(************** Content-type: application/mathematica ************** 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; * Copy the data starting with the line of stars above to the 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 phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 28490, 808]*) (*NotebookOutlinePosition[ 29427, 837]*) (* CellTagsIndexPosition[ 29383, 833]*) (*WindowFrame->Normal*) Notebook[{ Cell[BoxData[ \(\("\";\)\)], "Input"], Cell[BoxData[ \(\(\(\[IndentingNewLine]\)\(domain = Table[i, {i, 0, 99}]\)\)\)], "Input"], Cell[BoxData[ \(\(function[x_] = Sin[x/2];\)\)], "Input"], Cell[BoxData[ \(\(range = function[domain];\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(ListPlot[range, PlotJoined \[Rule] True]\)], "Input"], Cell[BoxData[ TagBox[\(\[SkeletonIndicator] Graphics \[SkeletonIndicator]\), False, Editable->False]], "Output"] }, Open ]], Cell[BoxData[{ \(\(\("\";\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(\(\(f\&~[ x] = \[Integral]\_\(-\[Infinity]\)\%\[Infinity] f[t]\ Exp[ 2\ Pi\ x\ \ t] \[DifferentialD]t\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(\(\(if\ f[t] = Sin[a\ t]\ then\)\(\ \)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(\ \(\(f\&~[ x] = \[Pi]\ \((\[Delta] \((a + 2\ \[Pi]\ x)\) - \[Delta] \((a - 2\ \[Pi]\ x)\))\)\)\(\[IndentingNewLine]\)\(\ \[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(\)}], "Input", Evaluatable->False], Cell[BoxData[{ \(\(\(Discrete\ Fourier\ Transform\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(y\_k = 1/\@N\ \(\[Sum]\+\(j = 0\)\%\(N - 1\)\(x\_j\) Exp[2\ \[Pi]\ I\ j\ k/N]\)\)}], "Input", Evaluatable->False], Cell[BoxData[{ \(\(n = 100;\)\), "\[IndentingNewLine]", \(\(ftransform = Table[Sum[ range[\([j]\)]\ Exp[2\ Pi\ I\ \ \((j - 1)\)\ \((k - 1)\)/n]/ Sqrt[n], {j, 1, n}], {k, 1, n}]\ ;\)\)}], "Input"], Cell[BoxData[{ \(\(N[ftransform];\)\), "\[IndentingNewLine]", \(\(xk = Table[\((k - 1)\)/n, {k, 1, n}];\)\), "\[IndentingNewLine]", \(\(ft = N[Thread[{\ xk, ftransform}]];\)\), "\[IndentingNewLine]", \(\)}], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(ListPlot[2\ Pi\ Re[ft]]\)], "Input"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.153107 0.257974 0.228702 [ [.17692 .24547 -3 -9 ] [.17692 .24547 3 0 ] [.33002 .24547 -3 -9 ] [.33002 .24547 3 0 ] [.48313 .24547 -3 -9 ] [.48313 .24547 3 0 ] [.63624 .24547 -3 -9 ] [.63624 .24547 3 0 ] [.78935 .24547 -3 -9 ] [.78935 .24547 3 0 ] [.94245 .24547 -3 -9 ] [.94245 .24547 3 0 ] [.01131 .02927 -12 -4.5 ] [.01131 .02927 0 4.5 ] [.01131 .14362 -24 -4.5 ] [.01131 .14362 0 4.5 ] [.01131 .37232 -18 -4.5 ] [.01131 .37232 0 4.5 ] [.01131 .48668 -6 -4.5 ] [.01131 .48668 0 4.5 ] [.01131 .60103 -18 -4.5 ] [.01131 .60103 0 4.5 ] [ 0 0 0 0 ] [ 1 .61803 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash .17692 .25797 m .17692 .26422 L s [(1)] .17692 .24547 0 1 Mshowa .33002 .25797 m .33002 .26422 L s [(2)] .33002 .24547 0 1 Mshowa .48313 .25797 m .48313 .26422 L s [(3)] .48313 .24547 0 1 Mshowa .63624 .25797 m .63624 .26422 L s [(4)] .63624 .24547 0 1 Mshowa .78935 .25797 m .78935 .26422 L s [(5)] .78935 .24547 0 1 Mshowa .94245 .25797 m .94245 .26422 L s [(6)] .94245 .24547 0 1 Mshowa .125 Mabswid .05443 .25797 m .05443 .26172 L s .08505 .25797 m .08505 .26172 L s .11567 .25797 m .11567 .26172 L s .1463 .25797 m .1463 .26172 L s .20754 .25797 m .20754 .26172 L s .23816 .25797 m .23816 .26172 L s .26878 .25797 m .26878 .26172 L s .2994 .25797 m .2994 .26172 L s .36065 .25797 m .36065 .26172 L s .39127 .25797 m .39127 .26172 L s .42189 .25797 m .42189 .26172 L s .45251 .25797 m .45251 .26172 L s .51375 .25797 m .51375 .26172 L s .54437 .25797 m .54437 .26172 L s .575 .25797 m .575 .26172 L s .60562 .25797 m .60562 .26172 L s .66686 .25797 m .66686 .26172 L s .69748 .25797 m .69748 .26172 L s .7281 .25797 m .7281 .26172 L s .75872 .25797 m .75872 .26172 L s .81997 .25797 m .81997 .26172 L s .85059 .25797 m .85059 .26172 L s .88121 .25797 m .88121 .26172 L s .91183 .25797 m .91183 .26172 L s .97307 .25797 m .97307 .26172 L s .25 Mabswid 0 .25797 m 1 .25797 L s .02381 .02927 m .03006 .02927 L s [(-1)] .01131 .02927 1 0 Mshowa .02381 .14362 m .03006 .14362 L s [(-0.5)] .01131 .14362 1 0 Mshowa .02381 .37232 m .03006 .37232 L s [(0.5)] .01131 .37232 1 0 Mshowa .02381 .48668 m .03006 .48668 L s [(1)] .01131 .48668 1 0 Mshowa .02381 .60103 m .03006 .60103 L s [(1.5)] .01131 .60103 1 0 Mshowa .125 Mabswid .02381 .05214 m .02756 .05214 L s .02381 .07501 m .02756 .07501 L s .02381 .09788 m .02756 .09788 L s .02381 .12075 m .02756 .12075 L s .02381 .16649 m .02756 .16649 L s .02381 .18936 m .02756 .18936 L s .02381 .21223 m .02756 .21223 L s .02381 .2351 m .02756 .2351 L s .02381 .28084 m .02756 .28084 L s .02381 .30371 m .02756 .30371 L s .02381 .32658 m .02756 .32658 L s .02381 .34945 m .02756 .34945 L s .02381 .3952 m .02756 .3952 L s .02381 .41807 m .02756 .41807 L s .02381 .44094 m .02756 .44094 L s .02381 .46381 m .02756 .46381 L s .02381 .50955 m .02756 .50955 L s .02381 .53242 m .02756 .53242 L s .02381 .55529 m .02756 .55529 L s .02381 .57816 m .02756 .57816 L s .02381 .0064 m .02756 .0064 L s .25 Mabswid .02381 0 m .02381 .61803 L s 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath .008 w .02381 .34438 Mdot .03343 .34441 Mdot .04305 .34451 Mdot .05267 .34468 Mdot .06229 .34491 Mdot .07191 .34522 Mdot .08153 .34561 Mdot .09115 .34608 Mdot .10077 .34664 Mdot .11039 .3473 Mdot .12001 .34807 Mdot .12963 .34895 Mdot .13925 .34998 Mdot .14887 .35116 Mdot .15849 .35252 Mdot .16811 .35409 Mdot .17773 .35591 Mdot .18735 .35803 Mdot .19697 .36051 Mdot .20659 .36344 Mdot .21621 .36692 Mdot .22583 .37111 Mdot .23545 .37623 Mdot .24507 .38259 Mdot .25469 .39067 Mdot .26431 .40123 Mdot .27393 .41553 Mdot .28355 .43592 Mdot .29317 .46717 Mdot .30279 .52079 Mdot .35089 .04078 Mdot .36051 .12552 Mdot .37013 .16931 Mdot .37975 .19589 Mdot .38937 .21362 Mdot .39899 .22618 Mdot .40861 .23544 Mdot .41823 .24247 Mdot .42785 .24791 Mdot .43747 .25216 Mdot .44709 .25549 Mdot .45671 .25809 Mdot .46633 .26007 Mdot .47595 .26154 Mdot .48557 .26255 Mdot .49519 .26314 Mdot .50481 .26334 Mdot .51443 .26314 Mdot .52405 .26255 Mdot .53367 .26154 Mdot .54329 .26007 Mdot .55291 .25809 Mdot .56253 .25549 Mdot .57215 .25216 Mdot .58177 .24791 Mdot .59139 .24247 Mdot .60101 .23544 Mdot .61063 .22618 Mdot .62025 .21362 Mdot .62987 .19589 Mdot .63949 .16931 Mdot .64911 .12552 Mdot .65873 .04078 Mdot .70683 .52079 Mdot .71645 .46717 Mdot .72607 .43592 Mdot .73569 .41553 Mdot .74531 .40123 Mdot .75493 .39067 Mdot .76455 .38259 Mdot .77417 .37623 Mdot .78379 .37111 Mdot .79341 .36692 Mdot .80303 .36344 Mdot .81265 .36051 Mdot .82227 .35803 Mdot .83189 .35591 Mdot .84151 .35409 Mdot .85113 .35252 Mdot .86075 .35116 Mdot .87037 .34998 Mdot .87999 .34895 Mdot .88961 .34807 Mdot .89923 .3473 Mdot .90885 .34664 Mdot .91847 .34608 Mdot .92809 .34561 Mdot .93771 .34522 Mdot .94733 .34491 Mdot .95695 .34468 Mdot .96657 .34451 Mdot .97619 .34441 Mdot % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{288, 177.938}, ImageMargins->{{43, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCache->GraphicsData["Bitmap", "\<\ CF5dJ6E]HGAYHf4PAg9QL6QYHgoooool2o`0006Gooooo000Kooooo`03 o`000?oooooooooo02Cooooo0ol0000Tooooo`Go00003_ooool2o`0001Gooooo0ol0000Sooooo`;o 00000_ooool3o`0002Cooooo0ol0000Vooooo`?o00004?ooool001_ooooo00?o0000ooooooooool0 9Oooool00ol0003oooooooooo`0Tooooo`04o`000?ooooooooooo`0002Cooooo00Go0000oooooooo ooooooooo`00000Wooooo`03o`000?oooooooooo02;ooooo00Go0000ooooooooooooooooo`00000T ooooo`05o`000?ooooooooooooooool000003oooool001_ooooo00?o0000ooooooooool09Oooool0 0ol0003oooooooooo`0Uooooo`03o`000?oooooooooo02Sooooo00?o0000ooooooooool08_ooool5 o`0002Oooooo00?o0000ooooooooool08_ooool01Ol0003ooooooooooooooooo000000oooooo000K ooooo`03o`000?oooooooooo02Gooooo00?o0000ooooooooool09_ooool00ol0003oooooooooo`0? ooooo`;o00005?ooool2o`00023ooooo0_l00003ooooo`04o`000?ooooooooooo`0002Sooooo00?o 0000ooooooooool08_ooool4o`00013ooooo000Kooooo`03o`000?oooooooooo02Gooooo00?o0000 ooooooooool08oooool01Ol0003ooooooooooooooooo0000013ooooo0_l0000Fooooo`03o`000?oo oooooooo01gooooo0_l00004ooooo`03o`000?oooooo000002Cooooo1?l0000Uooooo`03o`000?oo oooooooo017ooooo000Kooooo`;o00009Oooool2o`0002Gooooo00Go0000ooooooooooooooooo`00 000Cooooo`;o00003oooool01Ol0003ooooooooooooooooo000001gooooo0_l00007ooooo`;o0000 9?ooool00ol0003oooooooooo`0Wooooo`03o`000?oooooooooo013ooooo000Kooooo`03o`000?oo oooooooo02Gooooo00?o0000ooooooooool09?ooool3o`0001Cooooo0_l0000@ooooo`?o00007_oo ool2o`0000Sooooo00?o0000ooooooooool08_ooool5o`0002Kooooo0_l0000@ooooo`006oooool0 0ol0003oooooooooo`1Uooooo`;o0000;?ooool2o`0006gooooo000Kooooo`03o`000?oooooooooo 06Gooooo0_l000000ooooooo0000o`00000Wooooo`Co0000KOooool001_ooooo00?o0000oooooooo ool0J?ooool4o`0002;ooooo0_l000000ooooooo0000o`00001_ooooo`006oooool00ol0003ooooo ooooo`1Zooooo`;o000000?oooooo`000?l000007Oooool4o`0007;ooooo000Doooooooo00003Ol0 000001_ooooo00?o0000ooooooooool01Oooool00ol0003oooooooooo`05ooooo`03o`000?oooooo oooo00Gooooo00?o0000ooooooooool01Oooool00ol0003oooooooooo`06ooooo`03o`000?oooooo oooo00Gooooo00?o0000ooooooooool01Oooool00ol0003oooooooooo`05ooooo`03o`000?oooooo oooo00Gooooo00?o0000ooooooooool01Oooool00ol0003oooooooooo`05ooooo`03o`000?oooooo oooo00Kooooo00?o0000ooooooooool01Oooool00ol0003oooooooooo`05ooooo`;o000000Cooooo o`000?l0003ooooo1?l000000ooooooo0000o`000002o`000003ooooool0003o000000;o000000?o ooooo`000?l000000_l000000ooooooo0000o`000004ooooo`03o`000?oooooooooo00Kooooo00?o 0000ooooooooool01Oooool00ol0003oooooooooo`05ooooo`03o`000?oooooooooo00Gooooo00?o 0000ooooooooool01Oooool00ol0003oooooooooo`05ooooo`03o`000?oooooooooo00Gooooo00?o 0000ooooooooool01_ooool00ol0003oooooooooo`05ooooo`03o`000?oooooooooo00Gooooo00?o 0000ooooooooool01Oooool00ol0003oooooooooo`05ooooo`03o`000?oooooooooo00Gooooo00?o 0000ooooooooool01_ooool001_ooooo00?o0000ooooooooool09_ooool00ol0003oooooooooo`0U ooooo`03o`000?oooooooooo027ooooo0_l000000ooooooo0000o`000002o`000003ooooool0003o 000000;o000000?oooooo`000?l000000_l000000ooooooo0000o`000002o`0001Sooooo00?o0000 ooooooooool09Oooool00ol0003oooooooooo`0Vooooo`03o`000?oooooooooo00kooooo000Koooo o`03o`000?oooooooooo0?oooooo0oooool001_ooooo00?o0000ooooooooool0ooooool3ooooo`00 6oooool00ol0003oooooooooo`3oooooo`?ooooo000Kooooo`03o`000?oooooooooo0?oooooo0ooo ool001_ooooo0_l0003oooooo`Cooooo000Kooooo`03o`000?oooooooooo0?oooooo0oooool001_o oooo00?o0000ooooooooool0ooooool3ooooo`006oooool00ol0003oooooooooo`3oooooo`?ooooo 000Kooooo`03o`000?oooooooooo0?oooooo0oooool001_ooooo00?o0000ooooooooool0ooooool3 ooooo`006oooool2o`000?oooooo1?ooool001_ooooo00?o0000ooooooooool0ooooool3ooooo`00 6oooool00ol0003oooooooooo`3oooooo`?ooooo000Kooooo`03o`000?oooooooooo0?oooooo0ooo ool001_ooooo00?o0000ooooooooool0ooooool3ooooo`006oooool00ol0003oooooooooo`3ooooo o`?ooooo000Kooooo`;o0000ooooool4ooooo`006oooool00ol0003oooooooooo`3oooooo`?ooooo 000Kooooo`03o`000?oooooooooo0?oooooo0oooool001_ooooo00?o0000ooooooooool0ooooool3 ooooo`006_ooool2o`000003ooooool0003o000000;o000000Coooooo`000?l0003ooooo1?l00000 0ooooooo0000o`000002o`000=[ooooo0_l000000ooooooo0000o`000002o`000003ooooool0003o 000000;o000000Coooooo`000?l0003ooooo0_l00006ooooo`006_ooool2o`000003ooooool0003o 000000;o000000Coooooo`000?l0003ooooo1?l000000ooooooo0000o`000002o`000003ooooool0 003o000000;o000000?oooooo`000?l000000_l000000ooooooo0000o`000030ooooo`;o000000Co ooooo`000?l0003ooooo1?l000000ooooooo0000o`000002o`000003ooooool0003o000000;o0000 00?oooooo`000?l000000_l000001?oooooo0000o`000?ooool2o`0000Kooooo000Kooooo`;o0000 4_ooool4o`000003ooooool0003o000000;o000000Coooooo`000?l0003ooooo1?l0002fooooo`;o 000000?oooooo`000?l000000_l000001?oooooo0000o`000?ooool4o`000003ooooool0003o0000 01Sooooo000Kooooo`03o`000?oooooooooo01kooooo1?l000000ooooooo0000o`000002o`000:co oooo0_l000000ooooooo0000o`000002o`000003ooooool0003o000002Gooooo000Kooooo`03o`00 0?oooooooooo02?ooooo1?l000000ooooooo0000o`000002o`000:;ooooo0_l000000ooooooo0000 o`000002o`000003ooooool0003o000002[ooooo0008ooooo`;o00001?ooool00ol0003ooooooooo o`02ooooo`?o00001Oooool00ol0003oooooooooo`0Xooooo`Co000000?oooooo`000?l00000WOoo ool4o`000003ooooool0003o000002oooooo0007ooooo`04o`000?ooooooooooo`0000Oooooo00Go 0000ooooooooooooooooo`000004ooooo`03o`000?oooooooooo02gooooo1?l0002Hooooo`;o0000 00?oooooo`000?l00000=?ooool000Oooooo00Co0000oooooooooooo00002oooool00ol0003ooooo ooooo`02ooooo`03o`000?oooooooooo02oooooo0_l000000ooooooo0000o`00002Booooo`;o0000 00?oooooo`000?l00000=oooool000Oooooo00Co0000oooooooooooo00002oooool00ol0003ooooo ooooo`02ooooo`;o0000ooooo`006oooool00ol0 003oooooooooo`16ooooo`;o0000J_ooool2o`0004kooooo000Kooooo`03o`000?oooooooooo0?oo oooo0oooool001_ooooo0_l0003oooooo`Cooooo000Kooooo`03o`000?oooooooooo0?oooooo0ooo ool001_ooooo00?o0000ooooooooool0ooooool3ooooo`006oooool00ol0003oooooooooo`3ooooo o`?ooooo000Kooooo`03o`000?oooooooooo0?oooooo0oooool001_ooooo00?o0000ooooooooool0 ooooool3ooooo`006oooool00ol0003oooooooooo`3oooooo`?ooooo000Kooooo`;o0000ooooool4 ooooo`006oooool00ol0003oooooooooo`3oooooo`?ooooo000Kooooo`03o`000?oooooooooo0?oo oooo0oooool001_ooooo00?o0000ooooooooool0ooooool3ooooo`006oooool00ol0003ooooooooo o`3oooooo`?ooooo000Kooooo`03o`000?oooooooooo0?oooooo0oooool001_ooooo0_l0003ooooo o`Cooooo000Kooooo`03o`000?oooooooooo0?oooooo0oooool001_ooooo00?o0000ooooooooool0 ooooool3ooooo`001oooool3o`0000Cooooo00?o0000ooooooooool00_ooool3o`0000Gooooo00?o 0000ooooooooool0ooooool3ooooo`002?ooool00ol0003oooooooooo`07ooooo`05o`000?oooooo ooooooooool000001?ooool00ol0003oooooooooo`3oooooo`?ooooo0008ooooo`03o`000?oooooo oooo00_ooooo00?o0000ooooooooool00_ooool00ol0003oooooooooo`3oooooo`?ooooo0008oooo o`03o`000?oooooooooo00_ooooo00?o0000ooooooooool00_ooool2o`000?oooooo1?ooool000So oooo00?o0000ooooooooool01oooool4o`0000Gooooo00?o0000ooooooooool0ooooool3ooooo`00 1oooool2o`0000Wooooo00?o0000ooooooooool01_ooool00ol0003oooooooooo`3oooooo`?ooooo 0008ooooo`03o`000?oooooooooo00Oooooo1Ol00004ooooo`03o`000?oooooooooo0?oooooo0ooo ool001_ooooo00?o0000ooooooooool0ooooool3ooooo`00oooooolQooooo`00oooooolQooooo`00 oooooolQooooo`00oooooolQooooo`00oooooolQooooo`00oooooolQooooo`00\ \>"], ImageRangeCache->{{{91.5625, 320.938}, {238.5, 97.1875}} -> {-3.48482, \ 0.768899, 0.0245798, 0.0164552}}], Cell[BoxData[ TagBox[\(\[SkeletonIndicator] Graphics \[SkeletonIndicator]\), False, Editable->False]], "Output"] }, Open ]], Cell[BoxData[ \(Consider\ states\ \ | \(\(0\)\(>\)\)\ , \ \(\(|\)\(\(\(1\)\(>\)\) \ ... \)\(\ \)\(|\)\(\(N - 1\)\(>\)\)\), \ \ where\ N = 2\^n\)], "Input", Evaluatable->False], Cell[BoxData[ \(and\ the\ transformation\)], "Input", Evaluatable->False], Cell[BoxData[ \(\*"\"\<|j> \[RightArrow] 1/\!\(\@N\) \!\(\[Sum]\+\(k = 0\)\%\(N - \ 1\)\) Exp[2 \[Pi] I j k/N] |k>\>\""\)], "Input", Evaluatable->False], Cell[BoxData[{ \(\(\(A\ state\ in\ linear\ combination\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(\(\*"\"\< \!\(\[Sum]\+\(j = 0\)\%\(N - 1\)\) \!\(x\_j\) |k> \ \[RightArrow] \!\(\[Sum]\+\(k = 0\)\%\(N - 1\)\) \!\(y\_k\) |k>\>\""\(\ \[IndentingNewLine]\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(where\ \ y\_k\ is\ the\ FT\ of\ x\_j\)}], "Input", Evaluatable->False], Cell[BoxData[ \(Lets\ write\ the\ kets\ | k > \ in\ binary\)], "Input", Evaluatable->False], Cell[BoxData[{ \(\(\(|\)\(\(0\)\(>\)\)\)\ = \(\(|\)\(\(\(...\) \(0000\)\)\(>\)\)\)\), "\ \[IndentingNewLine]", \(\(\(|\)\(\(1\)\(>\)\)\)\ \ = \(\(|\)\(\(\(...\) \(0001\)\)\(>\)\)\)\), \ "\[IndentingNewLine]", \(\(\(\(\(|\)\(\(j\)\(>\)\)\)\ \ = \ \(\(\(|\)\(j\_1\ \(j\_\(\(2\ \ ... \)\(\ \)\)\) j\_n > \ \ \ \ where\ \(\(j\_k\)\_\ \)\)\) = 0\), 1\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(\(\(so\ that\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(j = \(j\_1\ 2\^\(n - 1\) + j\_2\ 2\^\(n - 2\) + ... \) + \ j\_n\ 2\^0\)}], "Input", Evaluatable->False], Cell[BoxData[{ \(\(\(We\ also\ define\ the\ fraction\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(\(0. j\_L\ j\_\(L + 1\) ... \) . j\_\(\(m\)\(\ \ \ \)\) = \(j\_L/2\ + \ j\_\(L + 1\)/4 + ... \) + \ j\_\(\(m\)\(\ \ \ \)\)/2\^\(m - L + 1\)\)}], "Input", Evaluatable->False], Cell[BoxData[ \(Now\)], "Input", Evaluatable->False], Cell[BoxData[{ RowBox[{ "\"\<|j> \[RightArrow] 1/\!\(\@N\) \!\(\[Sum]\+\(k = 0\)\%\(N - 1\ \)\) Exp[2 \[Pi] I j k/N] |k>\>\"", "\[IndentingNewLine]"}], "\[IndentingNewLine]", RowBox[{\(\(=\)\*"\"\<1/\!\(2\^\(n/2\)\) \!\(\[Sum]\+\(k\_1\)\)...\!\(\ \[Sum]\+\(k\_n\)\) Exp[2 \[Pi] I j \!\(\[Sum]\+\(L = 1\)\%n\)\!\(k\_L\) \!\ \(2\^\(-L\)\) ] | \!\(k\_1\) \!\(k\_2\)....\!\(k\_n\)>\>\""\), "\[IndentingNewLine]"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ "=", "\"\<1/\!\(2\^\(n/2\)\) \ \!\(\[Sum]\+\(k\_1\)\)...\!\(\[Sum]\+\(k\_n\)\) \!\(\* StyleBox[\" \",\nFontSize->18]\)\!\(\* StyleBox[ UnderoverscriptBox[ StyleBox[\"\[CircleTimes]\",\nFontSize->18], \(L = 1\), \"n\"],\n\ FontSize->18]\)[Exp[2 \[Pi] I j \!\(k\_L\) \!\(2\^\(-L\)\) ]| \ \!\(k\_\(\(L\)\(\\\ \\\ \)\)\)>]\>\""}], "\[IndentingNewLine]"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"=", "\"\<1/\!\(2\^\(n/2\)\) \!\(\* StyleBox[\" \",\nFontSize->18]\)\!\(\* StyleBox[ UnderoverscriptBox[ StyleBox[\"\[CircleTimes]\",\nFontSize->18], \(L = 1\), \"n\"],\n\ FontSize->18]\)\!\(\* StyleBox[\"(\",\nFontSize->18]\)\!\(\(\[Sum]\&1\)\+\(k\_L = 0\)\) Exp[2 \ \[Pi] I j \!\(k\_L\) \!\(2\^\(-L\)\) ]| \!\(k\_\(\(L\)\(\\\ \)\)\)>] \!\(\* StyleBox[\")\",\nFontSize->24]\)\!\(\* StyleBox[\"\\\"\",\nFontSize->24]\)"}], "\[IndentingNewLine]"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"=", "\"\<1/\!\(2\^\(n/2\)\) \!\(\* StyleBox[\" \",\nFontSize->18]\)\!\(\* StyleBox[ UnderoverscriptBox[ StyleBox[\"\[CircleTimes]\",\nFontSize->18], \(L = 1\), \"n\"],\n\ FontSize->18]\)\!\(\* StyleBox[\"(\",\nFontSize->18]\)\!\(\* StyleBox[\" \",\nFontSize->18]\)| \!\(\(0\_\\\ \)\)> + \!\(2\^\(2\\\ \[Pi]\ \\\ I\\\ j\\\ 2\^\(-L\)\)\) |1>)\>\""}], "\[IndentingNewLine]"}], "\[IndentingNewLine]", "Null"}], "Input", Evaluatable->False], Cell[BoxData[{"or", "\[IndentingNewLine]", "\"\<| \!\(j\_1\)\!\(j\_2\)....\!\(j\_n\)> \[RightArrow] \ \[IndentingNewLine]\n\!\(\(\(\(\* StyleBox[\"(\",\nFontSize->18]\* StyleBox[\" \",\nFontSize->18]\(|\)\(\\\ \)\(\(0\_\\\ \) > \\\ \\\ \(+\\\ \\\ \ \\\ 2\^\(2\\\ \[Pi]\\\ I\\\ 0. j\_n\)\)\)\(\\\ \\\ \ \)\(|\)\(\(1\)\(>\)\))\)\\\ \(\* StyleBox[\"(\",\nFontSize->18]\* StyleBox[\" \",\nFontSize->18]\(|\)\(\\\ \)\(\(0\_\\\ \) > \\\ \\\ \(+\\\ \\\ \ \\\ 2\^\(2\\\ \[Pi]\\\ I\\\ 0. \(j\_\(n - 1\)\) j\_n\)\)\)\(\\\ \\\ \ \)\(|\)\(\(1\)\(>\)\))\) ... \) \(\* StyleBox[\"(\",\nFontSize->18]\* StyleBox[\" \",\nFontSize->18]\(|\)\(\\\ \)\(\(0\_\\\ \) > \\\ \\\ \(+\\\ \\\ \ \\\ 2\^\(\(2\\\ \[Pi]\\\ I\\\ 0. \(j\_1\) j\_2 ... \) jn\)\)\)\(\\\ \\\ \ \)\(|\)\(\(1\)\(>\)\))\)\)\/2\^\(n/2\)\)\>\""}], "Input", Evaluatable->False], Cell[BoxData[{ RowBox[{\("\";\), "\[IndentingNewLine]"}], "\[IndentingNewLine]", RowBox[{ ButtonBox["Figure", ButtonData:>{ URL[ "http://www.physics.unlv.edu/~bernard/phy771/QFT.pdf"], None}, ButtonStyle->"Hyperlink"], "\[IndentingNewLine]"}], "\[IndentingNewLine]", RowBox[{\("\";\), "\[IndentingNewLine]"}], "\[IndentingNewLine]", \(\*"\"\<|\!\(k\_1\)> \ = |0> + Exp[2 \[Pi] I 0.\!\(j\_1\)\!\(j\_2\)...\!\(j\_n\)]|1>\>\"";\), "\ \[IndentingNewLine]", \(\*"\"\<|\!\(k\_2\)> = |0> + Exp[2 \[Pi] I 0.\!\(j\_2\ \)...\!\(j\_n\)]|1>\>\"";\), "\[IndentingNewLine]", \(\*"\"\<|\!\(k\_\(n - \ 1\)\)> = |0> + Exp[2 \[Pi] I 0.\!\(j\_\(n - 1\)\)\!\(j\_n\)]|1>\>\"";\), "\ \[IndentingNewLine]", RowBox[{\(\*"\"\<|\!\(k\_n\)> = |0> + Exp[2 \[Pi] I \ 0.\!\(j\_n\)]|1>\>\"";\), "\[IndentingNewLine]"}], "\[IndentingNewLine]", }], "Input"], Cell[BoxData[{ RowBox[{\(where\ we\ define\), "\[IndentingNewLine]"}], "\[IndentingNewLine]", RowBox[{ RowBox[{\(R\_k\), "=", RowBox[{"(", "\[NegativeThinSpace]", GridBox[{ {"1", "0"}, {"0", \(Exp[2\ \[Pi]\ I/2\^k]\)} }], "\[NegativeThinSpace]", ")"}]}], "\[IndentingNewLine]", "\[IndentingNewLine]"}], "\[IndentingNewLine]", \(and\ we\ swap\ \ the\ \ qubits\ at\ the\ end\ \(\((not\ showm)\)\(.\)\)\)}], "Input", Evaluatable->False], Cell[BoxData[{ \(\("\";\)\ \[IndentingNewLine]\), "\[IndentingNewLine]", \(gridn[n_] := Table[Random[Integer, n], {i, 1, n}]\)}], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(gridn[10]\)], "Input"], Cell[BoxData[ \({4, 9, 9, 5, 5, 2, 7, 0, 2, 1}\)], "Output"] }, Open ]], Cell[BoxData[ \(fft[n_] := Fourier[gridn[n]]\)], "Input"], Cell[BoxData[ \(complexity[n_] := \(Timing[fft[n]]\)[\([1]\)]\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(complexity[2000000]\)], "Input"], Cell[BoxData[ \(8.120000000000005`\ Second\)], "Output"] }, Open ]] }, FrontEndVersion->"4.1 for X", ScreenRectangle->{{0, 1280}, {0, 1024}}, WindowSize->{808, 616}, WindowMargins->{{187, Automatic}, {Automatic, 111}}, PrintingPageRange->{Automatic, Automatic}, PrintingOptions->{"PaperSize"->{612, 792}, "PaperOrientation"->"Portrait", "PostScriptOutputFile":>FrontEnd`FileName[{$RootDirectory, "a", "deimos", \ "1", "bernard", "web", "phy771"}, "QuantumFT.nb.ps", CharacterEncoding -> \ "ISO8859-1"], "Magnification"->1} ] (******************************************************************* Cached data follows. If you edit this Notebook file directly, not using Mathematica, you must remove the line containing CacheID at the top of the file. The cache data will then be recreated when you save this file from within Mathematica. *******************************************************************) (*CellTagsOutline CellTagsIndex->{} *) (*CellTagsIndex CellTagsIndex->{} *) (*NotebookFileOutline Notebook[{ Cell[1705, 50, 69, 1, 27, "Input"], Cell[1777, 53, 93, 1, 43, "Input"], Cell[1873, 56, 61, 1, 27, "Input"], Cell[1937, 59, 62, 1, 27, "Input"], Cell[CellGroupData[{ Cell[2024, 64, 73, 1, 27, "Input"], Cell[2100, 67, 130, 3, 27, "Output"] }, Open ]], Cell[2245, 73, 651, 15, 193, "Input", Evaluatable->False], Cell[2899, 90, 254, 6, 90, "Input", Evaluatable->False], Cell[3156, 98, 241, 5, 43, "Input"], Cell[3400, 105, 235, 4, 75, "Input"], Cell[CellGroupData[{ Cell[3660, 113, 56, 1, 27, "Input"], Cell[3719, 116, 17813, 505, 186, 5262, 346, "GraphicsData", "PostScript", \ "Graphics"], Cell[21535, 623, 130, 3, 27, "Output"] }, Open ]], Cell[21680, 629, 182, 3, 27, "Input", Evaluatable->False], Cell[21865, 634, 79, 2, 27, "Input", Evaluatable->False], Cell[21947, 638, 165, 3, 53, "Input", Evaluatable->False], Cell[22115, 643, 418, 8, 142, "Input", Evaluatable->False], Cell[22536, 653, 98, 2, 27, "Input", Evaluatable->False], Cell[22637, 657, 627, 13, 127, "Input", Evaluatable->False], Cell[23267, 672, 318, 6, 61, "Input", Evaluatable->False], Cell[23588, 680, 58, 2, 27, "Input", Evaluatable->False], Cell[23649, 684, 1876, 43, 364, "Input", Evaluatable->False], Cell[25528, 729, 827, 15, 116, "Input", Evaluatable->False], Cell[26358, 746, 971, 18, 214, "Input"], Cell[27332, 766, 515, 11, 131, "Input", Evaluatable->False], Cell[27850, 779, 193, 3, 59, "Input"], Cell[CellGroupData[{ Cell[28068, 786, 42, 1, 27, "Input"], Cell[28113, 789, 64, 1, 27, "Output"] }, Open ]], Cell[28192, 793, 61, 1, 27, "Input"], Cell[28256, 796, 78, 1, 27, "Input"], Cell[CellGroupData[{ Cell[28359, 801, 52, 1, 27, "Input"], Cell[28414, 804, 60, 1, 27, "Output"] }, Open ]] } ] *) (******************************************************************* End of Mathematica Notebook file. *******************************************************************)