Cluster state quantum computation in coupled cavity arrays

DimitrisG.Angelakis∗andAlastairKay

CentreforQuantumComputation,DAMTP,CentreforMathematicalSciences,

UniversityofCambridge,WilberforceRoad,CambridgeCB30WA,UK

Measurement-basedquantumcomputationhasrevolutionizedquantuminformationprocessing,andthephysicalsystemswithwhichitcanbeimplemented.Onesimplyneedstheabilitytoprepareaparticularstate,knownastheclusterstate,andsubsequentlytoperformsingle-qubitmeasurementsonit.Nevertheless,ascalableimplementationisyettoberealized.Hereweproposeahybridlight-mattersystemcomprisedofcoupledcavitiesinteractingwithtwolevelsystems.Util-isingthestable,individuallyaddressable,qubitsresultingfromthelocalisedlong-livedatom-photonexcitations,wedemonstratehowtousethenaturalsystemdynamicsto‘weave’thesequbitsintoaclusterstateandproposetheimplementationofquantumalgorithmsemployingjusttworowsofqubits.Finally,webrieflydiscusstheprospectsforexperimentalimplementationusingatoms,quantumdotsorCooperpairboxes.

arXiv:quant-ph/0702133v2 17 Jul 2007Quantumcomputationonclusterstates[1]hasbeenproposedinavarietyofsystems,includinglinearoptics,quantumdots,neutralatomsinopticallattices,andfly-ingatomschemes[2,3,4,5,6,7,8,9,10].Todate,experimentshavebeenperformedusingopticallattices[11],wheretheclusterstatecanbecreated,butthecur-rentlackofindividualaddressingremainsthestumblingblockandlinearoptics[12,13],wherescalabilityremainsaproblemduetotheneedtogeneratetheinitialmany-photonstatefrom,forexample,highordersofthepara-metricdownconversionproecss.Ontheotherhand,therehaverecentlybeentheoreticalandexperimentalbreakthroughsintothepossibilityofdirectcouplingofhighQcavitiesandinachievingstrongcouplingbetweenthecavitymodeandanembeddedtwo-levelsystem.Avarietyoftechnologieshavebeenemployed,namelyfibercoupledmicro-toroidalcavitiesinteractingwithatoms[14,15],arraysofdefectsinphotonicbandgapmateri-als(PBGs)[16,17,18]andsuperconductingqubitscou-pledthroughmicrowavestriplineresonators[19].Thishaspromptedproposalsfortheimplementationofopti-calquantumcomputing[20],theproductionofentangledphotons[21]andtherealizationofMottinsulatingandsuperfluidphases[22,23,24].Hereweproposetheuseofsucharraysfortherealizationofclusterstatequantumcomputation.

SystemDescription:Westartbedescribingthesys-temandshowinghowtoconstructqubitsfromthehy-bridlight-matterexcitations(polaritons).Forsimplicity,wedescribethesystemasalinearchainofNcoupledcavitiesdopedwithtwolevelsystems,althoughthisisreadilyadaptedtothetwo-dimensionalsettingthatwerequire.|g󰀉kand|e󰀉karetheatomicgroundandexcitedstatesatsitek(wehenceforthusetheterm‘atom’tore-fertoanyrelevanttwolevelsystem).TheHamiltoniandescribingthesystemisthesumofthreeterms;HfreeistheHamiltonianforthefreelightanddopantparts,HinttheHamiltoniandescribingtheinternalcouplingofthephotonanddopantinaspecificcavityandHhopfor

thelighthoppingbetweencavities.

H

free

=ωd=g

N󰀁k=1

a†kak

+ω0

󰀁

k

|e󰀉󰀈e|k

(1)(2)(3)

HH

int

N󰀁k=1

(a†k|g󰀉k󰀈e|k+ak|e󰀉k󰀈g|k)

†(a†kak+1+akak+1)

hop

=A

N󰀁k=1

ωdandAarethephotonfrequenciesandhopping

ratesrespectivelyandgisthelight-atomcouplingstrength.TheHfree+HintcomponentoftheHamil-toniancanbediagonalizedinabasisofcombinedpho-tonicandatomicexcitations,calledpolaritons(Fig.1).Thesepolaritonsaredefinedbycreationoperators(±,n)†Pk=|n±󰀉k󰀈g,0|k,wherethepolaritonsofthekthatom-cavity√systemaregivenby|n±󰀉k=(|g,n󰀉k±|e,n−1󰀉k)/n,and|n󰀉kdenotesthen-photonFockstate.Ashasbeenshownelsewhere,apolaritonicMottphaseexistsinthissystemwhereamaximumofoneexcitationpersiteisallowed[22].Thisoriginatesfromtherepulsionduetothephotonblockadeeffect[25,26].InthisMottphase,thesystem’sHamiltoniancanbewrittenintheinteractionpictureas

󰀂N−1†(−,1)†††

HI=Ak=1PkPk+1+PkPk+1,wherePk=Pk

(Fig.1).Asdoubleormoreoccupancyofthesitesispro-†y+xx

hibited,onecanidentifyPkwithσk=σk+iσk,whereσk

y

andσkarethestandardPaulioperators.Thesystem’sHamiltonianthenbecomesthestandardXYmodelofin-teractingspinqubitswithspinup/downcorrespondingtothepresence/absenceofapolariton.

HI=A

N−1󰀁k=1

yyxx

σkσk+1+σkσk+1.

(4)

SomeapplicationsofXYspinchainsinquantuminforma-tionprocessingcanthusbeenimplementedinthissystem

[27].2

FIG.1:Weworkwitha2Darrayofatom-cavitysystems.Whentheatomisonresonancewiththecavity,thegroundstate|g,0󰀄andthefirstexcitedstate|1−󰀄ofthecombinedatom-photon(polaritonic)systemineachsitecanbeusedasqubits.ByapplyingStarkshiftswithcontrolelectrodesorproperlytunedlaserfieldstosetsofqubits(thegatesshownunderthequbits),wedisabletheexchangeHamiltonianofaqubittoallofitsneighbours.ApplyinggatesB,C,DorequallyswitchingoffgateA(part(a))createschainsof3qubitswhichapplycontrolled-phasesandSWAPsbetweenthequbitsateitherendofthechain(indicatedbydashedlines).Repeatingthesequencewiththeother3gatesissufficienttoconnectthe3-chainsandgenerateaclusterstateinparallelacrossthewholedevice.Singlequbitrotationsandmeasurementsaremadebyproperlyapplyinglocalexternalfields,utilizingthefactthatthecavitiescanbewellseparated.

Clusterstategeneration:Thetypicalimplementationofclusterstatequantumcomputingrequiresinitializing√allqubitsina2Dlatticeinthe|+󰀉=(|0󰀉+|1󰀉)/

2A)andthenapply

ameasurementonthemiddle‘mediator’qubit(inthe

σzbasis).Dependingonthemeasurementresult,|0󰀉or|1󰀉,anonlocalgateisgeneratedbetweentheremainingtwoqubits,eitherSWAP.(σz⊗σz).CPorSWAP.CPrespectively[29,30].Inbothcases,thegatesinadditiontotheCPareCliffordoperationswhichcanberecordedandtakenintoaccountduringthemeasurement-basedcomputation.

Oursequencetogeneratetheclusterstateinitiatesbypreparingallqubitsinthe|+󰀉statethroughtheappli-cationofglobalπ/2pulse.Onequarterofthesiteswillbeusedaslogicalqubitsandtherestas“mediators”and“off”qubitsinterchangeably.AllqubitsaddressedbythegatesAtoD(Fig.1)are,bydefault,“off”,therebyisolat-ingallthequbits.Switchingonanyoneofthefourgatesthuscreateschainsof3qubits,whichweusetoenactaCPbetweenpairsofqubits(separatedbyamediatorqubit,whichwaspreviouslyoff).ConsecutiveuseofeachofthegatesAtoDservestoenactaCPgatebetweenaparticularqubitandallofitsnearest-neighbours,andthishappensinparallelacrossthewholedevice.ThisentiresequenceisillustratedinSupplementaryVideo1.Themeasurementsequenceisthenappliedasrequestedbytheclusterstatealgorithm,utilizingthelocalacces-sibilityofthesites(inanyimplementation,thecavity-atomsystemsarewellseparatedcomparedtotheres-olutionoftheexternalfieldusedforaddressingthem)[14,15,16,17,18,19].

InFig.2,wecalculatethefidelityofgenerationofaclusterstateona3x3arrayofcavities.Moresophisti-catedschemeshavethepotentialtofurtherreducetheexperimentalerrors.Forexample,standardHamiltoniansimulationtechniquesallowustonegatethesecondor-derexchangetermduetotheoff-resonancecavities,sim-plybyrepeatedlyapplyingσzgatestoeverysecondon-

FIG.2:Thefidelityofgenerationofaclusterstateona3x3gridofcavities,asthedetuning∆ofthemediatoroff-resonancecavitiesisvaried(inunitsofthehoppingA).Thedashedlineincludespost-selectionongetting|0󰀄outcomeswhenmeasuringoff-resonancequbits.Thegraylinesalsoincorporatespontaneousdecayandcavityleakageof0.05A(dark)and0.08A(light).

resonancetripletthroughouttheevolution.Onemightevenhopethatwecouldusethiscoherenteffecttoen-hancetheschemethroughtheuseof,forexample,op-timalcontroltechniques.Mostoftheerrorsconsideredhere(cavityleakage,spontaneousemissionoftheatom,andon-offdetuningofqubits)arelocaleffects,introduc-inglocalnoise,whichcanultimatelybeaddressedbyfault-toleranttechniques[31].

Implementingalgorithms:Initialexperimentalalgo-rithmicimplementationswithcoupledcavitiescanbeex-pectedtoutilizethemostbasicbuildingblockofourscheme,a3×3gridofcavities,whichallowsustogen-erateafour-qubitclusterstate.Aswiththefour-photonclusterstaterecentlyusedbyWaltheretal.[12],thisclusterstatewouldbesuitablefordemonstratingthepreparationofanarbitraryone-qubitstate,anentan-glinggatebetweentwoqubits,andeventheimplementa-tionofGrover’ssearchalgorithmontwoqubits[12].Forexample,byapplyingthelocalgatesH⊗H⊗σzistheHadamardrotation,weconvertof⊗‘box’σz,whereHclusterthatthe3×3gridpreparesintothe1Dclusterstateof4qubits,whichisgiventheinterpretationofasinglequbit,andmeasurementsonthestateyieldquan-tumgatesonthissinglequbit.Moreover,generationofthisfourqubitclusterstateissimplerthangenerationofanarbitrarilysizedclusterstatebecauseweonlyneedtwocontrolstepsinsteadoffour,therebykeepingusevenfurtherwithinthedecoherencetimeofthesystem.

PerhapsthenextimportantstepwouldthenbetodemonstrateShor’sfactoringalgorithm,thefactoringof15beingthestandarddemonstration.Toimplementasaclusterstatecomputation,thesixcomputationalqubits[33]translateintotherequirementofaclusterstatethatiselevenqubitswide.Hence,weneedanarraywhichis21cavitieswide.Thebreadthoftheclusterstate,whichcorrespondstotimeinthecircuitmodel,isaquantitythatwecantradeagainstthetimetakenforthecompu-tation.Atoneextreme,wecancreatethewholecluster

3

FIG.3:Sequenceforminimisingthenumberofqubitsre-quiredforaclusterstatecomputation.(a)Afterthefirstn−1stepsofthealgorithm,thefirstcolumnofqubitsisini-tialisedinthe|+󰀄state,andthethirdcolumn,withqubitsdenotedby∗,areinthestateofoutputforthefirstn−1stepsofthecomputation.(b)Weusecontrolsequences,bringingmediatorqubitsonresonance,toconvertthe|+󰀄statesintoaclusterstate,andtoentanglethemwiththeoutputqubits.TheSWAPintheentanglingoperationmovestheseoutputqubitstothefirstcolumn.(c)Measurethequbitsofthefirstcolumnascorrespondstothenthstepofthecomputation,andreinitialiseinthe|+󰀄state.Therightmostcolumncor-respondstotheoutput.Thesequencethenrepeats.

stateinonego,withthesimplesetoffourstepsalreadyoutlined,andwebenefitfromthelargedegreeofparal-lelismavailabletous.Thisrequiresa2Dgridofcavitiesofsize21×311[34].Attheotherextreme,agridof21×3cavitiessuffices.Inthiscase,onestartswiththe11×2clusterstate,andperformsonetimestepofmeasurement(i.e.measurethe11qubitsinonecolumn).Theresultremainsintheothercolumn.Wethenrepeattheclus-terstategenerationprocess,reinitialisingthemeasuredqubitsintheclusterstate,andperformingthenexttimestep(Fig.3).Thisrequires156consecutiveentanglingsteps,butthereinitialisingoftheclusterstateaftermea-surementeliminatestheeffectofdecoherenceoverthistimescale.Anycombinationbetweenthesetwoextremesisalsopossible,andisanecessarypropertyofanyscal-ableimplementationofclusterstatecomputationforthesakeofpreventingdecoherence.

Onceinitialclusterstateexperimentshavebeenper-formed,itsimplybecomesaquestionofhowmanycav-itiesonecanreasonablycoupletogether.Alternatively,sincethetwo-qubitgatethatwecangenerateisentan-gling(andhenceuniversalforquantumcomputation),wecanalsoconsiderusingitdirectlytoimplementthecir-cuitmodelofcomputation.Thishasamuchsmallerover-headofqubits,butinsteadrequiresmuchhigherqualitycavities.Forexample,tofactor15wewouldonlyneeda5×3gridofcavitiestogiveussixcomputationalqubits.However,wewouldneedapproximately15consecutiveentanglingsteps(wehaveattemptedtominimisethisnumberbyallowingasmanyofthegatestobeappliedinparallelaspossible,andbyoptimisingtheinitialla-belling√ofeachqubit),hencerequiringatimeoforder15π/(

4

fortheincreasedrunningtime.

Experimentalimplementations:Aspreviouslymen-tioned,therearethreeprimarycandidatetechnologies;fibercoupledmicro-toroidalcavities[14,15],arraysofdefectsinPBGs[16,17,18]andsuperconductingqubitscoupledthroughmicrowavestriplineresonators[19].Inordertoachievetherequiredlimitofnomorethanoneexcitationpersite[22],theratiobetweentheinternalatom-photoncouplingandthehoppingofphotonsdownthechainshouldbeoftheorderofg/A∼102−101(Acanbetunedwhilefabricatingthearraybyadjustingthedis-tancebetweenthecavitiesandgdependsonthetypeofthedopant).Inaddition,thecavity/atomicfrequenciestointernalcouplingratioshouldbeωd,ω0∼104g,105gandthelossesshouldalsobesmall,g/max(κ,γ)∼103,whereκandγarecavityandatom/otherqubitdecayrates.Thepolaritonicstatesunderconsiderationarees-sentiallyunaffectedbydecayforatime10/A(10nsforthetoroidalcaseand100nsformicrowavestriplineres-onators).Whilethedecaytimeof10/Amayseemun-comfortablyclosetothepreparationtimeforacluster√state,

5

anearest-neighbor,2-qubitgatealgorithm.Hence,thepossibilityforsomesmalldegreeofoptimisationinthe

numberofqubitsremains.