File:Qcircuit ToffolifromCNOT.svg

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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]
Date
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Own work

Created in LaTeX using Q-circuit
Author Geek3
SVG development
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 The SVG code is valid.
 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
This file is licensed under the Creative Commons Attribution-Share Alike 4.0 International license.
You are free:
  • to share – to copy, distribute and transmit the work
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Under the following conditions:
  • attribution – You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
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  1. ↑ On the CNOT-cost of TOFFOLI gates

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