Robust dynamical decoupling with bounded controls

LorenzaViola∗andEmanuelKnill†

LosAlamosNationalLaboratory,MailStopB256,LosAlamos,NewMexico87545

(Dated:July15,2002)

Weproposeageneralprocedureforimplementingdynamicaldecouplingwithoutrequiringarbitrarilystrong,impulsivecontrolactions.ThisisaccomplishedbydesigningcontinuousdecouplingpropagatorsaccordingtoEulerianpathsinthedecouplinggroupforthesystem.SuchEuleriandecouplingschemesoffertwoimportantadvantagesovertheirimpulsivecounterparts:theyareabletoenforcethesamedynamicalsymmetrizationbutwithmorerealisticcontrolresourcesand,atthesametime,theyareintrinsicallytolerantagainstalargeclassofsystematicimplementationerrors.

PACSnumbers:03.67.-a,02.70.-c,03.65.Yz,89.70.+c

2002Dynamicaldecouplingprovidesawell-definedframework foraddressingavarietyofissuesassociatedwiththemanipu-glationofopenquantumsystemsandinteractingquantumsub-uAsystems.Inspiredbycoherentaveragingmethodsinnuclearmagneticresonancespectroscopy[1],andcastincontrol- 9theoretictermsin[2,3],decouplingtechniquesareattract- inggrowinginterestfromthequantumcontrolandquantum1vinformationprocessing(QIP)communities.Significantappli-6cationshaveresultedintheareaofreliableQIP,wherede-5couplinghasbeeninstrumentalinthedevelopmentofquan-0tumerrorsuppressionschemes[3,4,5],withthepotentialfor8noise-tolerantuniversalquantumcomputationondynamically0generatednoiselesssubsystems[6].Inaddition,variantsof20thebasicdecouplingconceptsplayaroleinprotocolsforuni-/versalquantumsimulationofbothclosed-andopen-systemhpdynamics[7,8,9],withimplicationsforencodedsimula--tion[10,11].Inabroadercontext,applicationsofdynamicaltndecouplingtoproblemsthatrangefrominhibitingthedecayaofunstablestates[12],tosuppressingmagneticstatedecoher-uence[13],orreducingheatingeffectsinlineariontraps[14]havebeenrecentlyenvisaged.

:qvFromthepointofviewofimplementation,dynamicalde-iXcouplinghasreliedontheabilityofeffectingsequencesofrarbitrarilystrong,instantaneouscontrolpulses.Thatis,itre-aquiredtheabilitytoimpulsivelyapplyasetofcontrolHamil-tonianswithunboundedstrength(thebang-bang(b.b.)as-sumption[2]).Whileprovidingaconvenientstartingpoint,suchascenariosuffersfrombeingextremelyunrealisticforapplications.Inaphysicalcontrolsetting,additionaldisad-vantagesassociatedwithb.b.decouplingincludethediffi-cultyofsimultaneouslydescribingtheevolutionunderthenatural(drift)Hamiltonianandthecontrolterms,aswellasthepoorspectralselectivityofb.b.pulses,withsubstantialoff-resonanceeffects.Finally,althoughcompensationtech-niquesbasedoncompositerotationsexistforstabilizingcon-trolpulsesagainstoperationalimperfections[15],theyarehardtoreconcilewiththeb.b.framework,whichdoesnoteasilylenditselftoincorporatingrobustnessfeatures.

InthisLetter,weovercometheshortcomingsoftheb.b.formulationbyshowinghowtoimplementdynamicaldecou-plingbasedoncontinuousmodulationofbounded-strengthHamiltonians.IfGisthediscretegroupspecifyingthedesired

b.b.decoupler,thebasicideaistoconstrainthemotionofthecontrolpropagatorduringeachcyclealongapaththatinter-polatesbetweentheelementsofG.UndermildassumptionsonthecontrolHamiltonians,adecouplingprescriptioninduc-ingthesamesymmetrystructureasintheb.b.limitcanbeconstructedbyexploitingEuleriancyclesonaCayleygraphofG.Inadditiontosignificantlyweakeningtherelevantim-plementationrequirements,Euleriandecouplingturnsouttobelargelyinsensitivetocontrolfaults,openingthewaytotherobustdynamicalgenerationofnoise-protectedsubsystems.Decouplingsetting.−LetthetargetsystemSbedefinedonafinite-dimensionalstatespaceHS,andletEnd(HS)bethecorrespondingoperatoralgebra.Thus,HS≃Cd,End(HS)≃Matd(C)forsomed,withd=2nforann-qubitsystem.SmaybecoupledtoanuncontrollableenvironmentE,wherebytheevolutiononthejointstatespaceHS⊗HEisruled󰀈byatotaldriftHamiltonianH0=HS⊗11E+11S⊗HE+αSα⊗Eαforappropriatetracelessnoisegenera-torsSα∈End(HS)[3].Adecouplingproblemisconcernedwithcharacterizingtheeffectiveevolutionsthatcanbegener-atedfromH0viatheapplicationofacontrolfieldHc(t)⊗11EactingonSalone[3].Letthecontrolpropagatorbe

Uc(t)=Texp󰀊−i󰀇tdt′Hc(t′

)󰀋

,(1)

0

with󰀄=1.Inaframethatremovesthecontrolfield,thedy-namicsisgovernedbyatime-dependentHamiltonianH

˜(t)=Uc†(t)H0Uc(t),andtheoverallevolutionintheSchr¨odingerpictureresultsfromthenetpropagator

U(t)=Uc(t)Texp󰀊−i󰀇tdt′H˜(t′

)󰀋

.

(2)0

Assumingthatthecontrolactioniscyclic,Uc(t+Tc)=Uc(t)

forsomecycletimeTc>0andforallt,thestroboscopicdynamicsU(tM=MTc),M∈NtheeffectiveevolutioninducedbyH

˜,canbeidentifiedwith

(t)in(2).First-orderdecouplingaimsatgeneratingthedesiredevolutiontolowestorderinTc,U(tM)=exp(−i

H

(0)

=

1

Whilehigher-ordercorrectionscanbesystematicallyevalu-ated,theapproximation(3)tendstobecomeexactasthefastcontrollimitTc→0isapproached[1,3,5].

Inthesimplestb.b.decouplingsetting,thetime-averagein(3)mapsdirectlyintoagroup-theoreticalaverage.LetGbeadiscretegroupoforder|G|>1,G={gj},j=0,...,|G|−1,actingonHSviaafaithful,unitary,projectiverepresentationµ,µ(G)⊂U(HS).Letimagesofabstractquantitiesunderµbedenotedasµ(gj)=gˆj,andsoforth[16].Thenb.b.decou-plingaccordingtoGisimplementedbyspecifyingUc(t)overeachofthe|G|equalsub-intervalsdefiningacontrolcycle[3]:

Uc󰀁(ℓ−1)∆t+s󰀂

=ˆgℓ−1,s∈[0,∆t),(4)withTc=|G|∆tfor∆t>0,andℓ=1,...,|G|.Theresult-ingcontrolactioncorrespondstoextractingtheG-invariant

componentofH0,

|G|

󰀆

gˆ†jXgˆj,

X∈End(HS),(5)

gj∈G

istheprojectorontothecommutantC󰀄G′ofC󰀄GinEnd(HS)[3,

4].Accordingto(4),Uc(t)jumpsfromgˆℓ−1togˆℓ=

(ˆgℓgˆ†ℓ−1)ˆgℓ−1throughtheapplicationofanarbitrarilystrong,instantaneouscontrolkickattheℓ’thendpointtℓ=ℓ∆t,real-izingtheb.b.pulsepℓ=gˆℓgˆ†

Euleriandynamicaldecoupling.ℓ−1[5].

−WeseekawayforsmoothlysteeringUc(t)fromgˆℓ−1togˆℓbyacontrolactiondistributedalongthewholeℓ’thsub-interval.LetΓ={γλ},λ=1,...,|Γ|beageneratingsetforG.TheCayleygraphG(G,Γ)ofGwithrespecttoΓisthedirectedmultigraphwhoseedgesarecolouredwiththegenerators[17],wherever-texgℓ−1isjoinedtovertexgℓbyanedgeofonlyifgℓg−1

colourλifand

haveℓ−1=γλi.e.,gℓ=γλgℓ−1.Physically,imaginethatwetheabilitytoimplementeachgeneratorγˆλ,bytheapplicationofcontrolHamiltonianshλ(t)over∆t,γˆλ=Texp󰀊

−i

󰀇

∆t

dt′

hλ(t′

),

λ=1,...,|Γ|.(6)

0

󰀋

Thechoiceofhλ(t)isnotunique,allowingforadditionalimplementationflexibility.Onceachoiceismade,thecon-trolactionisdeterminedbyassigningacycletimeandaruleforswitchingtheHamiltonianshλ(t)duringthecyclesub-intervals.Weshowhowausefulruleresultsfromsequen-tiallyimplementinggeneratorssothattheyfollowaEuleriancycleonG(G,Γ).AEuleriancycleisdefinedasacyclethatuseseachedgeexactlyonce[17].BecauseaCayleygraphisregular,italwayspossessesEuleriancycles,whoselengthisnecessarilyL=|G||Γ|[17].

LetaEuleriancyclebeginningattheidentityg0ofGbegivenbythesequenceofedgecoloursused,PE=(pℓ)ℓ,withℓ=1,...,L,andpℓ=γλforsomeλ,foreveryℓ.Notethateachvertexhasexactlyonedepartingedgeofeachcolour,sothatPEdeterminesawelldefinedpath.Wedefine

2

EuleriandecouplingaccordingtoGbylettingTc=L∆tand

byassigningUc(t)asfollows:

Uc󰀁(ℓ−1)∆t+s󰀂

=uℓ(s)Uc󰀁(ℓ−1)∆t󰀂,

(7)wheres∈[0,∆t),anduℓ(s)=Texp(−idecoupling󰀉s

0dt′hℓ(t′)),uℓ(∆t)=pˆℓ,ℓ=1,...,L.Thisprescriptionmeansthatduringtheℓ’thsub-intervalonechoosesasacon-trolHamiltoniantheonethatimplementsthegeneratorγˆλ,withγλcolouringtheedgepℓinPE.TheeffectiveHamilto-nian

1

|Γ|

λ󰀆|Γ|

=1

Theb.b.limitisformallyrecoveredbylettingFtheEulerianapproach,attheexpenseΓ󰀃betheidentitymap.Inoflengtheningthecontrolcyclebyafactorof|Γ|,thesameG-symmetrizationcanbeattainedusingboundedcontrols.Themaximumstrengthsachievableinimplementingthegenera-tors(6)directlyaffectstheminimumattainableTc,andthere-foretheaccuracyoftheaveraging[3].Whiletheoverhead|Γ|dependsonthespecificgroup,itisworthnotingthat,similar

toΠG󰀃[4],QG󰀃satisfiesthepropertythatQG󰀃(X)=QG󰀂/G

0

(X)wheneverG0isanormalsubgroupofGandX∈CalreadyG0-invariant,Eulerian󰀅G0′[18].

Thus,ifthedynamicsisdecou-plingaccordingtoGcanbeaccomplishedbyusingaCayleygraphofthesmallerquotientgroupG/G0.

Robustnessanalysis.-ThefactthatcontrolactionsarenowdistributedalongfinitetimeintervalstranslatesintomajorgainsintermsofresilienceofEulerianschemesagainstimper-fectionsinthecontrolsthemselves.ImaginethatsystematicimplementationerrorsresultinafaultycontrolHamiltonianHc′(t),andpartitionHc′(t)into

Hc′

(t)=Hc(t)+∆Hc(t),

(10)

suchthatHc(t)∈C󰀄GistheintendedcontrolHamiltonian,and

∆Hc(t)istheerrorcomponent.Nowworkinthesameframeusedearlier,whichonlyremovestheidealcontrolpartfrom

theeffectiveHamiltonian.BecauseH(t)=H0+Hc′

(t)=[H0+∆Hc(t)]+Hc(t),thismapstheevolutionunderH0with

thefaultycontrolHwiththeidealcontrol.c′

(t)intotheevolutionunderH0+∆Hc(t)Thus,theneweffectivedynamicsisobtainedbyreplacingH0withH0+∆Hc(t)in(3).

Supposethatthefaultsareproperlycorrelatedwiththeun-derlyingpath,meaningthateverytimeaparticulargeneratorγˆλisimplemented,thesameimperfectionoccursatequiva-lenttemporallocationswithinthesub-interval,regardlessofthepositionofγλalongPE.Then∆Hc((ℓ−1)∆t+s)=∆hλ(s),λbeingthecolouroftheedgethatPEusesduringtheℓ’thsub-interval.Byasimilarcalculationasintheidealcase,thequantumoperationQGQ′󰀃ismodifiedasfollows:G

where󰀃(X)=ΠGQ󰀃(X)+QG󰀃(∆Hc),X∈End(HS),

(11)

G󰀃(∆Hc)canbecomputedasin(8)and(9),butwiththeoperatorXintheintegralreplacedbyonethatdependsonsandλ.Thus,Qtoryover[0,∆t],whichG󰀃(∆Hc)isafunctionalofthefaulthis-characterizestheresidualcontroler-rorsexperiencedbythesystem.Notably,twousefulfeaturesemerge:withoutextraassumptions,suchresidualcontroler-rorsbelongtoCCG,thenall󰀄G′.If,inaddition,∆Hc(t)isitself(asHc(t))

in󰀄controleffectsremaininC󰀄G,andtheresidual

controlerrorsbelongtothecenterZTheeffectsofQCG󰀃(∆Hc)maystill󰀄adverselyG

=C󰀄G∩Cimpact󰀄G′.theper-formanceofthesystem.However,theycanbecompensatedforbyencodingsinappropriatesubsystems[6].LetJ∈J

labeltheirreduciblecomponentsofC󰀄G.ThenHScanberep-resentedas

HS≃⊕JHJ≃⊕JCJ⊗DJ≃⊕JCnJ⊗CdJ,(12)

withnJ,dJ∈N,algebra󰀈

JnJdJ=d,andtheactionofthede-couplinggroupanditscommutantgivenbyCΠ󰀄G≃

⊕J11nJ⊗MatdJ(C),C

subsystemsarenoiselessG󰀄G′≃⊕JMatnJ(C)⊗11dJ,respec-tively.Becauseboth󰀃(Sα)andQandtheirdynamicalG󰀃(∆Hc)areinC󰀄G′,DJ-generationro-bustregardlessofwhether∆Hc(t)belongstoC󰀄Gornot.This

appliesinparticularifGactsirreduciblyonHS,inwhichcase3

arobustimplementationofmaximaldecouplingisachievable

byaveragingoveraniceerrorbasisonCd[3,7].Infact,en-codingintoDJ-subsystemsmaybevaluableeveninsituations

wheretheassumptionthatthecontrolsareinCG󰀄Gcannotbe

met:asQtationofthe󰀃(Sα)∈C󰀄G′,DJ-subsystemsremainunaffectedby

thenoise.Notethatforsuchsubsystems,boththeimplemen-decouplingschemeandtheexecutionofencodedcontroloperationsaretobeeffectedthroughfastmodulationofHamiltoniansalongthecontrolcycle[5,6].

WheneverQG󰀃(∆Hc)originatesfromfaultsinC󰀄G,addi-tionaloptionsareviable.Iftherepresentationµisprimary,

Zinated,Cpression󰀄G

=C11,thenanysystematicerroriseffectivelyelim-andnoencodingisnecessaryaslongasnoisesup-isensured,thatis,ΠG󰀃(Sα)=0forallα.Ifµisnotprimary,thenelementsinthecenterarediagonalovereachirreduciblecomponent.Thus,encodingsintoeitheraHJ-subspaceoraCJ-subsystemareinsensitivetothecontrolfaultsandprotectedagainstthenoisegeneratorifΠmayG󰀃(Sα)∈ZCaswell.Inpractice,choosingaCJ-subsystembees-pecially󰀄G

appealing,becausenotonlyisuniversalencodedcon-trolachievablebyless-demanding,slowapplicationtoniansinC󰀄ofHamil-G′[5],butaddedrobustnessagainsttrolerrorsinC󰀄arbitrarycon-Gisautomaticallyprovided[6].Next,weout-linesomeapplicationsrelevanttoQIP.

Example1:EulerianCarr-Purcelldecouplingonaqubit.-Considerasingledecoheringqubit,{Sα}={σz}[2].ThedecouplinggroupG=Z2={0,1}isrepresentedinU(C2)asGux(󰀃={11,σx}.Thereisonegenerator,γ1=1,henceL=2withnooverheads)=Texp(−i󰀉swithrespecttotheb.b.case.Let0dt′hx(t′

)),foraHamiltonianhx(t)∈C󰀄Grealizingγˆ1=σx(6).Ifhx(t)=f(t)σxforsomef(tanychoicesuchthat|󰀉),

∆t

0dsf(s)|=π/2isacceptable.OnG(G,Γ)choosePE=(γ1,γ1).ThenEuleriandecouplingisaccomplishedbylettingUc(t)=ux(t),fort∈[0,∆t),andUc(t)=ux(s)σxfort∈[∆t,∆t+s),s∈[0,∆t).ByexplicitcalculationofQG󰀃(∆Hc),oneseesthatsystematicer-rorsalongσy,σzproducenoeffect.EliminationofresidualcontrolerrorsinZC󰀄G

requiresusingthefullPauligroup.Example2:EulerianPaulidecouplingonqubits.-LetG󰀃={11,X,Y,Z}bethePaulierrorbasisforaqubit,withX=σx,Z=σz,andY=XZ.ThiscorrespondstoG=Z2×Z2,projectivelyrepresentedinU(C2).Ghastwogenerators,e.g.γ1=(0,1),γ2=(1,0),realizedasγˆ1=X,γˆ2=Z,respectively.AnEulerianpathonG(G,Γ)isPE=(γ1,γ2,γ1,γ2,γ2,γ1,γ2,γ1),oflengthL=8.The

assumptionthatbothhλand∆hλ,λ=1,2,areinCG=Mat2(C).Then(7)results󰀄Gisauto-maticallysatisfied,asC󰀄intoa

robustimplementationofmaximalaveraging,ΠG4nandsince2generatorsareneeded󰀃(σu)=0,u=x,y,z.Fornqubits,G=Zd×Zd,withd=2n.Thus,|G|=foreachqubit,L=n22n+1,causingtheproceduretobe(asintheb.b.limit[3])inefficient.

Example3:Euleriancollectivespin-flipdecoupling.-Fornqubits,letG=Z2×Z2actviathen-foldtensorpower

representationinU((C2)⊗n),whichisprojectivefornodd,

andregularforneven.Foranyn,G=⊗nk=1σx,Z=⊗toGaveragesk=1σz󰀃={11,X,Y,Z},whereX(k)n

(k),andY=XZ.De-couplingaccordingoutarbitrarylinearnoise,

Π󰀃(Sα)=0,Sα∈span{σ(uk)

G}[5].ForEulerianimplemen-tation,thesamepathofExample2maybeused,undertheappropriaterealizationofthecollectivegeneratorsγˆ1=X,γˆ2=Z.EnsuringG-symmetrizationrequiresthatthecontrol

Hamiltoniansh1,2(t)∈C󰀄G.BecausebothCisabelian󰀄G′andZnon-trivial,residualcontrolerrorsariseduetoQCGhence󰀃(∆Hc󰀄)G

are.The

situationissimplerforneven,asC󰀄Gsupport-ingfour(n−2)-dimensionalirreduciblesubspacesHJ.Be-sidesbeingnoiselessinthedecouplinglimitandinsensitivetoarbitrarycontrolerrorsinC󰀄G,encodingintoaHJ-subspace

isfurthermotivatedbythepossibilitytoachieveencodeduni-versalityviaslowapplicationoftwo-bodyHamiltoniansinC󰀄G′[6].Fornodd,bothCJandDJfactorsmayoccur.Leav-ingasidedetailshere,wenotethatDJ-subsystemsusefulifimplementingγˆ1,γˆ2viaHamiltoniansinC󰀄maybe

Gisdiffi-cultinpractice.

Example4:Euleriansymmetricdecoupling.-LetG=Snbethesymmetricgroupofordern,actingonHS≃(C2)⊗n

viagˆj⊗nk=1|ψk󰀛=⊗n

k=1|ψgj(k)󰀛,gj∈Sn.Inparticu-lar,theactioncorrespondingtoatransposition(k−1k),k∈{1,...,n},effectsanexchangegatebetweenqubitsk−1,k,denotedbyswapk−1,k.Symmetricdecouplingal-lows,inprinciple,toengineercollectiveerrormodelsonSstartingfromarbitrarylinearinteractionsbetweenSandE[4,6].AminimalgeneratingsetforSnisgivenbyγ1=(12),γ2=(12...n)i.e.,anadjacenttranspositionandthecyclic

shift,respectively.C=󰀟σk·󰀟σl.󰀅SncontainstheHeisenbergcouplings

h(k,l)Infact,everyoperatorinC󰀅Sncanbe

realizedbyapplyingHeisenbergHamiltonians[19].Focus,forinstance,onS3-symmetrization,whichmayberelevantforinducingcollectivedecoherenceonblocksof3qubits[20].Thenγˆ1=swap1,2andγˆ2=swap1,2swap2,3,withL=12.Becauseexp(−iπh(k,l)/4)=swapk,l,γˆ1canbeimplementedbychoosingh1=a1h(1,2),withstrengtha1=π/4∆t,whileγˆ2canberealizedbyapiecewise-constantHamiltonianh2(t)=a2h(2,3)fort∈[0,∆t/2),h2(t)=a2h(1,2)fort∈[∆t/2,∆t],a2=π/2∆t.AEule-rianpathonG(S3,Γ)isPE=(γ2,γ2,γ2,γ1,γ2,γ1,γ1,γ2,γ1,γ1,γ2,γ1).Euleriandecoupling(7)thenallowsforaro-bustdynamicalgenerationofthesmallestnon-trivialnoiselesssubsystem[21,22],supportedbyafactorDJ≃C2carryingthetwo-dimensionalirreduciblecomponentJ=[21]ofS3.Conclusion.-Wedevelopedanapproachtodynamicalde-couplingthatcombinesthegroup-theoreticalessenceofthe

4

b.b.settingwithgraph-theoreticalcontroldesignaccordingtoEuleriancycles.Besidesallowingforconsiderableleewayinthephysicalimplementationofthebasiccontrolgenerators,Euleriandecouplingeliminatestheneedforunfeasibleb.b.pulsesandnaturallyincorporatesrobustnessagainstrealisticcontrolfaults.Combinedwithquantumcodingtechniques,ourresultssignificantlyimprovetheprospectsthatdynamicaldecouplingbecomesapracticaltoolforreliablycontrollingquantumsystemsandquantuminformation.

SupportedbytheDOE(contractW-7405-ENG-36)andbytheNSA.L.V.alsogratefullyacknowledgessupportfromaJ.R.OppenheimerFellowship.

∗†

Electronicaddress:lviola@lanl.govElectronicaddress:knill@lanl.gov

[1]U.HaeberlenandJ.S.Waugh,Phys.Rev.175,453(1968).[2]L.ViolaandS.Lloyd,Phys.Rev.A58,2733(1998).

[3]L.Viola,E.Knill,andS.Lloyd,Phys.Rev.Lett.82,2417(1999).

[4]P.Zanardi,Phys.Lett.A258,77(1999).

[5]L.Viola,S.Lloyd,andE.Knill,Phys.Rev.Lett.83,4888(1999).

[6]L.Viola,E.Knill,andS.Lloyd,Phys.Rev.Lett.85,3520(2000).

[7]P.Wocjan,M.R¨otteler,D.Janzing,andT.Beth,Q.Inf.Comp.2,133(2002).

[8]L.Dodd,M.A.Nielsen,J.Bremner,andR.T.Thew,Phys.Rev.A65,R040301(2002).

[9]S.LloydandL.Viola,Phys.Rev.A65,R010101(2002).

[10]D.A.LidarandL.-A.Wu,Phys.Rev.Lett.88,017905(2001).[11]L.Viola,Phys.Rev.A66(2002),inpress.

[12]G.S.Agarwal,M.O.Scully,andH.Walther,Phys.Rev.Lett.86,4271(2001).

[13]C.SearchandP.Berman,Phys.Rev.Lett.85,2272(2000).[14]D.VitaliandP.Tombesi,Phys.Rev.A65,012305(2002).[15]R.Tycko,Phys.Rev.Lett.51,775(1983).[16]Foraprojectiverepresentation,g󰀅jgk=ω(gj,gk)ˆ

gjgˆkforsomephaseω(gj,gk),henceG

[17]B.Bollob´as,ModernGraph󰀃isnotagroupingeneral.Theory(Springer-Verlag,NewYork,1998).

[18]Here,thesetofoperatorsG󰀃/G0isdefinedwithrespecttoan

arbitrarychoiceofcosetrepresentativesofG0andisnotneces-sarilyaprojectiverepresentation.

[19]J.Kempe,D.Bacon,D.A.Lidar,andK.B.Whaley,Phys.Rev.A63,042307(2001).

[20]L.-A.WuandD.A.Lidar,Phys.Rev.Lett.88,207902(2002).[21]E.Knill,R.Laflamme,andL.Viola,Phys.Rev.Lett.84,2525(2000).

[22]

L.Viola,E.M.Fortunato,M.A.Pravia,E.Knill,R.Laflamme,andD.G.Cory,Science293,2059(2001).