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Cell[BoxData[
    \(\(\*"\"\<So f[r] is a periodic function, and has period  4. The \
order,r, of the function, for a given (a,N) is defined as the value in which\
\[IndentingNewLine]\!\(a\^r\) Mod=1, or looking at the above table it is \
equal\[IndentingNewLine]to the period of the function, or r=4\>\""\(.\)\)\)], \
"Input",
  Evaluatable->False],

Cell[BoxData[{
    \(What\ is\ the\ order, \ 
    r, \ \ of\ this\ function\[IndentingNewLine]\), "\[IndentingNewLine]", 
    \(\(11\^r\) Mod[15]\  = 
      1?\ \((lets\ try\ this\ longhand\ first)\)\)}], "Input",
  Evaluatable->False],

Cell[BoxData[
    \("\<It is hard to find the order of aMod[M]\>"\)], "Input",
  Evaluatable->False],

Cell[BoxData[{
    \(\(We\ have\ in\ register\ 1\ \ \ t - qubits\ so\ that\ \ 2\^t\  = 
        N\ ;\)\), "\[IndentingNewLine]", 
    \(and\ in\ register2\ \ r\ qubits\ so\ that\ 2\^r = 
      L . \ These\ choices\), "\[IndentingNewLine]", 
    \(depend\ on\ the\ domain\ of\ x, \ 
    and\ the\ range\ of\ the\ function\ \(\(f\)\(.\)\)\)}], "Input",
  Evaluatable->False],

Cell[BoxData[{
    RowBox[{"\[IndentingNewLine]", 
      ButtonBox[\(Shor\ circuit\),
        ButtonData:>{
          URL[ "http://www.physics.unlv.edu/~bernard/phy771/fig16.pdf"], 
          None},
        ButtonStyle->"Hyperlink"], "\[IndentingNewLine]", 
      "\[IndentingNewLine]"}], "\[IndentingNewLine]", \(Lets\ start\ the\ \
computation\ so\ that\ the\), "\[IndentingNewLine]", \(registers\ are\ loaded\
\ with\ the\ value\), "\[IndentingNewLine]", \(\(|\)\(0 > \[CircleTimes]\)\(|\
\)\(\(0\)\(>\)\)\)}], "Input",
  Evaluatable->False],

Cell[BoxData[{
    \(After\ going\ through\ the\ Walsh - 
      Hadamard\ gates\), "\[IndentingNewLine]", 
    \(the\ system\ is\ in\ state\[IndentingNewLine]\), "\[IndentingNewLine]", \

    \(\(\*"\"\<\!\(1\/\@2\^t\) \!\(\[Sum]\+\(x = 0\)\%\(N - 1\)\)|x>\
\[CircleTimes]|0>\>\"";\)\)}], "Input",
  Evaluatable->False],

Cell[BoxData[{
    \(After\ being\ processed\ by\ the\ control\ Uf\ gate\), "\
\[IndentingNewLine]", 
    \(it\ is\ in\ state\[IndentingNewLine]\), "\[IndentingNewLine]", 
    \(\(\*"\"\<\!\(1\/\@2\^t\) \!\(\[Sum]\+\(x = 0\)\%\(N - 1\)\)|x>\
\[CircleTimes]|f(x)>=\!\(1\/\@2\^t\) \!\(\[Sum]\+\(x = 0\)\%\(N - 1\)\)|x>\
\[CircleTimes]|\!\(a\^x\) ModM >\>\"";\)\)}], "Input",
  Evaluatable->False],

Cell[BoxData[{
    \(\(\*"\"\<We now make a measurement on the second register, we will\
\[IndentingNewLine]find one of the values for |\!\(a\^x\) Mod M>, and the \
system\nwill collapse (both register 1 & 2) into the state\n\n\>\"" 
          A \((\(|\)\(x > \[CircleTimes]\)\(\ \)\(|\)\(a\^x\ ModM\  > \
+\)\(|\)\(x + 
                  r > \ \[CircleTimes]\)\(|\)\(a\^x\ Mod\ M\  > \ +\)\(\ \)\(\
\n\)\(|\)\(x + 
                  2\ r > \[CircleTimes]\)\(\ \)\(|\)\(\(a\^x\ Mod\ M\  > + \
... \)\(+\)\)\(\ \)\(|\)\(x + \ 
                  n\ r > \ \[CircleTimes]\)\(|\)\(\(a\^x\ Mod\ M\)\(\ \)\(>\)\
\))\) = \[IndentingNewLine]\[IndentingNewLine]\*"\"\<A(|x> + |x+r> + |x+2 \
r>+...+ |x+ n r>) \[CircleTimes]|\!\(a\^x\) Mod M >\>\"";\)\n\), "\n", 
    \(where\ x + \ nr\  <= \ 2\^t - 1, \ 
    and\ A\ is\ chosen\ so\ that\ the\ collapsed\), "\[IndentingNewLine]", 
    \(state\ is\ normalized\ to\ unity\n\), "\[IndentingNewLine]", 
    \(can\ you\ see\ why\ that\ is\ the\ \(\(case\)\(?\)\)\)}], "Input",
  Evaluatable->False],

Cell[BoxData[
    \(Example : \ suppose\ M = 
      15, \ \(\(a\)\(=\)\(7\)\(\ \ \ \ \)\)\)], "Input",
  Evaluatable->False],

Cell[BoxData[
    \(\*"\"\<\!\(1\/\@2\^t\) \!\(\[Sum]\+\(x = 0\)\%\(N - 1\)\)|x>\
\[CircleTimes]|\!\(7\^x\) Mod15 > = \
\!\(1\/\@2\^t\)(|0>\[CircleTimes]|\!\(7\^0\) Mod15 \
>+|1>\[CircleTimes]|\!\(7\^1\) Mod15 >+|2>\[CircleTimes]|\!\(7\^2\) Mod15 \
>+|3>\[CircleTimes]|\!\(7\^3\) Mod15 >+|4>\[CircleTimes]|\!\(7\^4\) Mod15 >+ \
|5>\[CircleTimes]|\!\(7\^5\) Mod15 >\n\n+  |6>\[CircleTimes]|\!\(7\^6\) Mod15 \
> + ...|\!\(2\^t\)-1>\[CircleTimes]|\!\(7\^\(2\^t - 1\)\) Mod15 >\>\"" = \
\[IndentingNewLine]\[IndentingNewLine]\[IndentingNewLine]\*"\"\<\!\(1\/\@2\^t\
\)(|0>\[CircleTimes]|1>+|1>\[CircleTimes]|7>+|2>\[CircleTimes]|4>+|3>\
\[CircleTimes]|13>+|4>\[CircleTimes]|1> + |5>\[CircleTimes]|7>\n\n+  |6>\
\[CircleTimes]|4> + |7>\[CircleTimes]|13>+ |8>\[CircleTimes]|1> +...+ \
|\!\(2\^t\)-1>\[CircleTimes]|\!\(7\^\(2\^t - 1\)\) Mod15 >)\>\""\)], "Input",
  Evaluatable->False],

Cell[BoxData[{
    \(Suppose\ we\ measured\ the\ value\ "\<7\>"\ for\ the\ second\ register, \
\ the\ system\), "\[IndentingNewLine]", 
    \(wave\ function\ collapses\ into\[IndentingNewLine]\), "\
\[IndentingNewLine]", 
    \("\<A(|1>\[CircleTimes]|7>+|5>\[CircleTimes]|7>+|9>\[CircleTimes]|7>+|13>\
\[CircleTimes]|7> + ...)\>" = \
\[IndentingNewLine]\[IndentingNewLine]"\<|\[Psi]>= A (|1>+|5>+|9>+|13> + ...)\
\[CircleTimes]|7>

where A is chosen so that <\[Psi]|\[Psi]>=1\>"\)}], "Input",
  Evaluatable->False],

Cell[BoxData[{
    \(We\ note\ that\ the\ first\ register\ contains\ a\ linear\ \
combination\), "\[IndentingNewLine]", 
    \(\(\(of\ kets\ seperated\ by\ the\ order\ r . \ 
        In\ general\)\(,\)\)\)}], "Input",
  Evaluatable->False],

Cell[BoxData[
    \(\(\*"\"\<A(|x> + |x+r> + |x+2 r>+...+ |x+ n r>) \[CircleTimes]|\!\(a\^x\
\) Mod M >\>\"";\)\)], "Input",
  Evaluatable->False],

Cell[BoxData[
    \(We\ now\ apply\ a\ Fourier\ \((FT)\)\ gate\ onto\ the\ first\ register, \
\ \(\(i\)\(.\)\(e\)\(.\)\(\[IndentingNewLine]\)\(\[IndentingNewLine]\)\)\)], \
"Input",
  Evaluatable->False],

Cell[BoxData[{
    \(\*"\"\<A(FT(|x> + |x+r> + |x+2 r>+...+ |x+ n r>) \
)\[CircleTimes]|\!\(a\^x\) Mod M >\>\"" = \[IndentingNewLine]\
\[IndentingNewLine]\(\*"\"\<(\!\(A\/\@2\^t\))( \!\(\[Sum]\+\(k = 0\)\%\(2\^t \
- 1\)\) \!\(E\^\(2\\\ \[Pi]\\\ I\\\ k\\\ x/N\)\)|k>+\!\(E\^\(2\\\ \[Pi]\\\ \
I\\\ k\\\ \\\ \((x + r)\)/N\)\)|k>+\!\(E\^\(2\\\ \[Pi]\\\ I\\\ k\\\ \\\ \((x \
+ 2\\\ r)\)/N\)\)|k>+...)\[CircleTimes]|\!\(a\^x\) Mod M >\>\"" = \
\[IndentingNewLine]\[IndentingNewLine]\*"\"\< (\!\(A\/\@2\^t\)) \
\!\(\[Sum]\+\(k = 0\)\%\(2\^t - 1\)\)  \!\(\(\[Sum]\&nmax\)\+\(n = 0\)\) \
\!\(E\^\(2\\\ \[Pi]\\\ I\\\ k\\\ \
x/N\)\)\!\(c\_n\)(k,r)|k>\[CircleTimes]|\!\(a\^x\) Mod M >\>\""\)\
\[IndentingNewLine]\), "\[IndentingNewLine]", 
    \(where\ \ \ \(c\_n\) \((k, r)\) == 
      E\^\(2\ \[Pi]\ I\ k\ n\ r/N\)\)}], "Input",
  Evaluatable->False],

Cell[BoxData[
    \(\*"\"\<We now measure the contents of register 1, we can obtain any of \
the values  m=0,1,2,...\!\(2\^t\)-1\>\""\)], "Input",
  Evaluatable->False],

Cell[BoxData[{
    \(The\ probability\ \ amplitude\ \ for\ obtaining\ \ value\ \ m\
\[IndentingNewLine]\), "\[IndentingNewLine]", 
    \(\*"\"\<(<m|\[CircleTimes]<\!\(a\^x\) Mod M \
|)\!\(A\/\@2\^t\)\!\(\[Sum]\+\(k = 0\)\%\(2\^t - 1\)\) \!\(\(\(\\\ \)\(\[Sum]\
\+\(n = 0\)\)\)\&nmax\) \!\(E\^\(2\\\ \[Pi]\\\ I\\\ k\\\ \
x/N\)\)\!\(c\_n\)(k,r)|k>\[CircleTimes]|\!\(a\^x\) Mod M >\>\"" = \n\n\*"\"\<\
\!\(A\/\@2\^t\)\!\(\(\(\\\ \)\(\[Sum]\+\(n = 0\)\)\)\&nmax\)\!\(E\^\(2\\\ \
\[Pi]\\\ I\\\ m\\\ x/N\)\) \!\(c\_n\)(m,r) \>\""\[IndentingNewLine]\), "\
\[IndentingNewLine]", 
    \(and\ so\ the\ probability\ to\ measure\ value\ m\ is\[IndentingNewLine]\
\), "\[IndentingNewLine]", 
    \(\((\(|\)\(\(A\/\@2\^t\) \(\(\(\ \)\(\[Sum]\+\(n = 0\)\)\)\&nmax\) 
            E\^\(2\ \[Pi]\ I\ m\ x/N\)\ \(c\_n\) \((m, 
              r)\)\)\(|\)\()\^2\)\) = \
\[IndentingNewLine]\[IndentingNewLine]\(\(A\^2\/2\^t\) \(\(\ \)\(\(\(\[Sum]\)\
\(\ \ \)\)\+\(n = 0\)\)\)\&\(\(nmax\)\(\ \ \)\)\ \ \ \(\(\(\ \)\(\(\(\[Sum]\)\
\(\ \ \)\)\+\(k = 0\)\)\)\&\(\(nmax\)\(\ \ \)\)\) \(c\_n\) \((m, 
            r)\) \(\((\(c\_k\) \((m, 
                    r)\))\)\^*\) = \
\[IndentingNewLine]\[IndentingNewLine]\(A\^2\/2\^t\ \(\(\ \)\(\(\(\[Sum]\)\(\ \
\ \)\)\+\(n = 0\)\)\)\&\(\(nmax\)\(\ \ \)\)\ \ \ \(\(\(\ \)\(\(\(\[Sum]\)\(\ \
\ \)\)\+\(k = 0\)\)\)\&\(\(nmax\)\(\ \ \)\)\) \(c\_n\) \((m, 
              r)\) \(\((\(c\_k\) \((m, 
                      r)\))\)\^*\) = \
\[IndentingNewLine]\[IndentingNewLine]A\^2\/2\^t\ \(\(\ \)\(\(\(\[Sum]\)\(\ \ \
\)\)\+\(n = 0\)\)\)\&\(\(nmax\)\(\ \ \)\)\ \ \ \(\(\(\ \)\(\(\(\[Sum]\)\(\ \ \
\)\)\+\(k = 0\)\)\)\&\(\(nmax\)\(\ \ \)\)\) 
            E\^\(2\ \[Pi]\ I\ m \((n - k)\) r/N\)\)\)\)}], "Input",
  Evaluatable->False],

Cell[BoxData[{
    \(This\ probability\ is\ only\ appreciable\ \ if\ \ \ m\ r/
        N\ \ is\ a\ \ multiple\), "\[IndentingNewLine]", 
    \(of\ an\ \(\(integer\)\(.\)\)\)}], "Input",
  Evaluatable->False],

Cell[BoxData[{
    \(\(\("\<Define \>";\)\(\[IndentingNewLine]\)
    \)\), "\[IndentingNewLine]", 
    \(p[m_] := 
      Module[{t = 7, r = 4, N, A, nmax = 5}, A = 1/Sqrt[nmax + 1]; 
        N = \(\(2\)\(\ \)\)\^t\ ; \ \ term = \((A\^2/N)\)\ \ Sum[
              Exp[2\ Pi\ I\ m\ \((n - k)\) r/N], {n, 0, nmax}, {k, 0, 
                nmax}]]\)}], "Input"],

Cell[BoxData[{
    \(\(pgrid = 
        Table[Chop[N[p[m]]], {m, 0, 127}];\)\), "\[IndentingNewLine]", 
    \(\(mgrid = Table[m, {m, 0, 127}];\)\), "\[IndentingNewLine]", 
    \(\(measurements = Thread[{mgrid, pgrid}];\)\), "\[IndentingNewLine]", 
    \(\)}], "Input"],

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Cell[BoxData[
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      PlotJoined \[Rule] True]\)], "Input"],

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    \)\), "\[IndentingNewLine]", 
    \(\(\(m\ \((n - k)\) r/N = Integer\)\(\[IndentingNewLine]\)
    \)\), "\[IndentingNewLine]", 
    \("\<when n=k ; Integer=0\>"\), "\[IndentingNewLine]", 
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    \(\(\(m\ r/N\  = Integer\)\(\[IndentingNewLine]\)
    \)\), "\[IndentingNewLine]", 
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    \(we\ have\), "\[IndentingNewLine]", 
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    \("\<r=8,16,32...\>" for\ m = 64\[IndentingNewLine]\), "\n", 
    \(we\ have\), "\n", 
    \(r = 128/64, \ or\ 2\ \((128/32)\)\ or\ 3\ \((128/32)\)\), "\n", 
    \("\<r=8,16,32...\>"\[IndentingNewLine]\), "\n", 
    \(So\ the\ first\ peak\ corresponds\ \), "\n", 
    \(Integer = 1\), "\n", 
    \(and\ we\ guess\ that\ r = 
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    \(\(\(\((x\^\(r/2\))\)\^2 - 1 = 0\ Mod[M]\)\(\[IndentingNewLine]\)
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