; TeX output 2001.12.13:2049썠:-iӍ%O#}h!q cmsl12FNromGeometrytoQuanotumComputation#􍍍š&XQ ff cmr12Kazuyuki/FUJIdCI= !", cmsy10naDepartment/ofMathematicalSciences(Y4okohama/CityUniversity"6Y4okohama/236-0027JAP4ANˈ ."V 3 cmbx10AbstractMƍ-̻+K`y 3 cmr10The aimofthispapMeristoin!troduceourideaofHolonomicQuan!tumComputa- _tion(Computer).*OurmoMdelisbasedonbothharmonicoscillatorsandnon{linear_quan!tum͜optics,jnotonspinsofusualquantumcomputationandourmethoMdis_moreo!verfcompletelygeometrical.-̻WeefhopMethatthereforeourmodelma!ybestrongfordecoherence.VffH g^ O!cmsy7K`y cmr10E-mailUUaddress:qfujii@math.yokohama-cu.ac.jp*썠:-iӍ}-4NG cmbx121(Inutro=ductionb#XQ cmr12QuanrtumComputationisaveryattractiveandchallengingtaskinthiscenturyV.After!thebreakthroughbryPV.Shor[1]therehasbSeenremarkXableprogressinQuantumComputeru(orComputation(QCubrie y).aThisdiscorveryu(hadagreatin uenceonscien-tists.S;Thisdrivrednotonlytheoreticiansto ndingotherquantumalgorithms,butalsoexpSerimenrtaliststobuildingquantumcomputers.8See[2]and[3],[4]inoutline.Ontheotherhand,+GaugeTheoriesarewidelyrecognizedasthebasisinquanrtum eldDtheories.FThereforeitisvrerynaturaltointendtoincludegaugetheoriesinQCD:!", cmsy10adconstructionof\gaugetheoretical"quanrtumcomputationorof\geometric"quantumcomputation%inourterminologyV.8XThemeritofgeometricmethoSdofQCmarybestrongforthein uencefromtheenrvironment.In0[5]and[6]ZanardiandRasettipropSosedanattractivreideaaHolonomicQuantumComputation>[(Computer)|usingthenon-abSelianBerryphase(quanrtumholonomyinthezmathematicallanguage).{Seealso[7]and[8]asanotherinrterestinggeometricmoSdels.Inu"theirmoSdelaHamiltonian(includingsomeparameters)mrustbedegeneratedbe-cause3anadiabaticconnectionisinrtroSducedusingthisdegeneracy[9].7Inotherwords,adquanrtumcomputationalbundleisintroSducedonsomeparameterspaceduetothisdegeneracy(see[5])andthecanonicalconnectionofthisbundleisjusttheabSorve.They̛garveafewsimplebutinterestingexamplestoexplaintheiridea..TVomaketheirwrorksmmoremathematicalandrigoroustheauthorhasgiventhemathematicalreinforce-menrttotheirworks,see[13 ],[14],[15]and[16].8Buthiswrorksarestillnotsucient.InWAthistalkwrewillintroSduceourHolonomicQuantumComputationanddiscusssomeproblemstobSesolvred.WVestronglyhopSethatyroungmathematicalphysicistswillenterthisattractive eld.'2(MathematicalzPreliminariesWVestartwithmathematicalpreliminaries.LetHbSeaseparableHilbertspaceorver=N cmbx12C.FVor7g cmmi12mUR2N,wresetNN'St82cmmi8mb(goH)yURf`u cmex10n V=UR(v5|{Ycmr81;;vm)Bvl2URHPUNHjVp ;K cmsy8yV=1mğf`o`;:(1)[(Vp2yV%[=1m ()hvidjvjf i=ijJ)_where1m #isaunitmatrixinM@(Ҩm;CG)3p.WThisiscalleda(univrersal)ĝStiefelmanifold.,2NotethattheunitarygroupU@(m)actsonStm_(#H)/fromtherighrt:s[ZStm(H)OU@(Ҩm)"`!URStm(H)2(:?( ѾV;a)&7!URVpa::(2)(!Nextwrede nea(universal)GrassmannmanifoldSLGrmcA(gH)yURf`n XF2URM@(ҨH)#WjX 2 =URXJg;X y=X+andtr$؇X=mf`o;:(3)TwhereK2M@(H)denotesaspaceofallbSoundedlinearoperatorsonH. Z}ThenwrehaveaprojectionzË:URStm(H).=!URGrmM(|H)18; n9(Vp)*ZURVpV y 8;:(4)compatiblewiththeaction(2)(n9(Vpa)#2e=URVpa(Va)2yr=Vaa2yG V2y 8=VV2y 8=n9(V)d).@1(썠:-iӍ}-Norwtheset}vf}wU@(Ҩm)#;Stm/(H))K;n9;Grm(&H)+g-}z;:(5)S2@cmbx80MbSea xedreferencebpSoinrtofM.BLetH beafamilyofHamiltoniansparameterizedbryMwhichactaonaFVoScrkspaceH. WesetH0!=Hq?Etcmbx60 forsimplicitryandassumethatthishasam-folddegeneratevXacuum:-#H0vj\=UR0; j%=1m:Z(11)These'vjf 'sformam-dimensionalvrectorspace.CWVemayassumethathvidjvjf i<=ijJ.CThen(v1;;vm)?!2URStm(H).Gand#l퍒8F0VUR8 UR< UR:m  a@X 8jv=1exjf vjjxj\2URC9 = ;Pe/7԰eH=qC m:NamelyV,F0isavrectorspaceassoSciatedwitho.n.basis(|lv1;;vm)?p.Next|wreassumeforsimplicitythatafamilyofunitaryopSeratorsparameterizedbyMፍrW:URM!U@(H); W(0 A)=id :Z(12)isgivrenandH `abSoveisgivenbythefollowingisospSectralfamilyHʬURW()H0W() 1 \|:Z(13)InthiscasethereisnolevrelcrossingofeigenvXalues.*IMakinguseofW()wecande neaprojectorU_WgP:URM!GrmM(|H)-M; P()W()fZ0 @^m  ^X jv=1vjf vOn9yqjfZY1 YA:MW() 1Z(14)@2썠:-iӍ}-andharvethepullbackbundlesoverM{Af`nU@(Ҩm)lS;}fSt*;Íe)?Sr}t /;Mf`oO; f`njC m;eE f;ÍDe)?E-;Mf`odLh:Z(15)FVorthelatterwresetjvn9acUi0=UR(v1;;vm)A!:Z(16)InthiscaseacanonicalconnectionformAoff`n TU@(Ҩm)';;}fSt*;Íe)?Sr}t /;Mf`ol isgivrenby@sAUR=hvn9acUjPW() 1 \|dW()jvn9acUiU;Z(17)wheredisadi erenrtialformonM,anditscurvXatureformbyFcURdA+A^A;Z(18)see[9]and[10 ].Let bSealoopinMat0., o:S[0;1]!M; (0)= (1). 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a(:numbSeroperator),thenf?[N;a yG ]UR=a y1;[N;a]=a;[a;a yG ]=1:Z(21)LetC H`bSeaFVocrkspacegeneratedbyaanda2yG ,Y#andfjnijn2N[f0ggC bSeitsbasis.BTheactionsofaanda2y1onHParegivrenbys6ajniX=URϓp UTϓz 0mnXjn1,i;Ѳ;a yG jni=UR_p UT_z\] n+1(jn+1\]iO;Z(22)where4j0ieis4avXacuum(aj0i=$0). ,InthefollorwingwetreatcoherentopSeratorsandsqueezedopSerators.?CoherenrtOpSerator]DS( )UR=expHҟf`A a yԜ 7paf`_[forqOk h2C;Z(23)?SqueezedOpSerator]S( O)UR=expHҟf^F ō݂1݂Qmfe  2(a yG ) 2jW[* ō *H1 *HQmfe  2=wa 2f^ҧfor) 2C:Z(24)@4.Ӡ썠:-iӍ}-FVorthedetailssee[13 ].3DNextwreconsiderthesystemofn{harmonicoscillators.Ifwreset*+ai,=UR1 UN 1 a 1  1; aiOџ yC=UR1  1 a y 1  1;Z(25)for1URin,thenitiseasytosee5[aid;ajf ]UR=[ai ۟ y ;aj y)]=0;[aid;aj y)]=ijJ:Z(26)WVealsodenotebryNi,=URai -2y,MaiO(1URin)numbSeroperators."3ÍBNff cmbx12B3.1$cTwo{QubitffCase@Since wrewanttoconsidercoherentstatesbasedonLiealgebrassu(2)andsu(1;1),wemakreuseofSchwinger'sbSosonmethod,see[18 ]and[19].8Namelyifwreset۪C\6su(2)UR:JJ+ q=aOyO1a2;J=URaOyO2a1;J3V=ō1Qmfe  2 f`aOyO1a1jaOyO2a2f`U3;Z(27)4C\6su(1;1)UR:JK+ q=aOyO1aOyO2;K=URa2a1;K3V=ō1Qmfe  2 f`aOyO1a1j+aOyO2a2+1f`j 8;Z(28)Fthenwrehave=su(2)UR:J[J3;J+x]=J+;[J3;J]=J;[J+;J]=2J3;Z(29)=su(1;1)UR:J[K3;K+x]=K+;[K3;K]=K;[K+;K]=2K3:Z(30)In thefollorwingwetreatunitarycoherentopSeratorsbasedonLiealgebrassu(2)andsu(1;1). U@()B=expf`J+ W'* NJxf`Oafor BUR2C;Z(31)Vp()B=exp(`K+  kKx)gfor$UR2C:Z(32)FVorthedetailsofU@()andVp()see[17 ]and[18].Let{H0bSeaHamiltonianwithnonlinearinrteractionproducedbryaKerrmedium.,CPthatis&H0zx=th ^XN@(N1),4whereX&isacertainconstanrt,see[11 ]and[12].TheeigenrvectorsofTH0XcorrespSondingto0isf Uj0iR;j1ig0,ݘsoitseigenspaceisVVectf"j0i0;j1igPM ԰M#=YC22.3oThespaceVVectfj0i,;j1igHisucalled1-qubit(quanrtumbit)space,3Lsee[2]or[3].zSincewreareconsideringthesystemoftrwoparticles,theHamiltonianthatwretreatinthefollowingis)H0V=URh XN1(N1j1)+h N4XN2(N2j1):Z(33)Theeigenspaceof0ofthisHamiltonianbSecomestherefore"F0V=URVVect-f"-j0i0 ;j1igKA VVectif!jj0i/bg;j1ig=URVVect-f"-j0;0i;0;j0;1i;j1;0i;j1;1igP[԰s=C 4:Z(34)WVeOsetjvn9acUis=UR(j0;0i ;j0;1i;j1;0i;j1;1i)Q.NextwreconsiderthefollowingisospSectralfamilyofH0:8}H( q6Aacmr61*; q1;;; q2; q2)Er;=URW( 1; 1;;; 2; 2)H0W( 1; 1;;; 2; 2) 1 \|;Z(35)8}W( 1; 1;;; 2; 2)UR=W1( 1; 1)O12 (;)W2( 2; 2):Z(36)where7@O12 (;)UR=U@()Vp(); Wjf ( j; j)UR=Djf ( j)Sj( j) for%j%=UR1;2:Z(37)@5H썠:-iӍ.+`<~<~5p%ߟpϴpfd"$\O line10-pfd"$-R12CW1G4ǟW2(g\O12InthiscasebMUR=f`n ( 1; 1;;; 2; 2)UR2C 6f`oZ(38)andwrewanttocalculateAUR=hvn9acUjPWƟ 1 BdWjvn9acUi;Z(39)wheresd&=3kd 1ō @7Qmfe=  @ 1O+d, 1ō@ lpQmfe=  @ӽ 1$^+d 1ō H@7Qmfe="  @ 14+dW* 1ōFf@ UQmfe=" $@Wh/* 1"R+dō@33Qmfe  @ƴ+dW}U*ō {,@ ٟQmfe $@W$q*Z+dō@33Qmfe ß  @+drō @DQmfe ß  @!&+3kd 2ō @7Qmfe=  @ 2O+d, 2ō@ lpQmfe=  @ӽ 2$^+d 2ō H@7Qmfe="  @ 24+dW* 2ōFf@ UQmfe=" $@Wh/* 2#R:Z(40) yThecalculationof(39)isnoteasyV,see[16 ]forthedetails.Problem IstheconnectionformirreducibleinU@(4)?Our`analysisin[16 ]shorwsthattheholonomygroupgeneratedbyAmaybSeSU@(4)notU@(4).TVo>obtainU(4)asophisticatedtricrk}higherdimensionalholonomies[22 ]marybSenecessaryV.8Afurtherstudyisneeded."ʫB3.2$cN{QubitffCase@AreferenceHamiltonianisinthiscaseqʍyH0V=URX>n X Ii=1cHNid(Ni1); X+isaconstanrt"andtheeigen{spaceto0{eigenrvXalue(n{qubits)bSecomes,aF0V=URVVect-fj0;;0;01ji9j;j0;;0;11ji;j;;j1;;1;01ji;j1;;1;11jigPUR԰n:=C 2-:9;cmmi6nZ(41)andsetjvn9acUi =UR(j0;;0;01ji9j;j0;;0;11ji;j;;j1;;1;01ji;j1;;1;11ji).Theu(n){algebraisde nedbry&generatorsUWE:?fEij 5j1URi;j%ng; relations@:[EijJ;Ek6lZ]=jvk EilJlKiEk6j;Z(42)andfEin j1Tkin1gaWVeylbasis.}Bosonrepresenrtationofu(n){algebraiswell{knorwntobSe5Eij 6=URai - y,Maj&1URi;j%n:Z(43)Theu(n1;1){algebraisalsode nedbrygeneratorsTq:?fEij 5j1URi;j%ng; relations@:[EijJ;Ek6lZ]=jvk EilJlKiEk6j;Z(44)@6\r썠:-iӍ}-whereʨË=URdiag(1;;1;1),BandfEin Wj1URin^h1gaWVeylbasis.Bosonrepresenrtationofu(n1;1){algebraisgivrenbyQeCEij 6=URai - y,Maj&1URi;j%n1; Enn%=URan( yoanR+1eCEin |=URai - y,Man y%1f; Eni=URanPai$1in1:Z(45)AfamilyofHamiltoniansthatwretreatis n(HB=URWH0W 1;W="mn SY jv=1WjvnZ(46)#ōandWjvn yisvAWjvn Z( jf ; j;j;j)UR=Wjf ( j; j)Ojvn Z(j;j);Z(47)!andQWjf ( j; j)UR=Djf ( j)Sj( j); Ojvn Z(j;j)UR=Ujf (j)Vj(j);Z(48)andfor1URj%n1Q5.Ujf (j)UR=expH(jf aj y>5anRW'*j Xan yajf ); Vj(j)UR=exp(jf aj y>5an y&Nj"uanPajf )Z(49)makinguseofeacrhWVeylbasis.8WenotethatOnn%=UR1.6G!U,zDID1:ǟO"^fd"$\άqƟ^fd"$-}U,z0D2Ja7U,.۟U,.U,.*韰U,zUn^fd"$ά-3J^fd"$\_]Dn1U,z}D DnQ"$feG! fd8Fǟ"$feG! 6}TfdJa"fe} 6*]ȄfdU[** afe* 6sQXBInthiscased"MUR=f US(; jf ; j;j;j; jv+1B; jv+1;)gP9԰!=őC 4n2Z(50)!andMAUR=hvn9acUjPWƟ 1 BdWjvn9acUil;Z(51)whereClh| BNff cmbx12?Etcmbx6>2@cmbx8=N cmbx12;K cmsy8:!", cmsy109;cmmi682cmmi87g cmmi126Aacmr65|{Ycmr84NG cmbx12."V 3 cmbx10+K`y 3 cmr10&XQ ff cmr12#}h!q cmsl12XQ cmr12 !", cmsy10 O!cmsy7K`y cmr10< lcircle10O line10u cmex10