---
title: "Toffoli gate"
firstLetter: "T"
publishDate: 2021-04-09
categories:
- Quantum information

date: 2021-04-09T14:44:43+02:00
draft: false
markup: pandoc
---

# Toffoli gate

The **Toffoli gate** or **controlled-controlled-NOT (CCNOT) gate**
is a logic gate that is *reversible* (no information is lost)
and *universal* (all reversible logic circuits can be built using Toffoli gates).

It takes three input bits $A$, $B$ and $C$,
of which it returns $A$ and $B$ unchanged,
and flips $C$ if both $A$ and $B$ are true.
In circuit diagrams, its representation is:

<a href="toffoli.png">
<img src="toffoli.png" style="width:19%;display:block;margin:auto;">
</a>

This gate is reversible, because $A$ and $B$ are preserved,
and are all you need to reconstruct to $C$.
Moreover, this gate is universal,
because we can make a NAND gate from it:

<a href="nand.png">
<img src="nand.png" style="width:38%;display:block;margin:auto;">
</a>

A NAND is enough to implement every conceivable circuit.
That said, we can efficiently implement NOT, AND, and XOR using a single Toffoli gate too.
Note that NOT is a special case of NAND:

<a href="not.png">
<img src="not.png" style="width:32%;display:block;margin:auto;">
</a>

<a href="and.png">
<img src="and.png" style="width:35%;display:block;margin:auto;">
</a>

<a href="xor.png">
<img src="xor.png" style="width:35%;display:block;margin:auto;">
</a>

Using these, we can, as an example, make an OR gate
from three Toffoli gates,
thanks to the fact that $A \lor B = \neg (\neg A \land \neg B)$,
i.e. OR is NAND of NOT $A$ and NOT $B$:

<a href="or.png">
<img src="or.png" style="width:50%;display:block;margin:auto;">
</a>

Thanks to its reversibility and universality,
the Toffoli gate is interesting in quantum computing,
where it is often referred to as the CCNOT gate.
In the basis $\ket{A} \ket{B} \ket{C}$, its matrix is:

$$\begin{aligned}
    \boxed{
        \mathrm{CCNOT} =
        \begin{bmatrix}
            1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
            0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
            0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\
            0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\
            0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\
            0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\
            0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\
            0 & 0 & 0 & 0 & 0 & 0 & 1 & 0
        \end{bmatrix}
    }
\end{aligned}$$

If we apply this gate to an arbitrary three-qubit state $\ket{\psi}$,
it swaps the last two coefficients:

$$\begin{aligned}
    \mathrm{CCNOT} \ket{\psi}
    &= \mathrm{CCNOT} \big( c_{000} \ket{000} + c_{001} \ket{001} + c_{010} \ket{010} + c_{011} \ket{011} \\
    &\qquad\qquad\quad\:\; c_{100} \ket{100} + c_{101} \ket{101} + c_{110} \ket{110} + c_{111} \ket{111} \big)
    \\
    &= c_{000} \ket{000} + c_{001} \ket{001} + c_{010} \ket{010} + c_{011} \ket{011} \\
    &\quad\,\, c_{100} \ket{100} + c_{101} \ket{101} + c_{111} \ket{110} + c_{110} \ket{111}
\end{aligned}$$