Editing Quantum logic gate (section)

=== Toffoli (CCNOT) gate ===
{{Main|Toffoli gate}}
[[Image:Toffoli gate.svg|upright=0.6|thumb|Circuit representation of Toffoli gate]]
The Toffoli gate, named after [[Tommaso Toffoli]] and also called the CCNOT gate or [[#Deutsch gate|Deutsch gate]] <math>D(\pi/2)</math>, is a 3-bit gate that is [[Functional completeness|universal]] for classical computation but not for quantum computation. The quantum Toffoli gate is the same gate, defined for 3 qubits. If we limit ourselves to only accepting input qubits that are <math>|0\rangle</math> and {{nowrap|<math>|1\rangle</math>,}} then if the first two bits are in the state <math>|1\rangle</math> it applies a Pauli-''X'' (or NOT) on the third bit, else it does nothing. It is an example of a CC-U (controlled-controlled Unitary) gate. Since it is the quantum analog of a classical gate, it is completely specified by its truth table. The Toffoli gate is universal when combined with the single qubit Hadamard gate.<ref name="Aharonov">{{cite arXiv|last=Aharonov|first=Dorit|date=2003-01-09|title=A Simple Proof that Toffoli and Hadamard are Quantum Universal|eprint=quant-ph/0301040}}</ref>

{|
|-
! Truth table !! Matrix form
|-
|
{| class="wikitable"
|-
! colspan="3" | Input
! colspan="3" | Output
|- style="text-align:center;"
| 0 || 0 || 0 || 0 || 0 || 0
|- style="text-align:center;"
| 0 || 0 || 1 || 0 || 0 || 1
|- style="text-align:center;"
| 0 || 1 || 0 || 0 || 1 || 0
|- style="text-align:center;"
| 0 || 1 || 1 || 0 || 1 || 1
|- style="text-align:center;"
| 1 || 0 || 0 || 1 || 0 || 0
|- style="text-align:center;"
| 1 || 0 || 1 || 1 || 0 || 1
|- style="text-align:center;"
| 1 || 1 || 0 || 1 || 1 || 1
|- style="text-align:center;"
| 1 || 1 || 1 || 1 || 1 || 0
|}
|
<math>
\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}
</math>
|}

The Toffoli gate is related to the classical [[Logical conjunction|AND]] (<math>\land</math>) and [[exclusive or|XOR]] (<math>\oplus</math>) operations as it performs the mapping <math>|a, b, c\rangle \mapsto |a, b, c\oplus (a \land b)\rangle</math> on states in the computational basis.

The Toffoli gate can be expressed using [[Pauli matrices]] as

:<math> \mbox{Toff} = e^{i\frac{\pi}{8}(I-Z_1)(I-Z_2)(I-X_3)}= e^{-i\frac{\pi}{8}(I-Z_1)(I-Z_2)(I-X_3)}. </math>