quantum gate directory
Toffoli Gate
Symbol
$\mathrm{CCX}$
Description:
The Toffoli gate, also known as the Controlled-Controlled-NOT ($\mathrm{CCX}$) gate, flips the state of a target qubit if and only if the two control qubits are in the $|1\rangle$ state. It is a universal gate for classical reversible computation and plays an important role in quantum algorithms.
| Notation(s) | $\mathrm{CCX}$, $\mathrm{CCNOT}$, $\mathrm{TOFF}$ |
| Matrix | $\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \end{bmatrix}$ |
| Arity | 3 |
| Group | $\mathsf{U}(8)$ |
Properties
- Classical Universality: The Toffoli gate can implement any classical reversible computation when combined with single-qubit NOT gates.
- Reversibility: The Toffoli gate is its own inverse, i.e., $\text{CCX} \cdot \text{CCX} = I$.
- Decomposition: The Toffoli gate can be decomposed into single-qubit and two-qubit gates (e.g., Hadamard, T, and CNOT gates) on a universal quantum gate set.
Applications
- Classical reversible computation.
- Controlled operations involving multiple control qubits.
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