{VERSION 6 0 "IBM INTEL NT" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Input" 2 19 "" 0 1 255 0 0 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 1 16 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 1 12 0 0 0 0 0 2 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 261 "" 1 12 0 0 0 0 0 2 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Tim es" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 256 43 "Quantum Computation and Q uantum Information" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 73 "Michael A. Nielsen, Isaac L. Chuang (Cambridge, 2000) ISB N 0 521 63235 8." }}{PARA 0 "" 0 "" {TEXT -1 128 "Maple implementation by Marko Horbatsch (Computational Physics using Maple; www.yorku.ca/m arko) August 2002, modified Oct. 2003." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 208 "We implement the matrix representati on for a few quantum logic gates, such as the conditional NOT, derive \+ the swap, the Hadamard gate, the formation of entangled states and wor k through quantum teleportation." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "with(LinearAlgebra):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 129 "We intr oduce a basis for a 2-qubit; the first is the control qubit, the secon d is the target qubit; they form an entangled state." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "s00:=Vector([1,0,0,0]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$s00G-%'RTABLEG6%\")7B$=#-%'MATRIXG6#7&7#\"\"\"7# \"\"!F/F/&%'VectorG6#%'columnG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "s01:=Vector([0,1,0,0]): s10:=Vector([0,0,1,0]): s11:=Vector([0 ,0,0,1]):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 130 "The conditional NOT gate of quantum computing leaves the control qubit as is, and adds th e target and control bit in binary logic." }}{PARA 0 "" 0 "" {TEXT -1 56 "The following picture is taken from the text (Fig. 1.6):" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {OLE 1 26114 1 "[xm]Br=WfoRr B:::wk;nyyI;G:;:j::>:B>N:F:nyyyyy]::yyyyyy:::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::fyyyyyI@nYnyyyYE:G:I:K:M:O: Q:S:UJ:n;v;;JBB:]:_:a:c:e:g:i:k:m:o:q:s:u:w:y:;[:F>N>V>^>f>n>v>>?F?N?V ?^?f?n?v?>`:B:wAwAyA:::::::::::::::::::::::::::::::::::::::::::::::::: :NDYmq^H;C:ELq^H_mvJ::::::::gj:;H:<:T><::[u@mEgmW_AX:AB:> Z@Wlj^HMMufF;J:::::::N=?R:yyyyyy:>:<::::::C:SSPB:F:AlqfG [maNFO=;::::::::_J;vyyuy:>:<::::::Jb:wAwAC:E:G:IJ::;`:Z@?;> d[WfeGfagFK::::^xsNpk[::JgaVO=RouSGTygqYUhuyDXoxi=he@Fr^vxw^tUVcRvvwhk r:e]:N:s :;j<>:Qb:B:E:Sb:_k[^=VyD:s:qQBvZ:B:;JXE::Uk :V:Z=OaVgg[?h fgfr^eKWbgncCgbxG`VGd=ofk:B:HJ:t;HNW:Zg:Jm:Rnk\\yAuyAV[bLGTLGl:Dj:@JP@ JPry=I>CWKkUTG]GgCO;Kj=@JCV:;j:Z;N=E:Z:VZZHZ:V:FZyyy:>ZZPZ:^Z:Z:>:KHbAJZCFZyAuy]D;B:KL^>ZyiWSRJCuUve=gC?ryWQA@Nlx@;BS;F:\\j>\\J@ `j>ljBhJ?hjBPkAlJ@xjD\\kCHkDHkC\\kBlJ@lJ@R>SB=l::::j:pj?hj@tj@Z:B:r:;R:?B;Eb;IB:FZ:NZ=^Z?n:JBFZ>FZ =NZ>NZ?vZ@>[?VZFV`;b:]r:CRB_RFERJ]TRabB]UZ_c;cR^=Wf?tBCS?_Tj?uFccV?VnA t;Ec^aDv=YvAuVec??gfAfCCt^cDSEdjAGWgs?_XO?YS_yfCvjaXWAyvGtCcFcEFgeFkcH scYceWkeXseiOGeWGf_GggGhoGiwisKitSiu[ivciwkixsiY>DJ;@J;LJ_ZJItdP[R?qmQ\\Q@oonmAW^\\ymTQdgFglnb[h^e>iuapGoqHomt^ohInQI`j^_ splopwhg^=v_hnREDOyFOgsogwWmhuGximxJUuY]W:yI:mTVQElgUGkfwggpQV[;IU?Sey fR]eqotyDWyxROExfdyAXOxyLqQLOqS:qy:@lU\\T>TyouyderqmtdIOAyL_MVUUUTUPYD X[YV?etZyqMqPeIvgiuwxQ^dVoQYihttukWYsomO`AMnqo@QO;eyphPhPN`qMWdyY\\neq pspo?MwXhyYxQyaJLpUBPURaJbAkJ`MvFihxhuVwWWZI>_\\wdmw_oGeyNfohxIhchqrhw beXfqGqgXhcWpQFyqXyyAZEwbtycKIumXwG>iri`_oayhdO`gwQn`Yswn\\TpiKqf=W_uw iRooiXghifivuXwacxgciiYFcg^dfyl;Pnj?v@nqeNZyhogNppXupqhThyV_eeWchn[wqa ]?_u`]]Wyy>e@iyqW]oymKyi@Pptxa=wwyPtmPxmnyJq_Aifgvro`tLVsGOltasSWncXnS YyYxx>alXq\\IO]AIjHge<>cL?jNH[Y`ZJXrrgupx^tXe^ypFyYEBD=BDMuCEXCysy[Xw; 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Maple's " }{TEXT 19 5 "evalb" }{TEXT -1 119 " is unreliable when it performs this compa rison, the results change between versions. Thus we define our own pro cedure " }{TEXT 19 5 "Evalb" }{TEXT -1 32 ", which compares equations \+ with " }{TEXT 19 6 "Vector" }{TEXT -1 46 " objects on both sides compo nent by component:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "Evalb :=proc(VectorEqn::`=`) local L,R,dL,dR,i,res; L:=lhs(VectorEqn): R:=rh s(VectorEqn):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "if type(L,Vector) and type(R,Vector) then dL:=op(1,L): dR:=op(1,R): if dR<>dL then RETU RN(false) fi: " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 103 "for i from 1 to \+ dL do: res:=simplify(R[i]-L[i]): if res <> 0 then RETURN(false): fi: o d: RETURN(true): " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "else evalb(Vec torEqn): fi: end:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "Now apply it to our case:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "Evalb(U_cn ot . s00 = s00);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "Evalb(U_cnot . s01 = s01);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 26 "Evalb(U_cnot . s10 = s11);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "E valb(U_cnot . s11 = s10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 211 "We see that: when the control \+ qubit (qubit 1) is zero, the target qubit is left unchanged, and when \+ qubit 1 has value 1, then qubit 2 is flipped. This represents the cond itional NOT. We have verified eq. (1.18)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 141 "Now one should verify using binar y arithmetic that given the notation |A> and |B> for the control and t arget qubits that |B> goes into |A+B>." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "true and true, true and false, false and true, false \+ and false;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%%trueG%&falseGF$F$" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "true or true, true or false, false or true, false or false;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%%t rueGF#F#%&falseG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "This does not implement 0 + 0 = 0, 0 + 1 = 1 + 0 = 1, 1 + 1 = 0 !" }}{PARA 0 "" 0 "" {TEXT -1 61 "But the XOR between boolean states does do the desired thing:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "true xor true," }{TEXT -1 1 " " }{MPLTEXT 1 0 15 "true xor false," }{TEXT -1 1 " " } {MPLTEXT 1 0 15 "false xor true," }{TEXT -1 1 " " }{MPLTEXT 1 0 16 "fa lse xor false;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%&falseG%%trueGF$F# " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 261 "Note: the linearity of matri x multiplication will guarantee that we only need to know the action f or the transformation matrix on basis states. The expansion coefficien ts, which measure how much probability amplitude we have in either sta te factor out linearly." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 109 "Now let us understand how three conditional NOTs ca n form a swap, as explained in eq. (1.20), and figure 1.7." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {OLE 1 17410 1 "[xm]Br=WfoRrB: ::wk;nyyI;G:;:j::>:B>N:F:nyyyyy]::yyyyyy:::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::fyyyyyA>;F;N;V ;^;f;;JAjA>:[Z:F=F=N=V=^=f=n=v=nYnYvY::::::::::::::::::: ::::::::::::::::::::::::::::::::::::::::::::_lqvGcMJ:::::: :JEf:yyyxI^:^;AJ:>:;H:<:T><::;edWGgmW_AX:AB:>Z@Wlj^HMMuf F;J:::::::N=?R:yyyyyy:>:<::::::C:[tGB:F:AlqfG[maNFO=;::::::::_J;vyyuy: >:<::::::Jb:wAwAC:E:G J:v::;`:Z@?;>d[WfeGfagFK::::^xsNpk[::: :::::::j:b:B::::::::::vYxy;J:vZAj\\P:Oa:F:MZ=fH:EJ:V[ l;B:@:s :qQBvZ:B:;JXE::Uk:V: Z;N=E:Z: VZZHZ:V:FZyyy:>ZZPZ:^:B:;R:;DbAJCFZyAuy]D;B:KL^>ZyiWSRJCuUve= gC?ryWQA@Nlx@;BS;F:\\j>\\J@`j>ljBhJ?hjBPkAlJ@xjD\\kCHkDHkC \\kBlJ@lJ@R>SB=l::::j:pj?hj@tj@Z:B:r:;R:?B; Eb;IB:FZ:NZ=^Z?n:JBFZ>FZ=NZ>NZ?vZ@>[?VZFV`;b:]r:CRB_RFE RJ]TRabB]UZ_c;cR^=Wf?tBCS?_Tj?uFccV?VnAt;Ec^aDv=YvAuVec??gfAfCCt^cDSEd jAGWgs?_XO?YS_yfCvjaXWAyvGtCcFcEFgeFkcHscYceWkeXseiOGeWGf_GggGhoGiwisK itSiu[ivciwkixsiY>DJ;@J;LJTYIdyuO_f^cvUsuKmhUYfxyhyeVKYGwUFqkdtWvbQeryEtm SuCTBUC>aty[XwCdpIcDyWpSwy`uUxQjyQwanLPMY=ugqnhHkTTwXQUTEWuiV>]PCMMpPS JiJ^LJotJJuroYlOMQhirNyQMiYshjNEuHArdAu[mqohoHiooXyyPRyYNbUOOEvU=wspkB qP]pVler_qmLQx]MYqLyyHmu`ynaLq@spEVrDL>xuMxnx]jXqme`bAIwSghUie?otnPdTIjEiuSwucQaAIqFqfAyedavtysuybcpyPxuLiuxi\\cVuKI_ Bv]aWpxivUwy:FjyUtCcPkhsisTwvtYruosCwyZkdZmFtawkGteUyAKUwSyaaDoqhfKVaw hcuxqIUwguyXwXxNvQNyHOLQTnTs>yUaX[iDumuAiiHyqs]pyounOgeghfOuuieuonq_qnf]yOvSX dy>w^xnyIuyiwEN_J?gisUaETrFWCF;BH?TLUD;E;;JBZ:FZ :NZ;^?ZrNbCC;G[:>:W:;:rI;^r:U:;J<>:wBH?TLF=>:C:w:HjsK:F[:>:;:@ZrNBLJ?J :F:;BHrC[J:<:Oj:B:;:?BHr:eI:>:KD:YcSGm::GB:>:=jTHxyywY::Z=>W;J:xIyA::r:J:xIyA::r:J :xIyA::r:BHB;KJ:\\:B:=B:DJZb:B:\\=n=>:C:;BHJGJ:HjZ::]L:@JZB: GjbVBGmR>Hi[:FZ:B:;:::=:;:R:;\\:FD;J:::@J:B:::B>>Z;Z:JqPQpl@DJ:b;?uUee UCuVYdROuRZ@VhfGg]gfmodf_cm[;vc\\OGJ:B:PjslpnB:N\\:>:IrVSUTFc>_GJ:<: XjmDqqPQr`N\\l@J:J=xPs`pm`PqPPqPqpJ@PnclpnrM_UTB:f[[ghnwgAgfhgflGfngf^ [:^ds?h_gfhgHbZ@NFt@:@:<:_tRZ>Z:^di;Z::S<:<:NB:RN B:_tS:FZ:K:HjtDQpLo\\\\Odj:<:;Zh[NgBj:J:RMauSEEOEDPj:J:nZ[_gn_d[wfNgbR ;=:;JAHpllPpffr_hCWhB_dG_c\\wgi_g;ZBVf__GJdPNfZ;Z:RZ:r;CUSGUReEUcM:F[pgfl_h;_gckl:x:JePPs:<:I::hJkXp[xPpxPsLo v:=R:;BHJD>:;J>@J:CF>Z?>Gkj:xYf?V[bLnBDj:<:KL:xyp_[;_FvEU<\\;SJ@R>B=::::=:ZH:WJDn;WB?]:SjB@;l:]B=^;:::ZyAr:B< ?j:R:J:\\:>:PZ;R:>Z:B:r:;R:Lj\\:FZ>r:L:TJ>VZF Z@FDA:=YvA=bCHc^gbRjMuD:RVRurg:iCw^XtD[XU:ryXOCKxXnfcxRKfKkc>JJuBQOhiBrZw:FWhg?otnPDhfS;:CauUdjhS DF?X@vy:FJ`HJIsuo[=<:ZNM>;ZQ=w=:DUf[n?FxrP[LS?O=fBeYXl;tH;bs>:t]AtQ_lj RojmjKZldHridqqw=;;jdj:@J:r=;RJL`i?Z\\?Z_ ?ZZ>ZJCT;rULaisV;BTBV>ZMRJZ\\;:LAHABV<\\q>Zh>ZjJ:@J:\\=n:>:IB:?B:cb_hn?htv_ingmo_[_fiKQHpqppQ\\qjit>i[?hggfnGfZ>ig_ghWht>iwv^[_fiKWtpsd mpPPtv_a>^rOarKthpXXLfpNbGc=XquXQQXqStlqDqmxlR`mV`mQ^vNh^cHCTSa uR_UT[EWMuUWEB=eRYUWpmMPqt`PldMUDmTXMTDMmDmPDPUi[SMUTIICT:Zi;BX=mQZgggiv^xBB;CIkUR[UMdM\\xp kPqpPPqLq^LNY@PlJWLpqHpnLPW^Z>^Z::::::::::::::::N[Z>>;;::: ::::::::::::bJ;B:=ry^j:>[@nZ@n:Kb;GBUR=W R=UB>]B=SJ@R>B=::::=R]B=^;]R>B=F\\FJ@:: ::xYZ=ZBRN=BJAR:?C:?j:F\\;:;B?yIk=bbr;=R:=R::B;Er:?b;=B:IB:FZ:NZ=J=XJ>JB>Z?F^LF`;Gd@V \\[O^kGjNN`KHnAf\\kPb]jbAxSTKkTl;aJ?\\kFLRDQN]LTTUocQONdKsIxsMVdMXtQptPrDQtTQvdcSge[JeUXuY=IuWX sMwrQuwkCLkDTkE\\kFdkGlkHtkYDmSLmTTmU\\mVdmWlmXtmiDocLodToe\\ofdogloht oY>NZ;r:ER;CR;GB;Kr:ARU=B:?R>AB;F@ERNaRR?c;]UZ]fF=Wf[I?=Xn=yB?T?_dR_Uv aCCcsJ?f?_vNEcbAef?x;aenecrCtZiBOaFdW_UXm]OgXndxodPsYHsIamE`mqMxQiSXPw=`sixn@TCinOMkB]kJyoSe mTlWbInFTX[xmqISdYxwEQw\\QNpTY@tJQxUqNyuOFtKmUxiynUipOaUe\\LkdOXtTZDJL =J=hWEymlTNDxY:\\rtxRwMstynvqkLyiXybDVeS_thhpmi^AiloH_GgjvPhT?r^WtCnkC ou^OlmWeyAZIWl^X_N^cU^tIhy]^wtwuv``chmq`mnWkShssx\\P?nrVkOaqn@qpxjMP_t X^E_aC_]LOaCy]a`]GY\\Eisw_ayNv:_\\^^xOf`i`e@@eeyxmpueolIivgOcZwtbn_Zf_ `@l=NuUwoa@^u?xIhsAw_:PqjwioYtYNneQqghpvpxCfvgPdb_gaAi[oahPmSW^Zhl[NfT yj?AwCfa?NvIAx@_mYptaqkWy_\\yqH`gyHkbHdnixvI`;qiGybSqoaIwBYysYlPO[?aic PoPXZ[Owc_pihsiossob]YxIvsivox>ptipNydeI_xNju`oBGbxYvdartqiJXyxVlaNyrI tWWqxYexxsXqqoYu_Im?GZcpwt_ylI`]ovGPguW]coovIochn>Il]ncbptVVZlPjvabQHo i>\\AIv`AkyAZ^AvYHxXryDOYituaxyoxWgkXVvlyeBHy>XyHXyiAtIxmWAmihetAlxOwn Wyj`jsWlKwanYsdyqjQeygyQpfX^fyxanisdwqnivqhyxVfA?qvQkxV^`Y\\\\qnoxojXs qpwy:Ncxyvwy] Gn]]ePayAiceERcsCJIWxiikCWSyT_QUJODYqyoKIluh[yg;yVTeB_odM_WtORv_RMUEC_ VFGCe_xH]wSiuCaFRyVvoDWuy=Owy;bnqrYsDGqu]Kxb?iIMut?V;=TdqRZSFOOUVeC?CB OoRw?xM]ISud=]fdyI:LkIIvU\\YUIZugglY>]H][WiqtxEUhkdCiG;[fBmG]EHB=WcOdGqiyWh<;gcuhhAeLyI:yInYY paWS=BMedeuDIWRQyepQXymTd=fPYwMsEimHfOW[EXgLxApptDKhhO_QtLTrdtJ@YTO?pd NswOxi^vGV^DOtnYoaXtf@noHgof]Gg][ikKGm[^thFf?@]bxoRNfOokwny>Fsx?\\s?b; 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IEkhaOvxaixsFi[i?SeXQwucwuayISWsYWpoyYWFXYbG[rewwmety;BPwybWV@;IEdmOyx YUxvyQhHUdmqA=yaAJmMvOGrYyvO`uywm@@iMn`dVyuA_qGyaAjmHfQwuEgtx@ipIwXPZG YjwxexokyAZvfu@@iEn`OfiwikyVgbys;nVcyOwXtgyMoIpYWWmIa[iASU:ytYEe=mFrSk dXynMvgqxxIltYx:IUkxTYYWuuuvtqFZ;^[:>gTW]q?oSW]q? dMWdn?Z:>Z:>kKy\\:FZ:>:SXniB:;B:;::::=B::kcPqJDdoGXQ:doGXQnaoG^tSW]q?r RW]qC:<::=:SYniJXIJ:>xHx\\:>kIx\\:>mHx\\:fb^obf?Z:>Z:^rTW=[gPqJ:d:Nh:fFaE:[C:[C:>Z::::::::kB:?R:AB:;j:jyyyyyY:Z:B:F:Y<>Z:>:::::::::J?:<::::::wqyyI<:::::::::: ::jysy:>:<:::::::::::::::::::vYxI:;Z:::::::::1:" }{TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 324 "The prescription is to apply the conditi onal NOT gate twice as done above, but sandwiched in-between is one ac tion of the gate reversed (where the target quibit controls the contro l qubit). First we have to find the matrix that will do that! Let's ma ke a guess, and transpose the two non-zero 2-by-2 blocks that form U_c not." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "U_rcnot:=Matrix([[0 ,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(U_rcnotG-%'RTABLEG6%\")_qyA-%'MATRIXG6#7&7&\"\"!\"\"\"F.F.7&F /F.F.F.7&F.F.F/F.7&F.F.F.F/%'MatrixG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "Let us verify the action of this transformation matrix:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "U_rcnot . s00, U_rcnot . s01 , U_rcnot . s10, U_rcnot . s11;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&-%' RTABLEG6%\")GJ\"G#-%'MATRIXG6#7&7#\"\"!7#\"\"\"F+F+&%'VectorG6#%'colum nG-F$6%\")k#QG#-F(6#7&F-F+F+F+F/-F$6%\")W#pG#-F(6#7&F+F+F-F+F/-F$6%\") GX*G#-F(6#7&F+F+F+F-F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 111 "The qu estion is: did we flip qubit 1 depending on qubit 2 being 0 or 1, and \+ leave qubit 2 alone? Let's compare:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "s00, s01, s10, s11;" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6&-%'RTABLEG6%\")7B$=#-%'MATRIXG6#7&7#\"\"\"7#\"\"!F-F-&%'VectorG6#%'c olumnG-F$6%\")GJ,A-F(6#7&F-F+F-F-F/-F$6%\")sC$=#-F(6#7&F-F-F+F-F/-F$6% \")C;+A-F(6#7&F-F-F-F+F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "What \+ did we generate above?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "s 01,s00,s10,s11;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&-%'RTABLEG6%\")GJ,A -%'MATRIXG6#7&7#\"\"!7#\"\"\"F+F+&%'VectorG6#%'columnG-F$6%\")7B$=#-F( 6#7&F-F+F+F+F/-F$6%\")sC$=#-F(6#7&F+F+F-F+F/-F$6%\")C;+A-F(6#7&F+F+F+F -F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 155 "We kept qubit 1, while fl ipping qubit 2 when qubit 1 was |0>, rather than |1>. So this matrix \+ was not what we wanted! How should we find the right matrix?" }}{PARA 0 "" 0 "" {TEXT -1 21 "We want the outcomes:" }}{PARA 0 "" 0 "" {TEXT -1 57 "|00> -> |00> , |10> -> |10> , |01> -> |11> , |11> -> |01>" }} {PARA 0 "" 0 "" {TEXT -1 219 "We could set up a system of equations to determine the matrix (cannon approach to swat a fly). It makes more s ense to write the source and target spinors side-by-side, and to deter mine the required coupling coefficient." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "s00,s00;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$-%'RTABLEG 6%\")7B$=#-%'MATRIXG6#7&7#\"\"\"7#\"\"!F-F-&%'VectorG6#%'columnGF#" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 69 "The first row better have a 1 in \+ its first entry (and be [1,0,0,0] ?)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "s10,s10;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$-%'RTABLEG6%\")sC$=#-%'MATRIXG6#7&7#\"\"!F+7#\"\"\"F +&%'VectorG6#%'columnGF#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 75 "The t hird row better be [0,0,1,0], or at least have a 1 in the third entry. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "s01,s11;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$-%'RTABLEG6%\")GJ,A-%'MATRIXG6#7&7#\"\"!7#\"\"\"F+ F+&%'VectorG6#%'columnG-F$6%\")C;+A-F(6#7&F+F+F+F-F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 206 "The second row (producing the action for |01> ) should have a zero in the second column. The fourth row has to produ ce the 1 in the fourth column, while it acts on s01. So it needs a 1 i n the second column." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "s11, s01;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$-%'RTABLEG6%\")C;+A-%'MATRIXG6 #7&7#\"\"!F+F+7#\"\"\"&%'VectorG6#%'columnG-F$6%\")GJ,A-F(6#7&F+F-F+F+ F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "U_try:=Matrix([[1,0,0 ,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%&U_tryG-%'RTABLEG6%\")WE,B-%'MATRIXG6#7&7&\"\"\"\"\"!F/F/7&F/F/F/F .7&F/F/F.F/7&F/F.F/F/%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "Evalb(U_try . s00 = s00);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#% %trueG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "Evalb(U_try . s01 = s11);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "Evalb(U_try . s10 = s10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "Evalb(U_try . s11 = s01);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%tru eG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 91 "OK, so apparently this does what we want! Now let's compute the action of swapping circuit:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "U_swap:=U_cnot . U_try . U_c not;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'U_swapG-%'RTABLEG6%\")Sx>B- %'MATRIXG6#7&7&\"\"\"\"\"!F/F/7&F/F/F.F/7&F/F.F/F/7&F/F/F/F.%'MatrixG " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 82 "Indeed, this is the matrix li sted on page XXV for the swap! What does the swap do?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "U_swap . s00,s00;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$-%'RTABLEG6%\")GaAB-%'MATRIXG6#7&7#\"\"\"7#\"\"!F-F- &%'VectorG6#%'columnG-F$6%\")7B$=#F'F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "U_swap . s01, s10;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 $-%'RTABLEG6%\")wlEB-%'MATRIXG6#7&7#\"\"!F+7#\"\"\"F+&%'VectorG6#%'col umnG-F$6%\")sC$=#F'F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "U_ swap . s10, s01;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$-%'RTABLEG6%\")'>1 L#-%'MATRIXG6#7&7#\"\"!7#\"\"\"F+F+&%'VectorG6#%'columnG-F$6%\")GJ,AF' F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "U_swap . s11, s11;" } }{PARA 11 "" 1 "" {XPPMATH 20 "6$-%'RTABLEG6%\")sdMB-%'MATRIXG6#7&7#\" \"!F+F+7#\"\"\"&%'VectorG6#%'columnG-F$6%\")C;+AF'F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 132 "Note that swapping is not the same as invertin g both qubits. This would agree on the 2nd and 3rd case, but not the f irst and fourth." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 257 10 "Exercise1:" }}{PARA 0 "" 0 "" {TEXT -1 120 "Find the mat rix which inverts both qubits. Construct this matrix carefully by comp aring source and target qubit spinors." }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 258 22 "Solution to Exercise 1" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "s00,s11;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$-%'RTABLEG 6%\"'#HC&-%'MATRIXG6#7&7#\"\"\"7#\"\"!F-F-&%'VectorG6#%'columnG-F$6%\" )'*Rx?-F(6#7&F-F-F-F+F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "first \+ column in fourth row needs a 1." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "s10,s01;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$-%'RTABLEG6%\")C\") f?-%'MATRIXG6#7&7#\"\"!F+7#\"\"\"F+&%'VectorG6#%'columnG-F$6%\")g?b?-F (6#7&F+F-F+F+F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "third column i n second row needs a 1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "s0 1,s10;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$-%'RTABLEG6%\")g?b?-%'MATRIX G6#7&7#\"\"!7#\"\"\"F+F+&%'VectorG6#%'columnG-F$6%\")C\")f?-F(6#7&F+F+ F-F+F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "second column in third \+ row needs a 1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "s11,s00;" } }{PARA 11 "" 1 "" {XPPMATH 20 "6$-%'RTABLEG6%\")'*Rx?-%'MATRIXG6#7&7# \"\"!F+F+7#\"\"\"&%'VectorG6#%'columnG-F$6%\"'#HC&-F(6#7&F-F+F+F+F/" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "fourth column in first row needs a 1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "U_inv:=Matrix([[0,0 ,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&U_invG-%'RTABLEG6%\"'OB$*-%'MATRIXG6#7&7&\"\"!F.F.\"\"\"7&F.F .F/F.7&F.F/F.F.7&F/F.F.F.%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "U_inv . s00, s11;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$ -%'RTABLEG6%\"'C2l-%'MATRIXG6#7&7#\"\"!F+F+7#\"\"\"&%'VectorG6#%'colum nG-F$6%\")'*Rx?F'F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "U_in v . s10, s01;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$-%'RTABLEG6%\"'K/u-%' MATRIXG6#7&7#\"\"!7#\"\"\"F+F+&%'VectorG6#%'columnG-F$6%\")g?b?F'F/" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "U_inv . s01, s10;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$-%'RTABLEG6%\"'S#H(-%'MATRIXG6#7&7#\" \"!F+7#\"\"\"F+&%'VectorG6#%'columnG-F$6%\")C\")f?F'F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "U_inv . s11 , s00;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$-%'RTABLEG6%\"'!y.(-%'MATRIXG6#7&7#\"\"\"7#\"\"!F- F-&%'VectorG6#%'columnG-F$6%\"'#HC&F'F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 116 "The Hadamar d gate matrix can be used to create so-called Bell states, or entangle d states as explained in Fig. 1.12." }}{PARA 0 "" 0 "" {TEXT -1 104 "F irst one applies the 2-by-2 Hadamard gate to the control qubit, and th en carries out a conditional NOT." }}{PARA 0 "" 0 "" {TEXT -1 104 "Ent angled states are such 2-qubit states which are not factorizable as a \+ product of single-qubit states." }}{PARA 0 "" 0 "" {TEXT -1 18 "Here i s Fig. 1.12:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {OLE 1 21505 1 "[xm]Br=WfoRrB:::wk;nyyI;G:;:j::>:B>N:F:nyyyyy]::yyyyyy::::: 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gb=>fXftb?[Iqu]^uN?hmGhZ^w<>vNgx=rouyLystibqaty;BboV[MfNge^ycVIi]mSmGE GMt^CikaEyybnSv=GcG;Xl_ivuUXeBBosnGY\\awmWxryVekC:iCrAe\\;wIyInuINwtcK YDsB=]SkITJ]FSKFo?vYgxGaIGqiYSVIYBwcB:uT:gYXWb;?TFIrnOfesfmmvy;baIvy;B JcbhIvJyYnuiLCvaUsYoRLch>=t:EbKyUFmTFWfMWUMJBTn\\jyIu:>Z:>:::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::?B:YD:SE:EE:[D:_ml Z:vyyyyyAZ:>Z::::::j;<:WS:;B:<:=ja<:>Z:>:::::::::J?:<::::::w yyyyyB:::::::::::::yay=J:B:::::::::::::::::::jysy:>:<::::::::2:" } {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "H2:=Matrix( [[1,1],[1,-1]])/sqrt(2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#H2G-%'R TABLEG6%\")%o6M#-%'MATRIXG6#7$7$,$*&\"\"#!\"\"F0#\"\"\"F0F3F.7$F.,$*&F 0F1F0F2F1%'MatrixG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 273 "In order t o be able to work this out with our 2-qubit states, we need to embed t he Hadamard matrix so that it will act on qubit 1 in the 2-qubit state in spinor representation. We want to mix the components associated wi th qubit 1 irrespective of what's going on in qubit2." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "s00,s10,s01,s11;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6&-%'RTABLEG6%\")7B$=#-%'MATRIXG6#7&7#\"\"\"7#\"\"!F-F- &%'VectorG6#%'columnG-F$6%\")sC$=#-F(6#7&F-F-F+F-F/-F$6%\")GJ,A-F(6#7& F-F+F-F-F/-F$6%\")C;+A-F(6#7&F-F-F-F+F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "s00+s10;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG 6%\")oGHA-%'MATRIXG6#7&7#\"\"\"7#\"\"!F+F-&%'VectorG6#%'columnG" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "s01+s11;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\")sjeB-%'MATRIXG6#7&7#\"\"!7#\"\"\"F+F-&%' VectorG6#%'columnG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 121 "We really \+ have to construct the Hadamard matrix acting on qubit 1. Naive embeddi ng doesn't work! What is it that we want?" }}{PARA 0 "" 0 "" {TEXT -1 139 "H2 |00> = s2 (|00> + |10>) , H2 |10> = s2 (|00> - |10>) , H2 |01 > = s2 (|01> + |11>) , H2 |11> = s2 (|01> - |11>) ; where s2 = 1/sqrt (2)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "s00, s00 + s10;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$-%'RTABLEG6%\")7B$=#-%'MATRIXG6#7&7#\" \"\"7#\"\"!F-F-&%'VectorG6#%'columnG-F$6%\")+TiB-F(6#7&F+F-F+F-F/" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "s01, s01 + s11;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$-%'RTABLEG6%\")GJ,A-%'MATRIXG6#7&7#\"\"!7#\"\" \"F+F+&%'VectorG6#%'columnG-F$6%\")O4mB-F(6#7&F+F-F+F-F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "s10, s00 - s10;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6$-%'RTABLEG6%\")sC$=#-%'MATRIXG6#7&7#\"\"!F+7#\"\"\"F+ &%'VectorG6#%'columnG-F$6%\"))oMP#-F(6#7&F-F+7#!\"\"F+F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "s11, s01 - s11;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6$-%'RTABLEG6%\")C;+A-%'MATRIXG6#7&7#\"\"!F+F+7#\"\"\"& %'VectorG6#%'columnG-F$6%\"))[(zB-F(6#7&F+F-F+7#!\"\"F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "sq2:=1/sqrt(2):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "H2q1:=Matrix([[sq2,0,sq2,0],[0,sq2,0,sq2],[ sq2,0,-sq2,0],[0,sq2,0,-sq2]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%% H2q1G-%'RTABLEG6%\")kn#Q#-%'MATRIXG6#7&7&,$*&\"\"#!\"\"F0#\"\"\"F0F3\" \"!F.F47&F4F.F4F.7&F.F4,$*&F0F1F0F2F1F47&F4F.F4F7%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "H2q1 . s00, s00 + s10;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$-%'RTABLEG6%\")s:&Q#-%'MATRIXG6#7&7#,$*&\"\"#! \"\"F.#\"\"\"F.F17#\"\"!F+F2&%'VectorG6#%'columnG-F$6%\")kX(Q#-F(6#7&7 #F1F2F>F2F4" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "H2q1 . s01, \+ s01 + s11;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$-%'RTABLEG6%\"))y?R#-%'M ATRIXG6#7&7#\"\"!7#,$*&\"\"#!\"\"F0#\"\"\"F0F3F+F-&%'VectorG6#%'column G-F$6%\")G&)*Q#-F(6#7&F+7#F3F+F>F4" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "H2q1 . s10, s00 - s10;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$-%'RTABLEG6%\")%o()R#-%'MATRIXG6#7&7#,$*&\"\"#!\"\"F.#\"\"\"F.F1 7#\"\"!7#,$*&F.F/F.F0F/F2&%'VectorG6#%'columnG-F$6%\")k7+C-F(6#7&7#F1F 27#F/F2F7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "H2q1 . s11, s0 1 - s11;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$-%'RTABLEG6%\")[42C-%'MATR IXG6#7&7#\"\"!7#,$*&\"\"#!\"\"F0#\"\"\"F0F3F+7#,$*&F0F1F0F2F1&%'Vector G6#%'columnG-F$6%\"))yBT#-F(6#7&F+7#F3F+7#F1F7" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 167 "OK, so it worked. Now we can complete the action, b y following up on H2q1 by a cNOT gate, which is not decomposable, and \+ therefore leads to an entangled 2-qubit state:" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 17 "B:=U_cnot . H2q1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG-%'RTABLEG6%\");< " 0 "" {MPLTEXT 1 0 20 "B . s00 , s00 + s 11;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$-%'RTABLEG6%\")Gn>C-%'MATRIXG6# 7&7#,$*&\"\"#!\"\"F.#\"\"\"F.F17#\"\"!F2F+&%'VectorG6#%'columnG-F$6%\" )oN9C-F(6#7&7#F1F2F2F>F4" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "B . s01 , s01 + s10;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$-%'RTABLEG6% \")G&pU#-%'MATRIXG6#7&7#\"\"!7#,$*&\"\"#!\"\"F0#\"\"\"F0F3F-F+&%'Vecto rG6#%'columnG-F$6%\")OWCC-F(6#7&F+7#F3F>F+F4" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "B . s10 , s00 - s11;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$-%'RTABLEG6%\")saLC-%'MATRIXG6#7&7#,$*&\"\"#!\"\"F.#\" \"\"F.F17#\"\"!F27#,$*&F.F/F.F0F/&%'VectorG6#%'columnG-F$6%\")[#yV#-F( 6#7&7#F1F2F27#F/F7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "B . s 11 , s01 - s10;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$-%'RTABLEG6%\")SGUC -%'MATRIXG6#7&7#\"\"!7#,$*&\"\"#!\"\"F0#\"\"\"F0F37#,$*&F0F1F0F2F1F+&% 'VectorG6#%'columnG-F$6%\")KFZC-F(6#7&F+7#F37#F1F+F7" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 260 11 "Exercise 2:" }}{PARA 0 "" 0 "" {TEXT -1 134 "C onstruct the matrix for the Hadamard gate for qubit2, and then compute the matrix H4 which performs the Hadamard gate on both qubits." }}} {SECT 1 {PARA 3 "" 0 "" {TEXT 261 22 "Solution to Exercise 2" }} {EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "First, we have to state the goal: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "s00, s00 + s01;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$-% 'RTABLEG6%\")7B$=#-%'MATRIXG6#7&7#\"\"\"7#\"\"!F-F-&%'VectorG6#%'colum nG-F$6%\")w1^C-F(6#7&F+F+F-F-F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "s10, s10 + s11;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$-%'RTABLEG6 %\")sC$=#-%'MATRIXG6#7&7#\"\"!F+7#\"\"\"F+&%'VectorG6#%'columnG-F$6%\" )!ydX#-F(6#7&F+F+F-F-F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 " s01, s00 - s01;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$-%'RTABLEG6%\")GJ,A -%'MATRIXG6#7&7#\"\"!7#\"\"\"F+F+&%'VectorG6#%'columnG-F$6%\")))RhC-F( 6#7&F-7#!\"\"F+F+F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "s11, s10 - s11;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$-%'RTABLEG6%\")C;+A-%'M ATRIXG6#7&7#\"\"!F+F+7#\"\"\"&%'VectorG6#%'columnG-F$6%\"'/Fe-F(6#7&F+ F+F-7#!\"\"F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 10 "Therefore:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "H2q2:=Matrix([[sq2,sq2,0,0], [sq2,-sq2,0,0],[0,0,sq2,sq2],[0,0,sq2,-sq2]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%H2q2G-%'RTABLEG6%\")C8W@-%'MATRIXG6#7&7&,$*&\"\"#!\" \"F0#\"\"\"F0F3F.\"\"!F47&F.,$*&F0F1F0F2F1F4F47&F4F4F.F.7&F4F4F.F6%'Ma trixG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "Let's verify:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "H2q2 . s00, s00 + s01;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$-%'RTABLEG6%\")_.u@-%'MATRIXG6#7&7#,$* &\"\"#!\"\"F.#\"\"\"F.F1F+7#\"\"!F2&%'VectorG6#%'columnG-F$6%\"'gDe-F( 6#7&7#F1F>F2F2F4" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "H2q2 . \+ s10, s10 + s11;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$-%'RTABLEG6%\")g+)= #-%'MATRIXG6#7&7#\"\"!F+7#,$*&\"\"#!\"\"F0#\"\"\"F0F3F-&%'VectorG6#%'c olumnG-F$6%\")![];#-F(6#7&F+F+7#F3F>F4" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "H2q2 . s01, s00 - s01;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$-%'RTABLEG6%\"'/.%)-%'MATRIXG6#7&7#,$*&\"\"#!\"\"F.#\"\"\"F.F17# ,$*&F.F/F.F0F/7#\"\"!F5&%'VectorG6#%'columnG-F$6%\"'Kob-F(6#7&7#F17#F/ F5F5F7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "H2q2 . s11, s10 - s11;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$-%'RTABLEG6%\"'C5m-%'MATRIXG6 #7&7#\"\"!F+7#,$*&\"\"#!\"\"F0#\"\"\"F0F37#,$*&F0F1F0F2F1&%'VectorG6#% 'columnG-F$6%\")_nXB-F(6#7&F+F+7#F37#F1F7" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "OK, now take the matrix product:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "2* H2q1 . H2q2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\")g)4O#-%'MATRIXG6#7&7&\"\"\"F,F,F,7&F,!\"\"F,F.7&F,F, F.F.7&F,F.F.F,%'MatrixG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 557 "We mu ltiplied by 2 to have a clear view of the matrix. Without the factor o f 2 we have computed the Hadamard matrix W4, which represents the appl ication of a Hadamard gate to both qubits. It is obtained as an outer \+ product of two Hadamard gates W2 for each qubit (the order did not mat ter). An outer product represents a factorizable operation. The import ant thing to produce the Bell states was to apply U_cnot, the conditio nal NOT gate, which is represented by a matrix that cannot be factoriz ed into two matrices acting on qubit 1 and qubit 2 separately." }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 259 21 "Quantum Teleportation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 587 "According to Fig. 1.13 an entangled pai r shared between two remote communication partners A and B (A=Alice, B =Bob) can be used to transfer a quantum state from one end to the othe r by means of a few gates, and a classical communication channel by wh ich the outcome of a measurement done at A can be transmitted to B. Th e unknown state |psi> = a|0> + b|1> is being transferred from A to B, \+ i.e., while B will be able to reconstruct it, A will wind up with a de stroyed state. This has to be so to be compliant with the no-cloning t heorem, which does not permit to duplicate quantum states." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {OLE 1 27138 1 "[xm]Br=WfoRrB: ::wk;nyyI;G:;:j::>:B>N:F:nyyyyy]::yyyyyy:::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::fyyyyyY@nYnyyyYE:G:I:K:M:O:Q: S:UJ:n;v;;JBB:]:_:a:c:e:g:i:k:m:o:q:s:u:w:y:;[:F>N>V>^>f>n>v>>?F?N?V?^ ?f?n?v?>`:B:];_;wAwAyA:::::::::::::::::::::::::::::::::::::::::::::::: ::_lqvGcMJ:::::::JEf:yyyxI^:^;A:;JZ=Z:Z@[::JbqU;PuVQs[Aj;< :;b:>Z:j:vCSmlF@[KaFFcmnnHEM:>:::::::oJ;@jyyyyyI:;Z::::::Jd:B:j:vCS=>:::::::::J?N:ry?:yayA: <::::::?J:^<>:F:;JyKyKR<:TN>J b@pkhpjXpl>;:::JtaMSAA<::gwQ;cFgyBUb=OFceT\\cbqCfywWxcDBuvy\\PItsMxk`Y QuqTylrL=SUpy@alilq>doudPtqt:b:B:::cC\\cgeIYcGR;aTF_FOcDNMS?qcowWGmiRO F_ssdwWyocU]ei?GwOiT?Tu=y]WS?ATGyyyyyyA:C:=Z:vYxY;J:jgY:ET;^WF:MZ=nHN: s:;j<>Z:V[l;B:@:GKHjw?:sg:B:=J;Dlc`qsLqlp@OV:B:yayAmmq:McJKxWCJ:f?= 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ir]eOQcV]YjQem_xMYiR_sP;XvOHm_gXQuBoY=yRdqihcHgUrMciuMVYIgxUsR=EW_F@oh IUS;[Ggwvl]H\\oI_wuMcevyQuluypPLMYkxr@tjHUw_MP]LRmIWcPWtEM\\]kLXKuIvQe YUifqQbLWFQkWvySWwEAsUaweDixgWX`;Immh AMWYKIgIyuatw]uqcbXoyLOdoAwswv;UX[YRiiV[Id@aYbQyvQHAKGqQgwIDE[iKSeNqys wgq?riGvCECHwBIwgmkuDgCDuevcFwiv]]x_[InkuUiys=vVXsiyoZ::::::::kB:?R:AB:;j:Z:B:F:YD:J:, and a ta rget qubit line on which is encoded one half of the entangled 2-qubit \+ state. B has one line on which it has encoded the other qubit of the e ntangled state. The entire system, therefore, is a 3-qubit state, and \+ the objective is to transfer the information from qubit 1 (Alice) to q ubit 3 (Bob)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 322 "Alice will apply a cNOT followed by a Hadamard gate on q 1, and then measure probabilities on q1 and q2. These probabilities wi ll be transmitted classically to Bob, who will then know which gate to apply to q3, such that it will contain |psi> as originally stored on \+ q1. Our aim is to calculate the evolution of the states." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 145 "To be able to app ly the cNOT and Hadamard gates we need to set up A's 2-qubit state. Th e overall state is a product of |psi> = a|0> + b |1>, and" }}{PARA 0 "" 0 "" {TEXT -1 69 "1/sqrt(2) (|00> + |11>). There are eight (2^3) po ssible basis states." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 263 "s0 00:=Vector([1,0,0,0,0,0,0,0]): s001:=Vector([0,1,0,0,0,0,0,0]): s010:= Vector([0,0,1,0,0,0,0,0]): s011:=Vector([0,0,0,1,0,0,0,0]): s100:=Vect or([0,0,0,0,1,0,0,0]): s101:=Vector([0,0,0,0,0,1,0,0]): s110:=Vector([ 0,0,0,0,0,0,1,0]): s111:=Vector([0,0,0,0,0,0,0,1]):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 163 "Now we need to define the cNOT gate for q1 and q2. We need an 8-by-8 matrix that serves as an embedding of the 4-by- 4 cNOT gate, and which ignores the state of q3." }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 10 "s000,s001;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$ -%'RTABLEG6%\")C(R9#-%'MATRIXG6#7*7#\"\"\"7#\"\"!F-F-F-F-F-F-&%'Vector G6#%'columnG-F$6%\")?pa@-F(6#7*F-F+F-F-F-F-F-F-F/" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 43 "Both these states are to be left unchanged." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "#UcNOT:=Matrix([[1,0,],[0,1, ],[],[],[],[],[],[]]):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "Also th e next two should be left as is:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "s010,s011;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$-%'RTAB LEG6%\")+ma@-%'MATRIXG6#7*7#\"\"!F+7#\"\"\"F+F+F+F+F+&%'VectorG6#%'col umnG-F$6%\")gja@-F(6#7*F+F+F+F-F+F+F+F+F/" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 74 "#UcNOT:=Matrix([[1,0,0,0,],[0,1,0,0,],[0,0,1,0,],[0 ,0,0,1,],[],[],[],[]]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 " s100,s110,s101,s111;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&-%'RTABLEG6%\" )Sda@-%'MATRIXG6#7*7#\"\"!F+F+F+7#\"\"\"F+F+F+&%'VectorG6#%'columnG-F$ 6%\")sl_@-F(6#7*F+F+F+F+F+F+F-F+F/-F$6%\")[\"Q:#-F(6#7*F+F+F+F+F+F-F+F +F/-F$6%\")_g_@-F(6#7*F+F+F+F+F+F+F+F-F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 78 "In the above two pairs: the first spinor is to be transfo rmed into the second." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 156 "# UcNOT:=Matrix([[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[ 0,0,0,1,0,0,0,0],[0,0,0,0,0,0,],[0,0,0,0,0,0,],[0,0,0,0,1,0,0,0],[0,0, 0,0,0,1,0,0]]):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "and finally, t he remaining pair of pairs:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "s110,s100,s111,s101;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&-%'RTABLEG 6%\")sl_@-%'MATRIXG6#7*7#\"\"!F+F+F+F+F+7#\"\"\"F+&%'VectorG6#%'column G-F$6%\")Sda@-F(6#7*F+F+F+F+F-F+F+F+F/-F$6%\")_g_@-F(6#7*F+F+F+F+F+F+F +F-F/-F$6%\")[\"Q:#-F(6#7*F+F+F+F+F+F-F+F+F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 161 "UcNOT:=Matrix([[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0 ,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0 ,0,1],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&UcNOTG-%'RTABLEG6%\")KD=A-%'MATRIXG6#7*7*\"\"\"\"\"! F/F/F/F/F/F/7*F/F.F/F/F/F/F/F/7*F/F/F.F/F/F/F/F/7*F/F/F/F.F/F/F/F/7*F/ F/F/F/F/F/F.F/7*F/F/F/F/F/F/F/F.7*F/F/F/F/F.F/F/F/7*F/F/F/F/F/F.F/F/%' MatrixG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "Check it out:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "Evalb(UcNOT . s000 = s000) , Evalb(UcNOT . s001= s001);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%%trueGF #" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "Evalb(UcNOT . s010 = s 010) ,Evalb(UcNOT . s011 = s011);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$% %trueGF#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "Evalb(UcNOT . s 100 = s110) , Evalb(UcNOT . s101 = s111);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%%trueGF#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "Evalb(UcNOT . s110 = s100) , Evalb(UcNOT . s111 = s101);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%%trueGF#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "Now construct the Hadamard gate matrix representation for q1." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "sq2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"#!\"\"F%#\"\"\"F%F(" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 37 "s000, s000 + s100, s001, s001 + s101;" }}{PARA 11 " " 1 "" {XPPMATH 20 "6&-%'RTABLEG6%\")C(R9#-%'MATRIXG6#7*7#\"\"\"7#\"\" !F-F-F-F-F-F-&%'VectorG6#%'columnG-F$6%\")O*pI#-F(6#7*F+F-F-F-F+F-F-F- F/-F$6%\")?pa@-F(6#7*F-F+F-F-F-F-F-F-F/-F$6%\"(3;:\"-F(6#7*F-F+F-F-F-F +F-F-F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 114 "We will start with a \+ matrix filled by zeroes, and will replace the non-zero entries accordi ng to the requirements." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 " Hq1:=Matrix(8,8):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "Hq1[1, 1]:=sq2: Hq1[5,1]:=sq2: Hq1[2,2]:=sq2: Hq1[6,2]:=sq2:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "s010, s010 + s110, s011, s011 + s11 1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&-%'RTABLEG6%\")+ma@-%'MATRIXG6#7 *7#\"\"!F+7#\"\"\"F+F+F+F+F+&%'VectorG6#%'columnG-F$6%\")%o*e?-F(6#7*F +F+F-F+F+F+F-F+F/-F$6%\")gja@-F(6#7*F+F+F+F-F+F+F+F+F/-F$6%\")c4:C-F(6 #7*F+F+F+F-F+F+F+F-F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "Hq 1[3,3]:=sq2: Hq1[7,3]:=sq2: Hq1[4,4]:=sq2: Hq1[8,4]:=sq2:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "s100, s000 - s100 , s101, s001 - s1 01;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&-%'RTABLEG6%\")Sda@-%'MATRIXG6# 7*7#\"\"!F+F+F+7#\"\"\"F+F+F+&%'VectorG6#%'columnG-F$6%\"(K^+\"-F(6#7* F-F+F+F+7#!\"\"F+F+F+F/-F$6%\")[\"Q:#-F(6#7*F+F+F+F+F+F-F+F+F/-F$6%\") ;BcB-F(6#7*F+F-F+F+F+F9F+F+F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "Hq1[1,5]:=sq2: Hq1[5,5]:=-sq2: Hq1[2,6]:=sq2: Hq1[6,6]:=-sq2:" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "s110, s010 - s110 , s111, \+ s011 - s111;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&-%'RTABLEG6%\")sl_@-%' MATRIXG6#7*7#\"\"!F+F+F+F+F+7#\"\"\"F+&%'VectorG6#%'columnG-F$6%\")Sx] ?-F(6#7*F+F+F-F+F+F+7#!\"\"F+F/-F$6%\")_g_@-F(6#7*F+F+F+F+F+F+F+F-F/-F $6%\")3#RP#-F(6#7*F+F+F+F-F+F+F+F9F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "Hq1[3,7]:=sq2: Hq1[7,7]:=-sq2: Hq1[4,8]:=sq2: Hq1[8,8 ]:=-sq2:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "Hq1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\");N)G#-%'MATRIXG6#7*7*,$*&\"\" #!\"\"F.#\"\"\"F.F1\"\"!F2F2F,F2F2F27*F2F,F2F2F2F,F2F27*F2F2F,F2F2F2F, F27*F2F2F2F,F2F2F2F,7*F,F2F2F2,$*&F.F/F.F0F/F2F2F27*F2F,F2F2F2F7F2F27* F2F2F,F2F2F2F7F27*F2F2F2F,F2F2F2F7%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "Determinant(Hq1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "Let's check it out. .. we copy the lines used for the matrix construction, and turn them i nto equations to be checked:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "Evalb(sqrt(2)*Hq1 . s000 = s000 + s100), Evalb(sqrt(2)*Hq1 . s00 1 = s001 + s101);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%%trueGF#" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "Evalb(sqrt(2)*Hq1 . s010 = s 010 + s110), Evalb(sqrt(2)*Hq1 . s011 = s011 + s111);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%%trueGF#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "Evalb(sqrt(2)*Hq1 . s100 = s000 - s100) , Evalb(sqrt(2)*Hq1 . s1 01 = s001 - s101);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%%trueGF#" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "Evalb(sqrt(2)*Hq1 . s110 = s 010 - s110) , Evalb(sqrt(2)*Hq1 . s111 = s011 - s111);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%%trueGF#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 95 " Now we are able to put the teleportation gate together: Equation 1.29 \+ states the initial |psi>." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "psi:=sq2*(a*(s000 + s011) + b*(s100 + s111));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$psiG-%'RTABLEG6%\")#4)Q?-%'MATRIXG6#7*7#,$*(\"\"#!\" \"F0#\"\"\"F0%\"aGF3F37#\"\"!F5F-7#,$*(F0F1F0F2%\"bGF3F3F5F5F7&%'Vecto rG6#%'columnG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "psi:=simpl ify(psi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$psiG-%'RTABLEG6%\")[`Q ?-%'MATRIXG6#7*7#,$*(\"\"#!\"\"F0#\"\"\"F0%\"aGF3F37#\"\"!F5F-7#,$*(F0 F1F0F2%\"bGF3F3F5F5F7&%'VectorG6#%'columnG" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 19 "psi1:= UcNOT . psi;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%psi1G-%'RTABLEG6%\"(?*R7-%'MATRIXG6#7*7#,$*(\"\"#!\"\"F0#\"\" \"F0%\"aGF3F37#\"\"!F5F-F57#,$*(F0F1F0F2%\"bGF3F3F7F5&%'VectorG6#%'col umnG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "psi2:= Hq1 . psi1; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%psi2G-%'RTABLEG6%\"')[T)-%'MATR IXG6#7*7#,$*&\"\"#!\"\"%\"aG\"\"\"F37#,$*&F0F1%\"bGF3F3F4F-F-7#,$*&F0F 1F7F3F1F8F-&%'VectorG6#%'columnG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 572 "What is the significance of psi2 as compared to the initial psi, \+ i.e., what was the effect of applying the gates? Somehow, the informat ion (values of a,b) got transferred all over the place. Note, Alice ne ver knew what her amplitudes a and b were, and the claim is, that if s he measures the probability content of q1 and q2, and transmits it cla ssically to Bob, then Bob can infer what simple gate to apply in order to re-surrect the superposition a|0> + b|1> in 'his' qubit, i.e., in q3 (without knowing the values of a,b). How can we find out whether t his indeed works?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 482 "The claim is, that if Alice measures a |00>, then Bob si mply has a|0> + b|1> in q3. For Alice observing |01>, he gets b|0> + \+ a|1>, for Alice measuring |10> he gets a|0> - b|1>, and when she sees |11>, he obtains -b|0> + a|1>. This means that upon receiving inform ation from Alice on the classical channel as to what she measured on q 1/q2, Bob can apply a corresponding transformation (in the latter thre e cases) in order to rotate the state into a|0> + b|1>, thus receivin g psi." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 122 "How do we obtain the collapse of psi2 to a single-qubit wavefunct ion, i.e., what do we do to simulate Alice's measurement?" }}{PARA 0 " " 0 "" {TEXT -1 158 "It is a projection onto the 2-qubit state that sh ould do the trick. Two degrees of freedom would be contracted away, wi th a remaining two-spinor wavefunction." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 89 "One way to convince oneself, is to r ealize which superposition the state psi2 represents:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "phi2:=simplify((a*s000+b*s001) + (a *s011+b*s010) + (a*s100-b*s101) + (a*s111-b*s110));" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%%phi2G-%'RTABLEG6%\")#HOE#-%'MATRIXG6#7*7#%\"aG7#% \"bGF/F-F-7#,$F0!\"\"F1F-&%'VectorG6#%'columnG" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 19 "Evalb(phi2=2*psi2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 205 "The fo ur states above are grouped in such a way that they correspond to Alic e's four possible measurement outcomes, namely |00> (in which case qub it 3 is in the original state of q1), |01>, |10>, and |11>." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 81 "This calculatio n then carries out the matrix representation of eqs. 1.28 to 1.36." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 262 45 " Classical computing with qubits: Toffoli gate" }}{PARA 0 "" 0 "" {TEXT -1 259 "The idea is to demonstrate the a classical NAND gate can be implemented on a quantum computer. 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A NAND gate can be obtained by using the first two \+ qubits as input, setting the third input to 1, and using the q3 output as the result. 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