| BQSKit | — |
Groups
This gate is contained in the following groups:
The controlled-controlled-$Z$ gate applies a phase of $-1$ to the $|111\rangle$ state and acts as the identity on all others.
$$ \mathrm{CC}Z = \begin{bmatrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 \end{bmatrix} $$
Properties
- Hermitian and self-inverse: $\mathrm{CCZ}^2 = I$.
- Related to Toffoli by Hadamards on the target: $\mathrm{CC}X = (I \otimes I \otimes H)\mathrm{CC}Z(I \otimes I \otimes H)$.
- Symmetric under permutation of all three qubits, unlike Toffoli.
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