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Summary

Description
English: Quantum circuit diagram constructing a Toffoli gate from single-qubit operations and six controlled NOT (CNOT) gates. This is the minimum number of CNOTs required to build a Toffoli.[1]
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Created in LaTeX using Q-circuit
Author Geek3
SVG development
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This vector image was created with LaTeX, using Q-circuit
gate matrix:
Source code
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LaTeX code

LaTeX Qcircuit source code 
\documentclass[border=10pt]{standalone}
\input{Qcircuit} % available on https://github.com/CQuIC/qcircuit
\usepackage{graphicx}
\begin{document}
\newcommand{\multistate}[2]{*+{\hphantom{#2}} \POS[0,0].[#1,0] !C
  *{#2} \POS[0,0].[#1,0] \drop\frm{}}
\newcommand{\ghoststate}[1]{*+{\hphantom{#1}} }
\newcommand{\ccteqg}{\ghoststate{=}}
\newcommand{\ccteq}[1]{\multistate{#1}{=}}
\scalebox{5}{%
\Qcircuit @C=0.8em {
& \control \qw & \qw & \ccteq{2} & & \qw & \qw & \qw & \control \qw & \qw &
\qw & \qw & \control \qw & \qw & \control \qw & \gate{T} & \control \qw & \qw \\
& \control \qw \qwx & \qw & \ccteqg & & \qw & \control \qw & \qw & \qw \qwx &
\qw & \control \qw & \qw & \qw \qwx & \gate{T} & \targ \qwx & \gate{T^\dagger}
& \targ \qwx  & \qw \\
& \targ \qw \qwx & \qw & \ccteqg & & \gate{H} & \targ \qwx & \gate{T^\dagger}
& \targ \qwx & \gate{T} & \targ \qwx & \gate{T^\dagger} & \targ \qwx & \gate{T}
& \qw & \gate{H} & \qw & \qw }}
\end{document}

Bash code

postprocessing with pdf2svg, Inkscape and Scour 
x=Qcircuit_ToffolifromCNOT;
pdflatex $x.tex;
pdf2svg $x.pdf $x_.svg;
inkscape --without-gui --verb=EditSelectAll --verb=SelectionUnGroup --verb=FileVacuum --verb=FileSave --verb=FileClose $x_.svg;
scour -i $x_.svg -o $x.svg --renderer-workaround --remove-metadata --enable-id-stripping --create-groups --enable-viewboxing

Mathematica code

The following Mathematica code shows the equality: 
PauliMatrix = {{{0, 1}, {1, 0}}, {{0, -I}, {I, 0}}, {{1, 0}, {0, -1}}}[[#1]] &;
R[x_, y_, z_] := MatrixExp[-I/2 Sum[{x, y, z}[[i]]*PauliMatrix[i], {i, 3}]];
A_⊗B_ := KroneckerProduct[A, B];
A_⊗B_⊗C_ := KroneckerProduct[A, B, C];
Gn[G_, i_, n_] := Fold[#1⊗#2 &, {{1}}, ReplacePart[Table[I2, {n}], i -> G]];
I2 = IdentityMatrix[2];
X = R[Pi, 0, 0] I;
Z = R[0, 0, Pi] I;
H = {{1, 1}, {1, -1}}/Sqrt[2];
T = Exp[I Pi/8] R[0, 0, Pi/4];
Tdag = ConjugateTranspose[T];
Swap = {{1, 0, 0, 0}, {0, 0, 1, 0}, {0, 1, 0, 0}, {0, 0, 0, 1}};
CNOT = {{1, 0, 0, 0}, {0, 1, 0, 0}, {0, 0, 0, 1}, {0, 0, 1, 0}};
CNOT12 = (CNOT⊗I2);
CNOT23 = (I2⊗CNOT);
CNOT13 = (Swap⊗I2).CNOT23.(Swap⊗I2);

Toffoli = 
 CNOT12.(T⊗Tdag⊗H).CNOT12.(I2⊗T⊗T).CNOT13.Gn[Tdag, 3, 3].CNOT23.Gn[
    T, 3, 3].CNOT13.Gn[Tdag, 3, 3].CNOT23.Gn[H, 3, 3] // MatrixForm

Licensing

I, the copyright holder of this work, hereby publish it under the following license:
w:en:Creative Commons
attribution share alike
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You are free:
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Under the following conditions:
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  1. ↑ On the CNOT-cost of TOFFOLI gates

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