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B1DDBDC62C858B703DDE98CFF911D1B624B96D6D9FF56A2984139A10E315E8A1 F3DAE3F895ED94763821D547550E6892E9DA51276E7A8BB973F33447EF87A9C3 71F7C05D940968356910D1E72130B2772F6C972B34DAAD40A90C2AF7913456BC 0F48FF2BA29D2D3327B97DE8FC8AF3B3243816D713735C8A42F97CA94E179218 C679D102F16937F90305EC71C3BF2494823B25F609638AFE8778E2B7F880D836 30E2E607327E644A88BC1386096ECB61A82C2EB6A19401DB32BB1A7C7104467B BC37C0260F431BF90ACD09CF48CDCB7BE099B70387ECF01E50660EDE5F8A176D A4609D764B25F0058638056AABADC0093DC767E5A8EC6D0A76A72B88DB1B9033 14E5AAC2F50AABF3C80A117E3DE559251225A2BA36A3BCA5A00CD195436DDD2B 2A23E0F6DE325E6633CAC2735FAFB4C42AC2C0D53E26C4D03CE44CA711171CE3 FEB6D081BC24BB3A23A3038B80D59592BB16A3661B7935C721C039E4426B01AC 5F2C17AECB21EC3F406F376FCACBDB70976B5807E2938AD727A654D73AE524D5 247F3FD786AE09A00A522A239164F18E92DC8D8BBBBEA5F04A9709A647029429 A0DFB0B5A2307F6EA5935A785600BB0EDA45502F7299C6B6FFDA2651F41BD1C1 382636FC3CB75503B4E9A4BA56352177FF7F5DD8C2CA95FCE6101067B455E95C 5572636EF263628B9E1EC99E22797701110D966D54FF21C5D46A54F776991E69 5F3F2D4F3C0B5236D49626DACF7B69A5F1774EDFB591441811C0B048B3F6BEAF 085694599C8291F83403174E755838B32FA07921B3F4D3C8976482E938A88F1E 60D7A6BAA940225AEBC8A9ED70EE1ADE5BEBD6B8883F5782F02DFF8EC5D09DD9 1DB0E329A77AE6933AF371B05C1E59D68FD6523CC9DFB076C70BA3E593883D28 7AB87DD3119625ABB7B48A96AE29A740693D292E 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR10 %!PS-AdobeFont-1.1: CMR10 1.00B %%CreationDate: 1992 Feb 19 19:54:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 61 /equal put dup 91 /bracketleft put dup 93 /bracketright put readonly def /FontBBox{-251 -250 1009 969}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMEX10 %!PS-AdobeFont-1.1: CMEX10 1.00 %%CreationDate: 1992 Jul 23 21:22:48 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMEX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMEX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /parenleftbig put dup 1 /parenrightbig put dup 8 /braceleftbig put dup 9 /bracerightbig put dup 11 /angbracketrightbig put dup 12 /vextendsingle put dup 18 /parenleftbigg put dup 19 /parenrightbigg put dup 20 /bracketleftbigg put dup 21 /bracketrightbigg put dup 32 /parenleftBigg put dup 33 /parenrightBigg put readonly def /FontBBox{-24 -2960 1454 772}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5CF5B8CAC6A7BEB5D02276E511FFAF2AE11910 DE076F24311D94D07CACC323F360887F1EA11BDDA7927FF3325986FDB0ABDFC8 8E4B40E7988921D551EC0867EBCA44C05657F0DC913E7B3004A5F3E1337B6987 FEBC45F989C8DC6DC0AD577E903F05D0D54208A0AE7F28C734F130C133B48422 BED48639A2B74E4C08F2E710E24A99F347E0F4394CE64EACB549576E89044E52 EABE595BC964156D9D8C2BAB0F49664E951D7C1A3D1789C47F03C7051A63D5E8 DF04FAAC47351E82CAE0794AA9692C6452688A74A7A6A7AD09B8A9783C235EC1 EA2156261B8FB331827145DE315B6EC1B3D8B67B3323F761EAF4C223BB214C4C 6B062D1B281F5041D068319F4911058376D8EFBA59884BA3318C5BC95684F281 E0591BC0D1B2A4592A137FF301610019B8AC46AE6E48BC091E888E4487688350 E9AD5074EE4848271CE4ACC38D8CBC8F3DB32813DDD5B341AF9A6601281ABA38 4A978B98483A63FCC458D0E3BCE6FD830E7E09B0DB987A6B63B74638FC9F21A5 8C68479E1A85225670D79CDDE5AC0B77F5A994CA700B5F0FF1F97FC63EFDE023 8135F04A9D20C31998B12AE06676C362141AAAA395CDEF0A49E0141D335965F2 FB4198499799CECCC8AA5D255264784CD30A3E8295888EFBC2060ADDD7BAC45A EEEECDFF7A47A88E69D84C9E572616C1AC69A34B5F0D0DE8EE4EDF9F4ADE0387 680924D8D5B73EF04EAD7F45977CA8AD73D4DD45DE1966A3B8251C0386164C35 5880DD2609C80E96D1AB861C9259748E98F6711D4E241A269ED51FF328344664 3AF9F18DCE671611DB2F5D3EA77EE734D2BED623F973E6840B8DAD1E2C3C2666 DD4DD1C1CBB09EA0E12CF7E353D25984D0F6B782B023F64AC505BBE2CB2DF986 DFC4B6512FB2E9EFB2E34817A9DE05AED32B1755183E4DEF0B53D69F71CD2AE8 6738205D6FFB0AF80E3EDC6D423D27B4111D8B825697FAAD2E56747AA090747C 6D0C241778A517869D5EDE599AE0BA00938958D2CB2186CBE5436524307EE999 B6E4107AE898D2B1437CC7EF9079BF6FAE12E2A757A26D5FF00C12B921F49FA8 CF9738F65D40DBADACEAF852388FB5730AC58E5567CF1DD71F52AB9EB252D7C7 B251DDCCC7921D3C16A71950303DD42C1A6B81F66FE9DC690B6142E699684A06 19C72FCAC15BB6E67C2DB1F6213803AA3C7C638506E220F0F837103D999C6C53 0FAABF366D5011B301143B6D403D4F7DE5133805D76FD32F62C3E925699B587A 51D0C468A4F119F72FB13255A8FF981EE1870385A53CB4D55EA32494CC988E20 4D89715AEBE9E49E39F77C992847A68EF8F36DB986E6AC3BDDA6FC1951B1C317 407E1D6DC7DCE078583E72AB5F1A1303C3D2B8F07AC451FC64DC3A8D0AE4FAA9 528DA6ABC2C5D5FBF733F1E52025DA7E0FFE118099A11401351D272C75137C52 373B73AEECCADE39B85BAD092EFAE70F19B77F7E8CB8BCA7CFFF44C2108D2A3A A1C6A115E1A631EC2ED46A30062A9E5A5DCA80EA92F94B5DB0F2FEDEA81E908F 2028A7EF6D44EC13B3458E8DAE88651722F739CA546E628642C9A0FC1E8D5726 141FB2402EBB2A706E34888BE721EFBD02EDE74ACA7593C12F5C8191CD75192C 9803348E8C2BE54462210E34187A393DFD9448C1DAE713A80F96C57A46CDA5B7 47257086FDEF1FF1E16F9AF56BF5E88700CB724309EC1DB3F023FB06BF6E75F3 653CEC183A343EFFDAAE8FD69E8AB6B9D929CDD9A5D3B09054DA2D7F69F21CA2 67309C398394B0131207757872A89B5D8B5C912F35E7A1F44B4C9384A21F2E7B FE2AB93439177FC9BBE541124794A052404496CF585371BBA51D6EB7DA53B7BB F87A3D8D1B27DE4ADCCF22ACDF965370133F24D3ECC6CB09723B9A4858F282D5 1D2FD6283663F17179EBEB826B2F369ADF62330A2E4E52DF6E93392142F0C96E 71017CCBC51FEB1792F98B69D1CF8F93F5B4881C6307D3560F1366523A6A49D6 0CEEA3E89323E56C7B8FAF425E29EE6378BC055A5DABD823F80F5B7A9D4F1BF6 CCECBE16C491FA07D5B262DB5F2319C089CC51BBF450EBF1D58855064F98B780 8BB1147508C3CA453E2C3BF9AEF7D285B8CA8E0CB27C746DFFAF9ABE63B7965E 15935733ADEC51A3598CAE0BD4BEF9BF2E09A33A69E2222D481E1170BB05AD16 8DC928F46303EC9717BBF5CB14B1DCF9D91DA85B44F9CC83CCE4853A19C05AB2 3A48D59FC79E3D9B85C6C0C901E0CCF2D866B71457CC24E656922F9B654C73E5 9B0C46CABE580967151C0CF82BF8EAA663E2D3BA3269D3EBE77B897D48CA0D36 9A27B765A305EDC613607AEACAAA6E5431BF125DA2E838F9E90B9FEE5702D8C7 7B1BDFE98C1AA1B79A542F09B6878BCB42AAE823BB91F706A548191CFD5938A4 24F0C5C13730744A57F1C33375D8189526D06F6877EDB002CEFA255E5BA570A0 A29DB4E2B4979505246C8EFCB3B65B48A884564A7C15E3B78B039E9C600C2DC5 FBFEF5E2FEB16E5EA1E152B25C3C3C75572D2A650BD0743FCCBEE8AC5F5987A1 428A791D78CA1C7E28590461CA3E693C14CD0DABAA827AD8B8552B2F4C4D8A84 08D148CDB79A7C5A622CDF5E58FAAFF5E5CE38424CA8754DB9D97CA5DDA8B3D7 8E0D24CBE8D841BB38B86E6221BD094D7548D2961AD85B56658CC0DCB49030B5 479AC9D0607B6D56DFA6679C4733F8DDCB3BFA058E158D4C2085016934C294EB 043B173C72427548C2237DDFD33248BB087AB6DBBDFFA5DF88D5D731FAA42238 1AEDB2A4CB1E6102F46BA03290C6F3D5B26B36DBB3E45E73800310C03D7259E4 4F97D603D4659F73FFE9134376EE4F1895A0F711B4DBA5631AA6D132E37E67DE 0D4D53839D77DE08000093A8872F2E575FCFB8A5635CF7F9BE231D1F114A975C 54318218169C27D00BF56A7781D1BA67003A4B8BF55A5AFFAD516ED23B8E784D 5102D365F856078085278DD643CC2AFAF26BAD570C19F2A2F8850D00CD580C4F B601A9DABD3C37F2549484F04E32E3821888AEB6F56271FF4CC1C59E078A5D84 F05D804B3D1E413E1CCAC626CCBBC4C200C421EDCE4EB9AB3EA949BF0F424F23 53DFDA19F5BE689D84D482686C9396744FD0ADA9937CF16E8742D0F1265D623D D546F2736806FF47951C1539EEE202F3CE6F6C4D66EDFA84474D37F3B3230E5A 57F40184D041B9897AA7EB6A51BE3D528B3E8E26AD58B2937CD261A729759672 CA7376056409F5537EC02B6FE5ADFCE3E3D3B754A7C5877DE1B9834957E07698 8A3642BFBD03552E58B5B1D8C9F3881F96A7923B6262EAFDA85ACD6852B636B8 A5EB1483BF3D3D49B2DB20B233ECA5FFDE5D43745DDD7A11B165131670293B04 B24FEA8C95FC845FECD7CEF027D95EDB6E62B66596E48DD80EF0A27821A48630 AEF18B1D4855F7A57016E88B131FF4B88193B1CCE3F00EE15F98D751D7DB79C8 DFF3AD5F45BD87C372414F003E33C15058A2CDCE99B3E75C3063C155D3246878 92B1527ADB56B3095AEDA33B5839C38DA6F6F4959F95ECEC3135E4F33B2A61F9 799BCC126C4BABE2B6D195AA8E39272719A2B7B66A6B9AA5F417DF9BEFFC3D0D 00E910CD1BC5B228ACDB93C9 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI10 %!PS-AdobeFont-1.1: CMMI10 1.100 %%CreationDate: 1996 Jul 23 07:53:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 58 /period put dup 59 /comma put dup 61 /slash put dup 62 /greater put readonly def /FontBBox{-32 -250 1048 750}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: rsfs10 %!PS-AdobeFont-1.0: rsfs10 001.000 %%CreationDate: Sat Mar 21 18:47:14 1998 %%VMusage: 120000 150000 11 dict begin /FontInfo 14 dict dup begin /version (001.001) readonly def /Copyright (Conversion from mf curves by Metafog (c) 1995 Richard Kinch) readonly def /Notice (Copyright (c) Taco Hoekwater, 1998. All rights reserved.) readonly def /FullName (rsfs10) readonly def /FamilyName (rsfs10) readonly def /ItalicAngle -12 def /isFixedPitch false def /UnderlinePosition -100 def /UnderlineThickness 50 def /Weight (Roman) readonly def end readonly def /FontName /rsfs10 def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 72 /H put readonly def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /FontBBox {-2 -300 1240 728} readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMDUNH10 %!PS-AdobeFont-1.1: CMDUNH10 1.0 %%CreationDate: 1991 Aug 20 16:37:03 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMDUNH10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMDUNH10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 39 /quoteright put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 47 /slash put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 63 /question put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 73 /I put dup 76 /L put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put readonly def /FontBBox{-40 -528 1010 1028}readonly def currentdict end currentfile eexec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13.6 H.A end 2885 61 a 2885 61 a SDict begin [ /View [/XYZ H.V] /Dest (equation.5) cvn H.B /DEST pdfmark end 2885 61 a 377 269 a Fr(k)422 192 y Fk(\014)422 246 y(\014)453 269 y Fu(y)515 283 y Fl(k)550 195 y Fk(\013)616 269 y Fr(\000)700 192 y Fk(\014)700 246 y(\014)729 269 y Fu(y)791 283 y Fl(k)q Fg(+)p Fh(1)911 195 y Fk(\013)964 269 y Fr(k)21 b(\024)1163 208 y Fu(e)p 1131 248 112 4 v 1131 331 a Fv(4)p Fs(m)1252 269 y Ft(:)2667 b Fv(\(5\))150 531 y(Another)30 b(w)o(ay)e(to)h(see)g(this)g(is)g(that)g(applying)i (unitary)g(transformations)h(to)24 b Fs(U)2697 545 y Fl(m)2749 454 y Fk(\014)2749 509 y(\014)2779 531 y Fu(f)2836 458 y Fk(\013)2918 531 y Fv(and)g Fs(V)3123 545 y Fl(m)3175 454 y Fk(\014)3175 509 y(\014)3206 531 y Fu(f)3263 458 y Fk(\013)3344 531 y Fv(preserv)o(es)31 b(the)e(angle)150 644 y(between)c(them,)e(which)h(is)f(de\002ned)i(to)e(be)h(the)f(norm.) 150 799 y(W)-7 b(e)23 b(use)h(the)f(triangle)j(inequality)g(to)e (\002nish)f(to)h(proof.)377 1012 y Fr(k)422 934 y Fk(\014)422 989 y(\014)453 1012 y Fu(y)523 938 y Fk(\013)589 1012 y Fr(\000)673 934 y Fk(\014)673 989 y(\014)702 1012 y Fu(y)772 974 y Ff(0)794 938 y Fk(\013)847 1012 y Fr(k)84 b Fo(=)f Fr(k)1175 934 y Fk(\014)1175 989 y(\014)1205 1012 y Fu(y)1267 1026 y Fh(0)1305 938 y Fk(\013)1371 1012 y Fr(\000)1455 934 y Fk(\014)1455 989 y(\014)1484 1012 y Fu(y)1546 1026 y Fl(m)1599 938 y Fk(\013)1652 1012 y Fr(k)976 1227 y(\024)1130 1130 y Fl(m)p Ff(\000)p Fh(1)1149 1247 y Fj(\345)1144 1320 y Fl(i)p Fg(=)p Fh(0)1262 1227 y Fr(k)1307 1150 y Fk(\014)1307 1205 y(\014)1338 1227 y Fu(y)1400 1241 y Fl(i)1423 1154 y Fk(\013)1489 1227 y Fr(\000)1573 1150 y Fk(\014)1573 1205 y(\014)1602 1227 y Fu(y)1664 1241 y Fl(i)p Fg(+)p Fh(1)1772 1154 y Fk(\013)1825 1227 y Fr(k)976 1471 y(\024)g Fs(m)13 b Fr(\001)1287 1410 y Fu(e)p 1256 1451 V 1256 1534 a Fv(4)p Fs(m)1397 1471 y Fr(\024)1498 1410 y Fu(e)p 1498 1451 48 4 v 1499 1534 a Fv(4)1555 1471 y Ft(:)150 1747 y Fv(W)-7 b(e)27 b(ha)n(v)o(e)i(already)h(seen)g(that)e(quantum)i (computers)g(are)f(digital)h(computers,)h(and)e(therefore)h(a)e (reasonable)j(model)e(of)150 1859 y(computing.)66 b(But)35 b(we)g(also)h(established)j(that)c Fs(P)27 b Fr(\022)f Fs(BPP)g Fr(\022)g Fs(BQP)g Fr(\022)g Fs(PS)q(P)-8 b(A)g(C)r(E)7 b Fv(.)62 b(Since)35 b(we)g(do)g(not)h(kno)n(w)f(ho)n(w)g(to)150 1972 y(separate)24 b Fs(P)c Fv(from)i Fs(PS)q(P)-8 b(A)g(C)r(E)7 b Fv(,)19 b(it)i(follo)n(ws)h(that)h(we)e(cannot)i(unconditionally)j (pro)o(v)o(e)c(that)h(quantum)g(computers)g(are)f(more)150 2085 y(po)n(werful)28 b(than)g(classical)h(computers.)40 b(Instead)29 b(there)f(are)f(tw)o(o)f(w)o(ays)h(of)g(establishing)k (that)c(quantum)h(computers)h(are)150 2198 y(more)18 b(po)n(werful)i(than)f(classical)i(computers:)28 b(by)19 b(an)f(oracle)i(separation,)i(or)d(by)f(gi)n(ving)i(an)e(ef)n (\002cient)i(quantum)f(algorithm)150 2311 y(for)h(a)f(problem)i(belie)n (v)o(ed)h(to)e(be)f(hard)i(for)f(classical)i(algorithms.)29 b(Historically)-6 b(,)23 b(the)d(\002rst)g(demonstrations)j(that)e (quantum)150 2424 y(computers)k(are)d(more)h(po)n(werful)h(than)f (classical)i(computers)g(were)d(by)h(pro)o(ving)h(oracle)g (separations,)i(starting)e(with)f(the)150 2537 y(recursi)n(v)o(e)33 b(F)o(ourier)e(sampling)i(problem,)g(which)f(we)e(will)h(outline)h (belo)n(w)-6 b(.)51 b(W)-7 b(e)30 b(will)h(brie\003y)h(sk)o(etch)g (this)g(belo)n(w)f(and)150 2650 y(discuss)h(the)e(conjecture)j(that)d (recursi)n(v)o(e)i(F)o(ourier)f(sampling)g(does)g(not)f(lie)h(in)e(the) i(polynomial)h(hierarchy)-6 b(.)50 b(The)30 b(ne)o(xt)150 2763 y(oracle)20 b(separation,)i(Simon')-5 b(s)18 b(problem,)i(pro)o (vided)h(the)d(basic)i(template)f(that)g(Shor)f(follo)n(wed)i(in)e(his) h(quantum)g(algorithm)150 2876 y(for)24 b(f)o(actoring.)150 3015 y SDict begin H.S end 150 3015 a 150 3015 a SDict begin 13.6 H.A end 150 3015 a 150 3015 a SDict begin [ /View [/XYZ H.V] /Dest (section.8) cvn H.B /DEST pdfmark end 150 3015 a 166 x Fw(8)119 b(Quan)m(tum)41 b(Circuit)g(Implemen)m(tation)150 3389 y Fv(As)21 b(an)g(aside,)i(let)e(us)g(think)i(about)g(ho)n(w)d (one)i(might)g(implement)h(a)d(quantum)j(computer)-5 b(.)30 b(One)21 b(w)o(ay)g(to)g(do)h(it)f(w)o(ould)h(be)f(to)150 3501 y(ha)n(v)o(e)k(an)f(en)l(vironment)j(state)e(with)f Fs(n)g Fv(photons.)33 b(Then)24 b(for)g(a)g(bit)h(\003ip)e(operation,)k (the)d(qubit)i(either)f(absorbs)h(an)e(electron)150 3614 y(or)f(emits)h(one.)29 b(This)24 b(scheme)g(unfortunately)k(entangles)e (the)e(qubit)g(with)g(the)f(en)l(vironment)k(ho)n(we)n(v)o(er:)3510 3614 y SDict begin H.S end 3510 3614 a 3510 3614 a SDict begin 13.6 H.A end 3510 3614 a 3510 3614 a SDict begin [ /View [/XYZ H.V] /Dest (equation.6) cvn H.B /DEST pdfmark end 3510 3614 a 377 3799 a Fo(\()412 3722 y Fk(\014)412 3776 y(\014)443 3799 y Fv(0)488 3725 y Fk(\013)554 3799 y Fo(+)638 3722 y Fk(\014)638 3776 y(\014)667 3799 y Fv(1)712 3725 y Fk(\013)766 3799 y Fo(\))13 b Fr(\012)898 3722 y Fk(\014)898 3776 y(\014)927 3799 y Fs(n)972 3725 y Fk(\013)1026 3813 y Fl(e)1120 3799 y Fr(!)1272 3722 y Fk(\014)1272 3776 y(\014)1302 3799 y Fv(1)1347 3725 y Fk(\013)1400 3722 y(\014)1400 3776 y(\014)1431 3799 y Fs(n)g Fr(\000)g Fv(1)1618 3725 y Fk(\013)1670 3813 y Fl(e)1717 3799 y Fo(+)1801 3722 y Fk(\014)1801 3776 y(\014)1830 3799 y Fv(0)1875 3725 y Fk(\013)1929 3722 y(\014)1929 3776 y(\014)1959 3799 y Fs(n)g Fo(+)g Fv(1)2146 3725 y Fk(\013)2199 3813 y Fl(e)3944 3799 y Fv(\(6\))150 4025 y(This)31 b(seems)g(lik)o(e)h(an)f(unsurmountable)k(problem,)e(b)n (ut)f(one)f(can)g(imagine)i(that)e(the)g(en)l(vironment)j(state)e(w)o (ould)g(ha)n(v)o(e)150 4138 y(man)o(y)27 b(more)f(electrons)j(than)f (there)f(are)g(qubits.)40 b(Slightly)28 b(more)e(precisely)-6 b(,)30 b(if)c(the)h(en)l(vironment)j(is)d(in)f(state)3727 4061 y Fk(\014)3727 4116 y(\014)3757 4138 y Fu(f)3814 4065 y Fk(\013)3893 4138 y Fv(then)150 4251 y(after)d(a)f(bit)h(\003ip) f(it)g(w)o(ould)h(be)g(in)f(state)1369 4174 y Fk(\014)1369 4228 y(\014)1400 4251 y Fu(f)1457 4218 y Ff(0)1480 4177 y Fk(\013)1533 4251 y Fv(,)f(where)i Fu(f)1898 4226 y Fr(\030)1898 4255 y Fo(=)1988 4251 y Fu(f)2045 4218 y Ff(0)2068 4251 y Fv(.)k(In)22 b(f)o(act,)h(one)g(can)g(sho)n(w)g(that)g Fr(j)p Fu(f)f Fr(\000)12 b Fu(f)3300 4218 y Ff(0)3322 4251 y Fr(j)3367 4226 y(\030)3367 4255 y Fo(=)3456 4251 y Fv(1)p Ft(=)3546 4186 y Fr(p)p 3623 4186 46 4 v 3623 4251 a Fs(n)22 b Fv(as)h Fs(n)c Fr(!)g Fq(\245)p Fv(.)150 4406 y(W)-7 b(e)28 b(will)g(think)i(about)g(our)f(quantum)h (computations)h(as)e(an)g(array)g(of)g Fs(n)f Fv(qubits)i(and)f(a)g (classical)h(computer)h(as)d(a)g(con-)150 4519 y(troller)l(,)38 b(which)c(chooses)i(qubits)f(and)f(performs)i(`gates')f(on)e(them)h (sequentially)-6 b(.)64 b(It)33 b(is)h(still)g(not)g(clear)h(though)h (that)150 4631 y(quantum)25 b(mechanical)h(theory)f(will)e(hold)h(when) g(there)g(are)g(man)o(y)f(qubits.)150 4771 y SDict begin H.S end 150 4771 a 150 4771 a SDict begin 13.6 H.A end 150 4771 a 150 4771 a SDict begin [ /View [/XYZ H.V] /Dest (section.9) cvn H.B /DEST pdfmark end 150 4771 a 166 x Fw(9)119 b(Comm)m(unication)41 b(Complexit)m(y)g(of)e(Inner)i(Pro)s(duct)h(F)-10 b(unction)150 5144 y Fv(Suppose)26 b(Alice)e(has)h Fs(x)f Fv(and)h(Bob)f(has)h Fs(y)p Fv(.)31 b(Sho)n(w)23 b(that)i(it)f (requires)i(at)f(least)g Fq(W)p Fo(\()p Fs(n)p Fo(\))f Fv(communications)k(between)d(Alice)g(and)150 5257 y(Bob)e(to)h (compute)h Fs(x)13 b Fr(\001)g Fs(y)p Fv(.)28 b(\(Hint:)h(use)24 b(the)g(classical)i(algorithm)f(and)f(the)g(Hadamard)g(gate.\))p Black 150 5506 a Fc(CS)18 b(294,)i(Spring)e(2009,)i(Lecture)f(3)2915 b(5)p Black eop end %%Page: 6 6 TeXDict begin 6 5 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a Black 150 -350 a SDict begin H.S end 150 -350 a Black Black 150 -350 a SDict begin H.R end 150 -350 a 150 -350 a SDict begin [ /View [/XYZ H.V] /Dest (page.6) cvn H.B /DEST pdfmark end 150 -350 a Black 150 -143 a SDict begin H.S end 150 -143 a 150 -143 a SDict begin 13.6 H.A end 150 -143 a 150 -143 a SDict begin [ /View [/XYZ H.V] /Dest (section.10) cvn H.B /DEST pdfmark end 150 -143 a 116 x Fw(10)119 b(Simon's)41 b(Algorithm)150 181 y Fv(Recall)19 b(that)g(our)h(basic)f(primiti)n(v)o (e)h(for)f(designing)i(quantum)g(algorithms)f(is)f(F)o(ourier)g (sampling:)29 b(prepare)20 b(some)f(quantum)150 294 y(state)342 217 y Fk(\014)342 271 y(\014)373 294 y Fu(y)443 220 y Fk(\013)518 294 y Fo(=)610 301 y Fq(\345)675 313 y Fl(x)719 294 y Fu(a)776 308 y Fl(x)810 217 y Fk(\014)810 271 y(\014)840 294 y Fs(x)880 220 y Fk(\013)960 294 y Fv(on)27 b Fs(n)f Fv(qubits;)k(perform)e(a)e(Hadamard)h(transform,)i(resulting)g(in)e (the)g(superposition)3708 301 y Fq(\345)3773 313 y Fl(x)3817 294 y Fu(b)3867 308 y Fl(x)3901 217 y Fk(\014)3901 271 y(\014)3931 294 y Fs(x)3971 220 y Fk(\013)4025 294 y Fv(;)150 407 y(no)n(w)i(measure)h(to)f(sample)h Fs(x)f Fv(with)g(probability)k Fr(j)p Fu(b)1811 421 y Fl(x)1845 407 y Fr(j)1870 374 y Fh(2)1907 407 y Fv(.)45 b(The)29 b(point)h(is)f(that)h(classically)i(it)d(is)g(dif)n(\002cult)h(to)f (simulate)i(the)150 520 y(ef)n(fects)36 b(of)e(the)h(quantum)h (interference,)41 b(and)35 b(therefore)i(to)d(determine)j(for)e(which)g (strings)h Fs(x)e Fv(there)i(is)e(constructi)n(v)o(e)150 633 y(interference)27 b(and)d(are)g(therefore)h(output)g(with)f(high)g (probability)-6 b(.)150 787 y(W)f(e)23 b(no)n(w)g(consider)i(a)e(ne)n (w)g(w)o(ay)h(of)f(setting)i(up)f(the)g(initial)h(superposition)2566 710 y Fk(\014)2566 764 y(\014)2597 787 y Fu(y)2667 713 y Fk(\013)2740 787 y Fo(=)2830 794 y Fq(\345)2895 807 y Fl(x)2939 787 y Fu(a)2996 801 y Fl(x)3030 710 y Fk(\014)3030 764 y(\014)3061 787 y Fs(x)3101 713 y Fk(\013)3154 787 y Fv(.)150 883 y SDict begin H.S end 150 883 a 150 883 a SDict begin 13.6 H.A end 150 883 a 150 883 a SDict begin [ /View [/XYZ H.V] /Dest (subsection.10.1) cvn H.B /DEST pdfmark end 150 883 a 147 x Fx(10.1)99 b(Setting)33 b(up)h(a)f(random)g (pre-image)h(state)150 1221 y Fv(Suppose)25 b(we')-5 b(re)24 b(gi)n(v)o(en)g(a)f(classical)j(circuit)f(for)f(a)f Fs(k)14 b Fr(\000)f Fv(1)23 b(function)39 b Fs(f)34 b Fv(:)20 b Fr(f)q Fv(0)p Ft(;)10 b Fv(1)p Fr(g)2634 1179 y Fl(n)2692 1221 y Fr(!)20 b(f)p Fv(0)p Ft(;)10 b Fv(1)p Fr(g)3020 1179 y Fl(n)3057 1221 y Fv(.)150 1375 y(W)-7 b(e)20 b(will)g(sho)n(w)g(ho)n(w)g(to)h(set)f(up)h(the)g(quantum)h (state)1810 1298 y Fk(\014)1810 1352 y(\014)1840 1375 y Fu(f)1897 1301 y Fk(\013)1968 1375 y Fo(=)17 b Fv(1)p Ft(=)2146 1298 y Fr(p)p 2223 1298 42 4 v 2223 1375 a Fs(k)2275 1382 y Fq(\345)2340 1395 y Fl(x)p Fh(:)10 b Fl(f)g Fg(\()p Fl(x)p Fg(\)=)p Fl(a)2606 1298 y Fk(\014)2606 1352 y(\014)2636 1375 y Fs(x)2676 1301 y Fk(\013)2730 1375 y Fv(.)27 b(Here)20 b Fs(a)g Fv(is)h(uniformly)h(random)g(among) 150 1488 y(all)i Fs(a)f Fv(in)g(the)h(image)g(of)37 b Fs(f)13 b Fv(.)150 1643 y(The)40 b(algorithm)i(uses)f(tw)o(o)f(re)o (gisters,)47 b(both)41 b(with)f Fs(n)g Fv(qubits.)121 b(The)40 b(re)o(gisters)i(are)f(initialized)i(to)d(the)h(basis)g(state) 150 1755 y Fr(j)p Fv(0)10 b Fr(\001)g(\001)g(\001)i Fv(0)p Fr(i)e(j)q Fv(0)g Fr(\001)g(\001)g(\001)h Fv(0)p Fr(i)q Fv(.)94 b(W)-7 b(e)34 b(then)h(perform)h(the)e(Hadamard)i(transform)g Fs(H)2499 1722 y Ff(\012)p Fl(n)2620 1755 y Fv(on)f(the)g(\002rst)f(re) o(gister)l(,)39 b(producing)e(the)e(su-)150 1868 y(perposition)439 2030 y(1)p 387 2071 150 4 v 387 2157 a(2)432 2131 y Fl(n)p Fm(=)p Fh(2)627 2111 y Fj(\345)546 2192 y Fl(x)p Ff(2f)r Fh(0)p Fm(;)p Fh(1)p Ff(g)772 2162 y Fd(n)811 2092 y Fr(j)p Fs(x)p Fr(i)11 b(j)p Fv(0)f Fr(\001)g(\001)g(\001)i Fv(0)p Fr(i)e Ft(:)150 2367 y Fv(Then,)23 b(we)g(compute)39 b Fs(f)23 b Fo(\()p Fs(x)p Fo(\))h Fv(through)i(the)d(oracle)d Fs(C)1791 2381 y Fl(f)1846 2367 y Fv(and)k(store)g(the)g(result)h(in)e (the)h(second)h(re)o(gister)l(,)h(obtaining)g(the)e(state)439 2537 y(1)p 387 2577 V 387 2664 a(2)432 2638 y Fl(n)p Fm(=)p Fh(2)627 2618 y Fj(\345)546 2699 y Fl(x)p Ff(2f)r Fh(0)p Fm(;)p Fh(1)p Ff(g)772 2669 y Fd(n)811 2598 y Fr(j)p Fs(x)p Fr(i)11 b(j)j Fs(f)23 b Fo(\()q Fs(x)p Fo(\))q Fr(i)10 b Ft(:)150 2868 y Fv(The)29 b(second)i(re)o(gister)g (is)e(not)h(modi\002ed)g(after)g(this)g(step.)47 b(Thus)29 b(we)g(may)g(in)l(v)n(ok)o(e)j(the)d(principle)j(of)e(safe)g(storage)h (and)150 2981 y(assume)24 b(that)g(the)g(second)h(re)o(gister)g(is)f (measured)h(at)e(this)h(point.)150 3135 y(Let)i Fs(a)h Fv(be)g(the)g(result)h(of)f(measuring)i(of)e(the)g(second)h(re)o (gister)-5 b(.)41 b(Then)27 b Fs(a)f Fv(is)h(a)f(random)i(element)g(in) f(the)g(range)h(of)41 b Fs(f)13 b Fv(,)27 b(and)150 3248 y(according)g(to)d(rules)h(of)f(partial)i(measurement,)g(the)f(state)g (of)f(the)g(\002rst)g(re)o(gister)i(is)e(a)g(superposition)k(o)o(v)o (er)c(e)o(xactly)i(those)150 3361 y(v)n(alues)f(of)e Fs(x)h Fv(that)g(are)f(consistent)k(with)c(those)i(contents)g(for)f (the)g(second)h(re)o(gister)-5 b(.)30 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