Toffoli Gate
Last updated Jun 24, 2022
Chapter 4 - Multiple Quantum Bits > Toffoli Gate
# Toffoli Gate
- The Toffoli Gate is a three-qubit Quantum Gate that flips the right Qubit if the left and middle qubits are $1$ : $$\begin{aligned}
\text{Toffoli}\ket{000} &= \ket{000}, \
\text{Toffoli}\ket{001} &= \ket{001}, \
\text{Toffoli}\ket{010} &= \ket{010}, \
\text{Toffoli}\ket{011} &= \ket{011}, \
\text{Toffoli}\ket{100} &= \ket{100}, \
\text{Toffoli}\ket{101} &= \ket{101}, \
\text{Toffoli}\ket{110} &= \ket{111}, \
\text{Toffoli}\ket{111} &= \ket{110}.
\end{aligned}$$ - As we know that the Toffoli gate is universal for classical computing, and since it is a Quantum Gate, quantum computers can efficiently do everything a classical computer can efficiently do.
- Toffoli gate, represented as a Matrix: $$\text{Toffoli} =
\begin{pmatrix}
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \
\end{pmatrix}.$$ - Its circuit diagram is similar to the CNOT Gate: