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22394 lines (21794 loc) · 777 KB
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(************** Content-type: application/mathematica **************
CreatedBy='Mathematica 5.0'
Mathematica-Compatible Notebook
This notebook can be used with any Mathematica-compatible
application, such as Mathematica, MathReader or Publicon. The data
for the notebook starts with the line containing stars above.
To get the notebook into a Mathematica-compatible application, do
one of the following:
* Save the data starting with the line of stars above into a file
with a name ending in .nb, then open the file inside the
application;
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clipboard, then use the Paste menu command inside the application.
Data for notebooks contains only printable 7-bit ASCII and can be
sent directly in email or through ftp in text mode. Newlines can be
CR, LF or CRLF (Unix, Macintosh or MS-DOS style).
NOTE: If you modify the data for this notebook not in a Mathematica-
compatible application, you must delete the line below containing
the word CacheID, otherwise Mathematica-compatible applications may
try to use invalid cache data.
For more information on notebooks and Mathematica-compatible
applications, contact Wolfram Research:
web: http://www.wolfram.com
email: info@wolfram.com
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Notebook reader applications are available free of charge from
Wolfram Research.
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p\[LeftDoubleBracket]2\[RightDoubleBracket] -
1\[RightDoubleBracket] +
1\[IndentingNewLine]]\[IndentingNewLine]];\)\)], "Input"],
Cell[CellGroupData[{
Cell[BoxData[
\(\((hadm\ = \ clifford\ [3, 3])\) // stabForm\)], "Input"],
Cell[BoxData[
TagBox[\(Y\^C\),
DisplayForm]], "Output"]
}, Open ]],
Cell[CellGroupData[{
Cell[BoxData[
\(conj[hadm, I*pauliX] // stabForm\)], "Input"],
Cell[BoxData[
TagBox[\(\[ImaginaryI] Z\),
DisplayForm]], "Output"]
}, Open ]],
Cell[CellGroupData[{
Cell[BoxData[
\(\(\((pauliMultTbl\ = \[IndentingNewLine]{{pauliId, \ pauliX, \
pauliY, \ pauliZ}, \[IndentingNewLine]{pauliX, pauliId, \
I*pauliZ, \ \(-I\)*
pauliY}, \[IndentingNewLine]{pauliY, \(-I\)*pauliZ,
pauliId, \ I*pauliX}, \[IndentingNewLine]{pauliZ, \
I*pauliY, \(-I\)*pauliX, pauliId}})\)\ // \
stabForm\)\ // \ TableForm\)], "Input"],
Cell[BoxData[
TagBox[GridBox[{
{
TagBox[\( I\),
DisplayForm],
TagBox[\( X\),
DisplayForm],
TagBox[\( Y\),
DisplayForm],
TagBox[\( Z\),
DisplayForm]},
{
TagBox[\( X\),
DisplayForm],
TagBox[\( I\),
DisplayForm],
TagBox[\(\[ImaginaryI] Z\),
DisplayForm],
TagBox[\(\(-\[ImaginaryI]\) Y\),
DisplayForm]},
{
TagBox[\( Y\),
DisplayForm],
TagBox[\(\(-\[ImaginaryI]\) Z\),
DisplayForm],
TagBox[\( I\),
DisplayForm],
TagBox[\(\[ImaginaryI] X\),
DisplayForm]},
{
TagBox[\( Z\),
DisplayForm],
TagBox[\(\[ImaginaryI] Y\),
DisplayForm],
TagBox[\(\(-\[ImaginaryI]\) X\),
DisplayForm],
TagBox[\( I\),
DisplayForm]}
},
RowSpacings->1,
ColumnSpacings->3,
RowAlignments->Baseline,
ColumnAlignments->{Left}],
Function[ BoxForm`e$,
TableForm[ BoxForm`e$]]]], "Output"]
}, Open ]],
Cell[BoxData[
\(multPauli[a : pauliT, \
b : pauliT\ ] := \ \[IndentingNewLine]pauli\ [\[IndentingNewLine]a\
\[LeftDoubleBracket]1\[RightDoubleBracket]\ *\
b\[LeftDoubleBracket]1\[RightDoubleBracket]\ *\ \(pauliMultTbl\ \
\[LeftDoubleBracket]a\[LeftDoubleBracket]2\[RightDoubleBracket], \
b\[LeftDoubleBracket]2\[RightDoubleBracket]\[RightDoubleBracket]\
\)\[LeftDoubleBracket]1\[RightDoubleBracket], \
\[IndentingNewLine]\(pauliMultTbl\ \[LeftDoubleBracket]
a\[LeftDoubleBracket]2\[RightDoubleBracket], \
b\[LeftDoubleBracket]2\[RightDoubleBracket]\[RightDoubleBracket]\)\
\[LeftDoubleBracket]2\[RightDoubleBracket]\[IndentingNewLine]]\)], "Input"],
Cell[CellGroupData[{
Cell[BoxData[
\(multPauli[pauliX, pauliY]\ // stabForm\)], "Input"],
Cell[BoxData[
TagBox[\(\[ImaginaryI] Z\),
DisplayForm]], "Output"]
}, Open ]],
Cell[CellGroupData[{
Cell[BoxData[
\(\((allCliffords\ = \[IndentingNewLine]Flatten\ \
[\[IndentingNewLine]Table\ [\[IndentingNewLine]Table\ \
[\[IndentingNewLine]clifford\ [p, s], \[IndentingNewLine]{s, 1,
4}], \[IndentingNewLine]{p, 1,
6}]\[IndentingNewLine]])\)\ // \ stabForm\)], "Input"],
Cell[BoxData[
RowBox[{"{",
RowBox[{
TagBox[\(I\^A\),
DisplayForm], ",",
TagBox[\(X\^A\),
DisplayForm], ",",
TagBox[\(Y\^A\),
DisplayForm], ",",
TagBox[\(Z\^A\),
DisplayForm], ",",
TagBox[\(I\^B\),
DisplayForm], ",",
TagBox[\(X\^B\),
DisplayForm], ",",
TagBox[\(Y\^B\),
DisplayForm], ",",
TagBox[\(Z\^B\),
DisplayForm], ",",
TagBox[\(I\^C\),
DisplayForm], ",",
TagBox[\(X\^C\),
DisplayForm], ",",
TagBox[\(Y\^C\),
DisplayForm], ",",
TagBox[\(Z\^C\),
DisplayForm], ",",
TagBox[\(I\^D\),
DisplayForm], ",",
TagBox[\(X\^D\),
DisplayForm], ",",
TagBox[\(Y\^D\),
DisplayForm], ",",
TagBox[\(Z\^D\),
DisplayForm], ",",
TagBox[\(I\^E\),
DisplayForm], ",",
TagBox[\(X\^E\),
DisplayForm], ",",
TagBox[\(Y\^E\),
DisplayForm], ",",
TagBox[\(Z\^E\),
DisplayForm], ",",
TagBox[\(I\^F\),
DisplayForm], ",",
TagBox[\(X\^F\),
DisplayForm], ",",
TagBox[\(Y\^F\),
DisplayForm], ",",
TagBox[\(Z\^F\),
DisplayForm]}], "}"}]], "Output"]
}, Open ]],
Cell[BoxData[{
\(\(Clear\ [applyLocCliffs];\)\), "\[IndentingNewLine]",
\(applyLocCliffs\ [s : stabT, \
ops : {cliffordT, \
cliffordT}]\ := \[IndentingNewLine]Block\ [{s2}, \
\[IndentingNewLine]s2\ = \ \[IndentingNewLine]Table\ \
[\[IndentingNewLine]Table\ [\[IndentingNewLine]conj\ [
ops\[LeftDoubleBracket]c\[RightDoubleBracket],
s\[LeftDoubleBracket]r,
c\[RightDoubleBracket]], \[IndentingNewLine]{c, 1,
2}], \[IndentingNewLine]{r, 1, 2}]; \[IndentingNewLine]stab\ [
wellformStabRow[s2\[LeftDoubleBracket]1\[RightDoubleBracket]], \
wellformStabRow[
s2\[LeftDoubleBracket]2\[RightDoubleBracket]]]\[IndentingNewLine]]\
\)}], "Input"],
Cell[CellGroupData[{
Cell[BoxData[
\(applyLocCliffs\ [egs, \ {hadm, \ clifford[1, 3]}]\ // \
stabForm\)], "Input"],
Cell[BoxData[
TagBox[
RowBox[{"(", "\[NoBreak]", GridBox[{
{
TagBox[\( Z\),
DisplayForm],
TagBox[\( I\),
DisplayForm]},
{
TagBox[\(-I\),
DisplayForm],
TagBox[\( X\),
DisplayForm]}
}], "\[NoBreak]", ")"}],
Function[ BoxForm`e$,
MatrixForm[ BoxForm`e$]]]], "Output"]
}, Open ]],
Cell[CellGroupData[{
Cell[BoxData[
\(TableForm\ [\[IndentingNewLine]stabForm\ \
[\[IndentingNewLine]unconnectedStates\ = \ \[IndentingNewLine]Table\ [\
\[IndentingNewLine]Table[\[IndentingNewLine]applyLocCliffs[\[IndentingNewLine]\
egs, \[IndentingNewLine]{allCliffords\[LeftDoubleBracket]
op1\[RightDoubleBracket], \
allCliffords\[LeftDoubleBracket]
op2\[RightDoubleBracket]}\[IndentingNewLine]], \
\[IndentingNewLine]{op2, 1, 24}], \[IndentingNewLine]{op1, 1,
24}]], \[IndentingNewLine]TableDepth\ \[Rule] \ 2, \
TableHeadings\ \[Rule] \ {stabForm[allCliffords],
stabForm[allCliffords]}]\)], "Input"],
Cell[BoxData[
TagBox[GridBox[{
{"\<\"\"\>",
TagBox[\(I\^A\),
DisplayForm],
TagBox[\(X\^A\),
DisplayForm],
TagBox[\(Y\^A\),
DisplayForm],
TagBox[\(Z\^A\),
DisplayForm],
TagBox[\(I\^B\),
DisplayForm],
TagBox[\(X\^B\),
DisplayForm],
TagBox[\(Y\^B\),
DisplayForm],
TagBox[\(Z\^B\),
DisplayForm],
TagBox[\(I\^C\),
DisplayForm],
TagBox[\(X\^C\),
DisplayForm],
TagBox[\(Y\^C\),
DisplayForm],
TagBox[\(Z\^C\),
DisplayForm],
TagBox[\(I\^D\),
DisplayForm],
TagBox[\(X\^D\),
DisplayForm],
TagBox[\(Y\^D\),
DisplayForm],
TagBox[\(Z\^D\),
DisplayForm],
TagBox[\(I\^E\),
DisplayForm],
TagBox[\(X\^E\),
DisplayForm],
TagBox[\(Y\^E\),
DisplayForm],
TagBox[\(Z\^E\),
DisplayForm],
TagBox[\(I\^F\),
DisplayForm],
TagBox[\(X\^F\),
DisplayForm],
TagBox[\(Y\^F\),
DisplayForm],
TagBox[\(Z\^F\),
DisplayForm]},
{
TagBox[\(I\^A\),
DisplayForm],
TagBox[
RowBox[{"(", "\[NoBreak]", GridBox[{
{
TagBox[\( X\),
DisplayForm],
TagBox[\( I\),
DisplayForm]},
{
TagBox[\( I\),
DisplayForm],
TagBox[\( X\),
DisplayForm]}
}], "\[NoBreak]", ")"}],
Function[ BoxForm`e$,
MatrixForm[ BoxForm`e$]]],
TagBox[
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{
TagBox[\( X\),
DisplayForm],
TagBox[\( I\),
DisplayForm]},
{
TagBox[\( I\),
DisplayForm],
TagBox[\( X\),
DisplayForm]}
}], "\[NoBreak]", ")"}],
Function[ BoxForm`e$,
MatrixForm[ BoxForm`e$]]],
TagBox[
RowBox[{"(", "\[NoBreak]", GridBox[{
{
TagBox[\( X\),
DisplayForm],
TagBox[\( I\),
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{
TagBox[\(-I\),
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TagBox[\( X\),
DisplayForm]}
}], "\[NoBreak]", ")"}],
Function[ BoxForm`e$,
MatrixForm[ BoxForm`e$]]],
TagBox[
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{
TagBox[\( X\),
DisplayForm],
TagBox[\( I\),
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{
TagBox[\(-I\),
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TagBox[\( X\),
DisplayForm]}
}], "\[NoBreak]", ")"}],
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MatrixForm[ BoxForm`e$]]],
TagBox[
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{
TagBox[\( X\),
DisplayForm],
TagBox[\( I\),
DisplayForm]},
{
TagBox[\(-I\),
DisplayForm],
TagBox[\( Y\),
DisplayForm]}
}], "\[NoBreak]", ")"}],
Function[ BoxForm`e$,
MatrixForm[ BoxForm`e$]]],
TagBox[
RowBox[{"(", "\[NoBreak]", GridBox[{
{
TagBox[\( X\),
DisplayForm],
TagBox[\( I\),
DisplayForm]},
{
TagBox[\(-I\),
DisplayForm],
TagBox[\( Y\),
DisplayForm]}
}], "\[NoBreak]", ")"}],
Function[ BoxForm`e$,
MatrixForm[ BoxForm`e$]]],
TagBox[
RowBox[{"(", "\[NoBreak]", GridBox[{
{
TagBox[\( X\),
DisplayForm],
TagBox[\( I\),
DisplayForm]},
{
TagBox[\( I\),
DisplayForm],
TagBox[\( Y\),
DisplayForm]}
}], "\[NoBreak]", ")"}],
Function[ BoxForm`e$,
MatrixForm[ BoxForm`e$]]],
TagBox[
RowBox[{"(", "\[NoBreak]", GridBox[{
{
TagBox[\( X\),
DisplayForm],
TagBox[\( I\),
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TagBox[\( I\),
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TagBox[\( Y\),
DisplayForm]}
}], "\[NoBreak]", ")"}],
Function[ BoxForm`e$,
MatrixForm[ BoxForm`e$]]],
TagBox[
RowBox[{"(", "\[NoBreak]", GridBox[{
{
TagBox[\( X\),
DisplayForm],
TagBox[\( I\),
DisplayForm]},
{
TagBox[\(-I\),
DisplayForm],
TagBox[\( Z\),
DisplayForm]}
}], "\[NoBreak]", ")"}],
Function[ BoxForm`e$,
MatrixForm[ BoxForm`e$]]],
TagBox[
RowBox[{"(", "\[NoBreak]", GridBox[{
{
TagBox[\( X\),
DisplayForm],
TagBox[\( I\),
DisplayForm]},
{
TagBox[\(-I\),
DisplayForm],
TagBox[\( Z\),
DisplayForm]}
}], "\[NoBreak]", ")"}],
Function[ BoxForm`e$,
MatrixForm[ BoxForm`e$]]],
TagBox[
RowBox[{"(", "\[NoBreak]", GridBox[{
{
TagBox[\( X\),
DisplayForm],
TagBox[\( I\),
DisplayForm]},
{
TagBox[\( I\),
DisplayForm],
TagBox[\( Z\),
DisplayForm]}
}], "\[NoBreak]", ")"}],
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MatrixForm[ BoxForm`e$]]],
TagBox[
RowBox[{"(", "\[NoBreak]", GridBox[{
{
TagBox[\( X\),
DisplayForm],
TagBox[\( I\),
DisplayForm]},
{
TagBox[\( I\),
DisplayForm],
TagBox[\( Z\),
DisplayForm]}
}], "\[NoBreak]", ")"}],
Function[ BoxForm`e$,
MatrixForm[ BoxForm`e$]]],
TagBox[
RowBox[{"(", "\[NoBreak]", GridBox[{
{
TagBox[\( X\),
DisplayForm],
TagBox[\( I\),
DisplayForm]},
{
TagBox[\(-I\),
DisplayForm],
TagBox[\( X\),
DisplayForm]}
}], "\[NoBreak]", ")"}],
Function[ BoxForm`e$,
MatrixForm[ BoxForm`e$]]],
TagBox[
RowBox[{"(", "\[NoBreak]", GridBox[{
{
TagBox[\( X\),
DisplayForm],
TagBox[\( I\),
DisplayForm]},
{
TagBox[\(-I\),
DisplayForm],
TagBox[\( X\),
DisplayForm]}
}], "\[NoBreak]", ")"}],
Function[ BoxForm`e$,
MatrixForm[ BoxForm`e$]]],
TagBox[
RowBox[{"(", "\[NoBreak]", GridBox[{
{
TagBox[\( X\),
DisplayForm],
TagBox[\( I\),
DisplayForm]},
{
TagBox[\( I\),
DisplayForm],
TagBox[\( X\),
DisplayForm]}
}], "\[NoBreak]", ")"}],
Function[ BoxForm`e$,
MatrixForm[ BoxForm`e$]]],
TagBox[
RowBox[{"(", "\[NoBreak]", GridBox[{
{
TagBox[\( X\),
DisplayForm],
TagBox[\( I\),
DisplayForm]},
{
TagBox[\( I\),
DisplayForm],
TagBox[\( X\),
DisplayForm]}
}], "\[NoBreak]", ")"}],
Function[ BoxForm`e$,
MatrixForm[ BoxForm`e$]]],
TagBox[
RowBox[{"(", "\[NoBreak]", GridBox[{
{
TagBox[\( X\),
DisplayForm],
TagBox[\( I\),
DisplayForm]},
{
TagBox[\( I\),
DisplayForm],