Controlled-NOT for multiparticle qubits and topological quantum computation based on parity

onparitymeasurements

OdedZilberberg,BerndBraunecker,andDanielLoss

DepartmentofPhysics,UniversityofBasel,Klingelbergstrasse82,4056Basel,Switzerland

(Dated:February1,2008)Wediscussameasurement-basedimplementationofacontrolled-NOT(CNOT)quantumgate.Suchagatehasrecentlybeendiscussedforfreeelectronqubits.Hereweextendthisschemeforqubitsencodedinproductstatesoftwo(ormore)spins-1/2orinequivalentsystems.Thekeytosuchanextensionistofindafeasiblequbit-paritymeter.Wepresentageneralschemeforreducingthisqubit-paritymetertoalocalspin-paritymeasurementperformedontwospins,onefromeachqubit.TwopossiblerealizationsofamultiparticleCNOTgatearefurtherdiscussed:electronspinsindoublequantumdotsinthesinglet-tripletencoding,andν=5/2Isingnon-Abeliananyonsusingtopologicalquantumcomputationbraidingoperationsandnontopologicalchargemeasurements.

PACSnumbers:03.67.Lx,73.21.La,05.30.Pr,85.35.Be

arXiv:0708.1062v2 [cond-mat.mes-hall] 25 Jan 2008I.INTRODUCTION

Single-quantum-bit(qubit)operationsandatwo-qubitgatethatgeneratesentanglementaresufficientforuniver-salquantumcomputation[1].Onesuchtwo-qubitgateisthecontrolled-NOT(CNOT)whichflipsthestateofatargetqubitifthecontrolqubitisinthelogical|1󰀁state.AphysicalimplementationoftheCNOTgatetypicallyrequiresacontroloftheinteractionbetweenthequbits,e.g.forspinqubitssee[2,3,4].However,introducinginteractionsbetweenqubitsinevitablyintro-ducesadditionaldecoherencesourcesandisnotpossi-bleinsomequantumcomputationproposalssuchas,forinstance,inlinear-opticsquantumcomputationduetothefactthatphotonsinteractinanegligibleway.How-ever,Knill,Laflamme,andMilburn(KLM)haveshownthatmeasurementsratherthaninteractionscanprovidethemeanstoimplementaCNOTgateonphotonsus-ingnonunitaryoperations[5].Shortlythereafter,addi-tionalmeasurement-basedapproachesforquantumcom-putationwereproposed[6,7].

TheKLMmodelindeedservedasasteppingstoneforcoherentquantuminformationprocessing,butwasre-strictedtotheunderlyingphysicalsystem,relyingonthebosonicpropertiesofphotons.Attemptstodesignasim-ilarimplementationforfermionicsystemsencounteredsomedifficultiesintheformofano-gotheorem[8,9],whichshowedthatforfermions,single-electronHamil-toniansandsingle-spinmeasurementsaresimulatedeffi-cientlybyclassicalmeans.Thisno-gotheorem,however,wassidesteppedrecentlyinaworkbyBeenakkeretal.[10].Bytakingadvantageoftheadditionalchargedegreeoffreedomofanelectron,atwo-spinparitymeasurementwasproposed.Usingthisparitymeter,ameasurement-basedCNOTgateforfree“flying”electronswasde-signed.Followingthisresult,implementationsofaparitygateforspin[11]andchargequbits[12]havebeenpro-posed.

ThesetupinRef.[10]wasproposedforqubitsen-codedinthespinstatesoffreeelectrons,i.e.theelec-tronspinup(down)isinterpretedasalogical1(0)state.

Manyqubit-encodingschemes,however,encodeaqubitintwostatesfromaHilbertspacelargerthanthetwo-dimensionalspin-1/2Hilbertspace,specifically,fromaproductHilbertspaceoftwo(ormore)two-levelsystems.Anexampleisthesinglet-triplet(S−T0)encoding[13].Forsimplicity,werefertothecomposingparticlesofthiskindofqubitasspins-1/2,yetweemphasizethattheycanhavevariousphysicalorigins.Suchencodingschemesresultfromsystem-dependentconstraints,forinstance,seekingalessnoisyphysicalsystemasinthecaseofelectronspinsindoublequantumdots[13],orduetotopologicalconstraintsinthecaseofν=5/2Ising-typeanyons,wheretwoquasiparticlesformatwo-levelsystemequivalenttoaspin-1/2[14].

Inthispaperwediscussaqubit-paritymeasurement-basedimplementationofaCNOTgateforsuchmultipar-ticlequbits.Theimplementationisadirectextensionoftheschemesproposedin[5,10].Thekeytosuchanex-tensionistofindafeasiblequbit-paritymeasurement.Wepresentageneralschemetoreducethismeasurementtoalocalspin-paritymeasurementofarepresentativespinfromeachqubit.Forconcreteness,wespecificallydiscussqubitsbasedontheS−T0basisandpresentforthiscaseaproofofthelinearityofthemeasurement-basedCNOTgateoperation.Thelinearityproofisre-quiredduetothenonlinearnatureofthemeasurement-basedimplementationofthegateandcanbeusedsimi-larlyforthecaseofRef.[10].WealsoproposeapossiblerealizationofsuchaS−T0CNOTondoublequantumdotsusingarecentlyproposedspin-paritymeter[11].Forν=5/2Ising-typeanyons,ameterequivalenttoaspin-paritymeterinvolvesmeasuringthechargeoffourquasiparticles.Suchmeasurementshavebeenrecentlyproposed[15,16,17,18,19]andfirststepstowardtheirimplementationhavebeenpresented[20,21].InRef.[14]thistypeofparitymeasurementwasinvokedalongsidetopologicalbraidingoperationstoimplementatwo-qubitentanglinggate.Weusethisparitymetertoconstructthemeasurement-basedCNOTgateforthissystem.AcomparisontotheschemeofRef.[14]showsthefollow-ingdifferences:Thepresentschemerequiresonlylocal

braidingbetweentheanyonscomposingaqubitbutalsoadditionalanyonsforanancillaandanadditionalparitymeasurement.TheschemeinRef.[14]isthusmoreeffi-cientinanyonresourcesandusesoneparitymeasurementless,butitrequireslong-rangedanyonbraidingopera-tionsbetweenqubits,whichwillbeexperimentallychal-lenging.

Thepaperisstructuredasfollows:InSec.IIwepresenttheschemeforthequbit-paritymeterusingarepresentativespin-paritymeasurement.TheschemeispresentedfullyfortheS−T0qubitencoding.Wethenex-tendtheresultofRef.[10]andpresentthemeasurement-basedCNOTsetupusingtheS−T0qubit-paritymeter.InSec.IIIwediscusspossibleimplementationsoftheCNOTscheme,focusingontwophysicallyentirelydif-ferentsystems:doublequantumdotsandν=5/2Ising-typeanyons.IntheAppendixweprovethelinearityofthemeasurement-basedCNOTgate.

II.

SCHEMEFORQUBIT-PARITY

MEASUREMENTANDCNOT

Inordertoextendthemeasurement-basedCNOTgateproposedinRef.[10]toamultiparticlequbitencoding,onemustfindawaytomeasurethequbitparityoftwosuchqubits.Weproposeageneralschemeinwhichthequbitsarerotatedto“witness”statessuchthatarep-resentativespinparitymeasurementdemonstratestheirqubitparity.Weillustratethisschemeonaspecifictwo-spinsinglet-tripletqubitencodingwheretwoselectedBellstatesserveasthe√qubit’slogicalstate,i.e.|0󰀁=

|ThisT0󰀁=2.is(an|↑↓󰀁encoding+|↓↑󰀁)/

schemeusedforelectronspinsindouble-quantum-dotsetups[13,22,23,24,25,26,27].Weshowthataspin-paritymeasurementissufficientforaS−T0paritymeteranddetailtheCNOTimplemen-tation.

InordertodemonstratetheequivalencebetweenspinparityandS−T0qubitparity,werotatethequbitstatestowitnessstatesoverwhichaspinparitymea-surementwillmakethedistinctionofqubitparity.Animportantbuildingblockinthisschemeisthesingle-qubitHadamardgateH

ˆ.Appliedtothecomputationalbasis1states{|0󰀁,|1󰀁},ithasthematrixrepresentation2(111−1

),andityieldsfortheS−T0encodingH

ˆ|T0H

ˆ|S󰀁󰀁==|↑↓󰀁|↓↑󰀁,.(1)

Therefore,theleftspinintheright-handsideofEq.(1)canserveasawitnessfortheoriginaltwo-spinstate.Forexample,iftheleftspinisinthe|↑󰀁state,theoriginalprerotatedstatewasa|T0theleftspinsoftworotated󰀁.SHence,−T0thespinparityofqubitparity.IfP

ˆqubitsindicatesthe

sisaspin-paritygate(asusedinRefs.[10,28])weobtainaS−T0qubit-paritygatefromthe

2

a

Hˆ1Hˆ1cPˆsbHˆ2Hˆ2dFigure1:Agatethatusesaspin-paritymeasurementto

measuretheparityofS−T0qubits.ApairofS−T0qubitsentersthegateinarmsaandb.EachofthequbitsistatedbyaHadamardgateH

ˆro-.Thespinparityoftheleftspins(seeEq.(1))fromeachqubitisthenmeasuredintheP

ˆsbox.ThequbitsarerotatedbackbyHadamardgatesandtheparityofthespinsisequivalenttotheparityofthequbits.

operation

P

ˆ=Hˆ1Hˆ2PˆsHˆ1Hˆ2,(2)

whereH

ˆ1,Hˆ2aretheHadamardgatesqubits1and2,andP

ˆoperatingonsmeasuresthespinparitybetweenthetwoleftspinsofqubits1and2.AsketchofthisgateisshowninFig.1.

AsanexamplefortheoperationofP

ˆ,let|ψ󰀁=|T0󰀁1⊗(α|T0󰀁2+β|S󰀁2)beatwo-qubitstate.Once

rotatedbyHadamardgatesthestatebecomes|ψ

˜󰀁=|↑↓󰀁1⊗(α|↑↓󰀁2+β|↓↑󰀁2).Measuringthespinparityofthe

leftspinsineachqubitresultsin|ψ˜{1}󰀁=|↑↓󰀁1⊗|↑↓󰀁2

ifevenspinparityismeasured(ps=1)and|ψ˜|↑↓󰀁{0}1⊗|↓↑󰀁2ifoddspinparityismeasured(ps=0).󰀁Ro-=

tatingthequbitsbyHadamardgatesagainresultsintheprojectedqubitstateswithaqubitparityequivalenttothemeasuredspinparity.

ThefactthatHadamardgatesrotatetowitnessstatesandbackinthisS−T0encodingresultsfromthefactthatthecomputationalstatesareasuperpositionofthetwoproductspinstates{|↑↓󰀁,|↓↑󰀁}withequalampli-tudes.Thus,foran√x-alignedsingle-spinqubitencoding|±󰀁=(|↑󰀁±|↓󰀁)/1/

cinHˆHˆσˆccout

aPˆs,1in=|T0󰀁HˆmeasuredaoutPˆs,2tin

σˆttout

Figure2:Measurement-basedCNOTgateforS−T0qubits.

Theboxesrepresentspin-paritymeasurementsoftheleftspins(seeEq.(1))ofeachqubit.ThreeHadamardgatesrotatethequbitsenteringandleavingthefirstbox.Thein-putoftheCNOTgateconsistsofcontrolandtargetqubitsplusanancillawhichispreparedinthe|T0󰀁state.Thean-cillaismeasuredattheoutputina|S󰀁or|T0󰀁state.Theoutcomeofthismeasurementplusthetwomeasuredspinparitiesdeterminewhichoperatorsσˆc,σˆtonehastoapplyonthecontrolandtargetqubits,respectively,inordertocompletetheCNOToperation:Weapplyonthecontrolqubitσˆc=σˆzifp2=0andσˆc=1ifp2=1.Forthetargetqubit,σˆt=σˆxifp1=1andtheancillaismeasuredinthe|S󰀁state,orifp1=0andtheancillaismeasuredinthe|T0󰀁state.Otherwise,σˆt=1.See[10].

wayforaS−T0CNOTimplementation.Theresult-inggateisshowninFig.2.ThegatecanbeseenasaHadamard-rotatedversionofthegatefromRef.[10]thatoperatesontwo-spinqubitsinsteadoffreeflyingelectronqubits.Inaddition,thegatehasthefollowingadvantagesoverthegatefromRef.[10]:(1)Thean-cillaispreparedinapurecomputationalstateinsteadofasuperpositionofcomputationalstates,and(2)fewerHadamardoperationsarerequired.

Theparityandancillameasurements(seeFig.2)areprojectivenonlinearoperations.Eachmeasurementprojectsthestateontooneoftwopossibleoutcomestates.InFig.3wepresentthe“calculationtree”oftheCNOTgatewherethethreeconsecutivemeasure-mentsleadtoeightpossibleoutcomestates.Withthelasttuningstepofthegate,however,weobtainasingledeterministicresult,i.e.allbrancheshavetheoutcome(uptoaglobalphase),

|ψ󰀁c⊗|ψ󰀁t→αγ|T0󰀁c|T0󰀁t+αδ|T0|S󰀁S󰀁󰀁c|S󰀁t+

βγc|t+βδ|S󰀁c|T0󰀁t,

(3)

where|ψ󰀁controlc=α|T0and󰀁c+β|S󰀁targetcand|ψ󰀁inputstates,t=γ|T0respectively.󰀁t+δ|S󰀁arethetHenceEq.(3)describestheoperationofaCNOTgateonS−T0qubits.

IntheAppendixwefollowthecalculationtreeinFig.3whenthegateinFig.2isappliedtoanarbitrarytwo-qubitstate.TheresultyieldsEq.(3)andprovesthatthegateisindeedaCNOTgateandthatitsoperationislinear.

3

III.POSSIBLEIMPLEMENTATIONS

Wepresentheretwopossibleimplementationsofthe

CNOTgatefortwotypesofsystemsthathavebeenpro-posedforquantumcomputation.Inthefirstpartwediscusshowitmayberealizedondouble-quantum-dotqubits.Inthesecondpartweconsideranimplemen-tationfornon-AbelianIsing-typeanyonsthathavebeenproposedtoexistaselementaryexcitationsinafractionalquantumHallsystemwithfillingfactorν=5/2.

A.

Doublequantumdots

Sincetheintroductionofelectronspinsinquantumdots(QDs)asaplatformforquantuminformationpro-cessing[2],therehasbeenmuchresearchinthisdirection.SeveralproposalsspecificallyfocusonaS−T0qubiten-codingwheretwoelectronsinneighboringQDsformthe|qubitS󰀁andoperations|T0󰀁states.aswellPossibleasaCNOTimplementationsgatebasedonofcontrolsingle-ofthedesignandtheinteractionsinthesystemhavebeendiscussedinthelastfewyears[13,22,23,24,25,26,27].InFig.4weshowthatthemeasurement-basedCNOTgatecanberealizedinsuchsystemsaswell.Thespinparitycanbemeasuredusingarecentlyproposedspin-paritymeter[11].Thismeter,however,islocalandcan-notmeasurespinsindistantQDs,i.e.ifwelabeltheelectronsby1,2(firstqubit)and3,4(secondqubit),therequiredwitnessparityofspins1and3cannotbemea-sured.FromEq.(1)wesee,however,thatmeasuringevenparitybetweenspins2and3isthesameasmea-suringoddparitybetweenspins1and3,andviceversa.Uponthisreinterpretation,theCNOTgateremainsun-changed.WiththesuggestedgeometricarrangementofQDsshowninFig.4,itmayfurtherbepossiblethatasinglespin-paritymeter,couplingalternatelytotheleftorrightQDoftheancilla,issufficientfortheoperation.

B.Ising-typeanyons

Topologicalquantumcomputation(TQC)[29,30,31]proposesaschemeinwhichcoherentquantumcomputa-tionisdonebytopologicaloperationsperformedonnon-Abeliananyons.AphysicalsystemthatmayserveasaplatformforTQCisthetwo-dimensionalelectrongasinthefractionalquantumHallregime.Atfillingfractionν=5/2,localizedelementaryexcitations(quasiparticles)areproposedtohavenon-AbeliananyonstatisticsandaredubbedIsinganyons[32,33,34].

Twosuchquasiparticlesformatwo-levelsystemequiv-alenttoaspin-1/2.However,inRef.[14]itisshownthatduetotopologicalsuperselectionrulesthequbitisen-codedintwoproductstates|0󰀁=|0,0󰀁,|1󰀁=|1,1󰀁fromtheHilbertspaceformedbyfourquasiparticles.Thus,thissystemformsaHilbertspaceequivalenttothatofa

4

ˆcHˆ1PˆcHˆaHˆ2PˆaMσˆcσˆtz=1|ψ1,1,z󰀁p2=1z=0|ψ1,1,1󰀁|ψ1,1,0󰀁˜1,p,z󰀁|ψ2|ψ1,p2,z󰀁p2=0|ψ1,0,z󰀁p1=1|ψi󰀁

˜󰀁|ψp1=0|ψ0,1,z󰀁p2=1˜0,p,z󰀁|ψ2|ψ0,p2,z󰀁z=1|ψ1,0,1󰀁z=0|ψ1,0,0󰀁|ψf󰀁z=1|ψ0,1,1󰀁z=0|ψ0,1,0󰀁p2=0|ψ0,0,z󰀁z=1|ψ0,0,1󰀁z=0|ψ0,0,0󰀁Figure3:Calculationtreeofthemeasurement-basedCNOTgateinFig.2.Thecomputationsplitsinaccordancewiththe

parityandancillameasurements.DuetotheHadamardgaterotations,themeasurementsdonotdestroytheinitialstateandeachpathofthecomputationhasthesameprobabilityofoccurring.Weobtaineightpossibleresultstateswhichwedenoteas|ψp1,p2,z󰀁.IntheAppendixwefollowtheexecutionofthegateandshowthattheresultsineachofthecalculationarmsindeedmergeintoasingleresult|ψf󰀁whichisequaltotheresultoftheCNOToperation.

productHilbertspaceoftwospins-1/2[37].Inaddition,itisshowninRef.[14]thatinordertoimplementuniver-salquantumcomputationonthissystem,nontopologicalparity-likemeasurementsarerequired.Suchmeasure-mentsmaybecarriedoutbyaninterferometricdevicerecentlyproposedin[15,16,17,18,19]andfirststepshavebeentakentowarditsimplementation[20,21].Werefertothesereferencesformoredetails.

Themeasurement-basedCNOTschemecanbeimple-mentedoverthissystemaswell.Theparitymeterhereactsdirectlyonthecomputationalstatessothatnoro-tationpriortotheparitymeasurementisrequired.Ifwelabeltheanyonsformingthefirstqubit1,2,3,4andthoseofthesecondqubit5,6,7,8,theparityoftwoqubitscanbemeasuredbyaninterferometerwhichmeasuresthechargeofthefouradjacentanyons3,4,5,6.Thismea-surementisequivalenttothespin-paritymeasurementoftwoneighboringspins,onefromeachqubit,asdis-cussedinSec.IIIA.TherequiredHadamardrotationsbythemeasurement-basedCNOTscheme[10]canbeim-plementedusingtopologicalbraidingoftheIsinganyons[35,36].Ifweconsiderthequbitformedbytheanyons

π

1,2,3,4,thebraidingofanyons1,2resultsinaei

ˆx4σ[14].Since

ˆ=eiiH

π

4

σˆxiπ

e

Figure4:Double-quantum-dotimplementationsetupforthe

measurement-basedCNOTgate.Adotwithanelectroninitisrepresentedbyanemptycirclecontainingafilledcircle.Theancilladotsaresituatednexttoaspin-paritymeter(proposedinRef.[11]).Inordertomeasuretheparityoftheancillaandcontrolqubits,P1,thespinparityoftherightelectronspinofthecontrolandtheleftelectronspinoftheancillaismeasured.Theparityoftheancillaand

targetqubits,P2,ismeasuredbythespinparityoftherightelectronspinoftheancillaandtheleftelectronspinofthetarget.

1234

iHˆ=

TimeFigure5:Hadamardgateusingbraidingofν=5/2Ising

anyons[35,36].Thegateaddsaglobalπ/2phasewhichcanbeignored.

IV.CONCLUSION

Wehavepresentedageneralschemetomeasurethequbitparityoftwomultiparticlequbitsviaarepresen-tativespin-paritymeasurementinsomerotatedstate.Usingthisqubit-paritymeterwehaveextendedthemeasurement-basedCNOTsetupproposedinRef.[10]toadditionalencodingschemes.Asanexample,wedis-cussedtheS−T0qubitencodingcaseindetail.Inthisencoding,asshowninFig.2,therotationsusedbythequbit-paritymeterledtoaslightlysimplerrotatedsetupoftheCNOTgateascomparedto[10].Wealsousedthe

5

ControlAncillaTarget123456789101112Pˆ1TimePˆ2MˆaFigure6:Measurement-basedCNOTgateimplementedon

ν=5/2Isinganyonqubits.Thecontrol,ancilla,andtargetqubitsareshownfromlefttoright,e.g.thecontrolqubitisrepresentedbyanyons1,2,3,4.TherepresentativeparitymeasurementsareshownbytheP

ˆ“spin”-boxesandthean-cillameasurementbytheboxatthebottom.ThebraidingbetweenthemeasurementsrepresentsHadamardrotationsonthequbits.

S−T0setuptoprovideaproofofthelinearityofthegate(seeFig.3),whichisrequiredastheCNOTisim-plementedbynonunitaryoperations.Asanillustration,wepresentedtwopossibleimplementationsoftheCNOTgate.WehaveproposedapossiblesetupfortheS−T0encoding(seeFig.4).Forν=5/2Ising-typeanyons,theCNOTgatecanbeimplementedwithbraidingoperationsandtheparitymeterproposedin[14](seeFig.6).Incontrasttoasimilargatedescribedin[14],thepresentCNOTrequiresonemoreparitymeasurementandtheadditionalancilla.Butallbraidingoperationsremainstrictlylocalandconfinedwithintheindividualqubits.Bothschemeshavetheirstrengthsbutitisyetunknownwhichisamoreefficientrouteforimplementation.

Acknowledgments

WethankW.A.Coish,S.Bravyi,L.Chirolli,D.Stepa-nenko,andD.Zumb¨uhlforusefuldiscussions.FinancialsupportbytheNCCRNanoscienceandtheSwissNSFisacknowledged.

AppendixA:PROOFOFLINEARITY

Toprovethelinearityofthemeasurement-basedCNOTgateshowninFig.2,wefollowthegateexe-cutionthatisportrayedinFig.3whenthecontrolandtargetqubitsaretakeninitiallytobeinarbitrarystates:

|c|󰀁t󰀁==αγ|0|0󰀁󰀁++βδ||11󰀁󰀁==αγ||T0T󰀁+β|S󰀁(A1)0󰀁+δ|S󰀁

(A2)Theinitialstateoftheinputqubitsplustheancillais:

|ψi󰀁=|c󰀁⊗|a󰀁⊗|t󰀁.

(A3)

Thecalculationsplitsinaccordancewiththeparityandancillameasurements.Weobtaineightoptionalre-sultstateswhichwedenoteas|ψp1,p2,z3.Weprovethattheresultsineachof󰀁theasseencalculationinFig.armsfinallymergeintoasingleresult|ψftotheresultoftheCNOToperation.

󰀁whichisequalAtthefirststep,thecontrolispassedthroughaS−T0Hadamardgate,resultingin

|ψ˜󰀁=󰀁

α2

(|T0󰀁βc+|S󰀁c)+2(|T0󰀁c−|S󰀁c)󰀂⊗|a󰀁⊗|t󰀁.

(A4)

Thefirstparitymeasurementisperformedonthecon-trolandancillaqubits.Topresenttheresultofthespin-paritymeasurement,wefirstwritethecontrolandancilla

intheproductspinbasis{|↑↓󰀁,|↓↑󰀁},

|ψ˜󰀁=[α|↑↓󰀁c+β|↓↑󰀁c

]⊗|↑↓󰀁a+|↓↑󰀁a

2

⊗|t󰀁.(A5)Measurementofthespinparityoftheleftspinsofthecontrolandancillaqubitshastwopossibleoutcomes:|ψ˜1,p2,z󰀁=[α|↑↓󰀁c|↑↓󰀁a+β|↓↑󰀁c|↓↑󰀁a

)]⊗|t󰀁,(A6)|ψ

˜0,p2

,z󰀁=[α|↑↓󰀁c|↓↑󰀁a+β|↓↑󰀁c|↑↓󰀁a

]⊗|t󰀁.(A7)

TheancillaandcontrolqubitsarethenrotatedbyHadamardgates:

|ψ1,p2,z󰀁=[α|T0󰀁c|T0󰀁a+β|S󰀁c|S󰀁a]⊗|t󰀁,(A8)|ψ0,p2,z󰀁=[α|T0󰀁c|S󰀁a+β|S󰀁c|T0󰀁a]⊗|t󰀁.

(A9)

Nowtheancillaandtargetqubitsenteraspin-paritymeasurement.Oncemorewewritetheirstatesintheproductspinbasis:

|ψ1,p2,z󰀁=󰀃α|T0󰀃

󰀁c|↑↓󰀁a+|↓↑󰀁a2+β|S󰀁c

|↑↓󰀁a−2

|↓↑󰀁a

󰀄

⊗γ

|↑↓󰀁t+|↓↑󰀁t

+|↑↓󰀁t−|↓↑󰀁t󰀄,(A10)|ψ,p2,z󰀁=󰀃2δ2

α|T0|↑↓󰀁a−|↓↑󰀁a|↑↓󰀁a+|↓↑󰀁0a

󰀃󰀁c2+β|S󰀁c

2

󰀄

⊗γ|↑↓󰀁t+|↓↑󰀁t2+δ

|↑↓󰀁t−2

|↓↑󰀁t󰀄.(A11)6

Measuringthespinparityoftheleftspinsofthean-cillaandtargetqubitsagainsplitstheresultsetintotwopossiblebranches:|ψαγ

1,1,z󰀁=

2[|T0󰀁c|↑↓󰀁a|↑↓󰀁t+|T0󰀁c|↓↑󰀁a|↓↑󰀁t]+αδ

2[|T0󰀁c|↑↓󰀁a|↑↓󰀁t−|T0󰀁c|↓↑󰀁a|↓↑󰀁t]+βγ

2[|S󰀁c|↑↓󰀁a|↑↓󰀁t−|S󰀁c|↓↑󰀁a|↓↑󰀁t]+βδ

2

[|S󰀁c|↑↓󰀁a|↑↓󰀁t+|S󰀁c|↓↑󰀁a|↓↑󰀁t],(A12)|ψαγ

1,0,z󰀁=

2[|T0󰀁c|↑↓󰀁a|↓↑󰀁t+|T0󰀁c|↓↑󰀁a|↑↓󰀁t]−αδ

2[|T0󰀁c|↑↓󰀁a|↓↑󰀁t−|T0󰀁c|↓↑󰀁a|↑↓󰀁t]+βγ

2[|S󰀁c|↑↓󰀁a|↓↑󰀁t−|S󰀁c|↓↑󰀁a|↑↓󰀁t]−βδ

2

[|S󰀁c|↑↓󰀁a|↓↑󰀁t+|S󰀁c|↓↑󰀁a|↑↓󰀁t],(A13)|ψαγ

0,1,z󰀁=

2[|T0󰀁c|↑↓󰀁a|↑↓󰀁t−|T0󰀁c|↓↑󰀁a|↓↑󰀁t]+αδ

2[|T0󰀁c|↑↓󰀁a|↑↓󰀁t+|T0󰀁c|↓↑󰀁a|↓↑󰀁t]+βγ

2[|S󰀁c|↑↓󰀁a|↑↓󰀁t+|S󰀁c|↓↑󰀁a|↓↑󰀁t]+βδ

2

[|S󰀁c|↑↓󰀁a|↑↓󰀁t−|S󰀁c|↓↑󰀁a|↓↑󰀁t],(A14)|ψαγ

0,0,z󰀁=

2[|T0󰀁c|↑↓󰀁a|↓↑󰀁t−|T0󰀁c|↓↑󰀁a|↑↓󰀁t]−αδ

2[|T0󰀁c|↑↓󰀁a|↓↑󰀁t+|T0󰀁c|↓↑󰀁a|↑↓󰀁t]+βγ

2[|S󰀁c|↑↓󰀁a|↓↑󰀁t+|S󰀁c|↓↑󰀁a|↑↓󰀁t]−βδ

2

[|S󰀁c|↑↓󰀁a|↓↑󰀁t−|S󰀁c|↓↑󰀁a|↑↓󰀁t].(A15)Theancillaandtargetqubitscanberewritteninthe

S−T0basisusingthefollowingrelations:

|↑↓󰀁=

1

2[|T0󰀁+|S󰀁]|↓↑󰀁=1

2[|T0󰀁−|S󰀁]

(A16)

SubstitutingEq.(A16)intoEqs.(A12)-(A15)and

7

rewritingtheexpressionsweobtain:

αγ

[|T0󰀁c|T0󰀁a|T0󰀁t+|T0󰀁c|S󰀁a|S󰀁t]+|ψ1,1,z󰀁=2αδ

[|T0󰀁c|S󰀁a|T0󰀁t+|T0󰀁c|T0󰀁a|S󰀁t]+2βγ

[|S󰀁c|S󰀁a|T0󰀁t+|S󰀁c|T0󰀁a|S󰀁t]+2βδ

[|S󰀁c|T0󰀁a|T0󰀁t+|S󰀁c|S󰀁a|S󰀁t],(A17)2|ψ1,0,z󰀁=

αγ

[|T0󰀁c|T0󰀁a|T0󰀁t−|T0󰀁c|S󰀁a|S󰀁t]+2αδ

[|T0󰀁c|T0󰀁a|S󰀁t−|T0󰀁c|S󰀁a|T0󰀁t]+2βγ

[|S󰀁c|S󰀁a|T0󰀁t−|S󰀁c|T0󰀁a|S󰀁t]+2βδ

[|S󰀁c|S󰀁a|S󰀁t−|S󰀁c|T0󰀁a|T0󰀁t],(A18)2

|ψ1,1,0󰀁=αγ|T0󰀁c|T0󰀁t+αδ|T0󰀁c|S󰀁t+

βγ|S󰀁c|S󰀁t+βδ|S󰀁c|T0󰀁t,

(A22)

|ψ1,0,1󰀁=−αγ|T0󰀁c|S󰀁t−αδ|T0󰀁c|T0󰀁t+

βγ|S󰀁c|T0󰀁t+βδ|S󰀁c|S󰀁t,(A23)|ψ1,0,0󰀁=αγ|T0󰀁c|T0󰀁t+αδ|T0󰀁c|S󰀁t−βγ|S󰀁c|S󰀁t−βδ|S󰀁c|T0󰀁t,

|ψ0,1,1󰀁=αγ|T0󰀁c|T0󰀁t+αδ|T0󰀁c|S󰀁t+

βγ|S󰀁c|S󰀁t+βδ|S󰀁c|T0󰀁t,|ψ0,1,0󰀁=αγ|T0󰀁c|S󰀁t+αδ|T0󰀁c|T0󰀁t+

βγ|S󰀁c|T0󰀁t+βδ|S󰀁c|S󰀁t,|ψ0,0,1󰀁=αγ|T0󰀁c|T0󰀁t+αδ|T0󰀁c|S󰀁t−

βγ|S󰀁c|S󰀁t−βδ|S󰀁c|T0󰀁t,

(A24)(A25)

(A26)

αγ

|ψ0,1,z󰀁=[|T0󰀁c|S󰀁a|T0󰀁t+|T0󰀁c|T0󰀁a|S󰀁t]+

2αδ

[|T0󰀁c|T0󰀁a|T0󰀁t+|T0󰀁c|S󰀁a|S󰀁t]+2βγ

[|S󰀁c|T0󰀁a|T0󰀁t+|S󰀁c|S󰀁a|S󰀁t]+2βδ

[|S󰀁c|S󰀁a|T0󰀁t+|S󰀁c|T0󰀁a|S󰀁t],(A19)2|ψ0,0,z󰀁=

αγ

[|T0󰀁c|S󰀁a|T0󰀁t−|T0󰀁c|T0󰀁a|S󰀁t]+2αδ

[|T0󰀁c|S󰀁a|S󰀁t−|T0󰀁c|T0󰀁a|T0󰀁t]+2βγ

[|S󰀁c|T0󰀁a|T0󰀁t−|S󰀁c|S󰀁a|S󰀁t]+2βδ

[|S󰀁c|T0󰀁a|S󰀁t−|S󰀁c|S󰀁a|T0󰀁t].(A20)2

(A27)

|ψ0,0,0󰀁=−αγ|T0󰀁c|S󰀁t−αδ|T0󰀁c|T0󰀁t+

βγ|S󰀁c|T0󰀁t+βδ|S󰀁c|S󰀁t.(A28)

Applyingthegatesσˆc,σˆtonthecontrolandtarget

qubitsinaccordancewiththeresultsoftheparityandancillameasurements(asshowninFig.2)givesthesamestateinalleightcomputationbranches(uptoaglobalphase):

|ψf󰀁=αγ|T0󰀁c|T0󰀁t+αδ|T0󰀁c|S󰀁t+

βγ|S󰀁c|S󰀁t+βδ|S󰀁c|T0󰀁t.

(A29)

UnderthequbitencodingthisstateisindeedtheresultoftheCNOTgate:

|ψf󰀁=αγ|0󰀁c|0󰀁t+αδ|0󰀁c|1󰀁t+

βγ|1󰀁c|1󰀁t+βδ|1󰀁c|0󰀁t.

Thisprovesthelinearityofthegate.

Measuringtheancillainasingletortripletresultsinthefinaleightstates:

|ψ1,1,1󰀁=αγ|T0󰀁c|S󰀁t+αδ|T0󰀁c|T0󰀁t+

βγ|S󰀁c|T0󰀁t+βδ|S󰀁c|S󰀁t,

(A21)

(A30)

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Analternativequbitencodingschemetothefour-quasiparticlequbitistoencodethequbitinthreequasi-particles,forwhichaCNOTgatehasbeenproposedin[35].