Qubit teleportation between non-neighbouring nodes ... - Nature

Future quantum internet applications will derive their power from the ability to share quantum information across the network. Skiptomaincontent Thankyouforvisitingnature.com.YouareusingabrowserversionwithlimitedsupportforCSS.Toobtain thebestexperience,werecommendyouuseamoreuptodatebrowser(orturnoffcompatibilitymodein InternetExplorer).Inthemeantime,toensurecontinuedsupport,wearedisplayingthesitewithoutstyles andJavaScript. Advertisement nature articles article Qubitteleportationbetweennon-neighbouringnodesinaquantumnetwork DownloadPDF DownloadPDF Subjects QuantuminformationQuantumoptics AbstractFuturequantuminternetapplicationswillderivetheirpowerfromtheabilitytosharequantuminformationacrossthenetwork1,2.Quantumteleportationallowsforthereliabletransferofquantuminformationbetweendistantnodes,eveninthepresenceofhighlylossynetworkconnections3.Althoughmanyexperimentaldemonstrationshavebeenperformedondifferentquantumnetworkplatforms4,5,6,7,8,9,10,movingbeyonddirectlyconnectednodeshas,sofar,beenhinderedbythedemandingrequirementsonthepre-sharedremoteentanglement,jointqubitreadoutandcoherencetimes.Herewerealizequantumteleportationbetweenremote,non-neighbouringnodesinaquantumnetwork.Thenetworkusesthreeopticallyconnectednodesbasedonsolid-statespinqubits.Theteleporterispreparedbyestablishingremoteentanglementonthetwolinks,followedbyentanglementswappingonthemiddlenodeandstorageinamemoryqubit.Wedemonstratethat,oncesuccessfulpreparationoftheteleporterisheralded,arbitraryqubitstatescanbeteleportedwithfidelityabovetheclassicalbound,evenwithunitefficiency.Theseresultsareenabledbykeyinnovationsinthequbitreadoutprocedure,activememoryqubitprotectionduringentanglementgenerationandtailoredheraldingthatreducesremoteentanglementinfidelities.Ourworkdemonstratesaprimebuildingblockforfuturequantumnetworksandopensthedoortoexploringteleportation-basedmulti-nodeprotocolsandapplications2,11,12,13. MainQuantumteleportationisthecentralroutineforreliablysendingqubitsacrosslossynetworklinks3,aswellasakeyprimitiveofquantumnetworkprotocolsandapplications2,11,12.Usingateleporterintheformofapre-sharedentangledstate,thequantuminformationistransferredbyperformingajointBell-statemeasurement(BSM)onthesender’spartoftheentangledstateandthequbitstatetobeteleported.ThestateisrecoveredonthereceivingnodebyagateoperationconditionedontheBSMoutcome3.Becausethequantuminformationisnottransmittedbyaphysicalcarrier,theprotocolisinsensitivetolossintheconnectingphotonicchannelsandonintermediatenodes.AdeterministicBSMcombinedwithreal-timefeed-forwardenablesunconditionalteleportation,inwhichstatetransferisachievedeachtimeaqubitstateisinsertedintotheteleporter.Pioneeringexplorationsofquantumteleportationprotocolswereperformedusingphotonicstates4,5,6.Followingthedevelopmentofquantumnetworknodeswithstationaryqubits,remotequbitteleportationwasrealizedbetweentrappedions7,trappedatoms8,10,diamondnitrogen-vacancy(NV)centres9andmemorynodesbasedon atomicensembles14.Althoughfuturequantumnetworkapplicationswillwidelyuseteleportationbetweennon-connectednodesinthenetwork,thedemandingsetofrequirementsonthepre-sharedentanglement,theBSMandthecoherencetimesforenablingreal-timefeed-forwardhas,sofar,preventedtherealizationofteleportationbeyonddirectlyconnectedstationarynetworknodes.Hereweovercomethesechallengesbyasetofkeyinnovationsandachievequbitteleportationbetweennon-neighbouringnetworknodes(seeFig.1a).Ourquantumnetworkconsistsofthreenodesinalineconfiguration,Alice,BobandCharlie.EachnodecontainsaNVcentreindiamond.UsingtheNVelectronicspinasthecommunicationqubit,weareabletogenerateremoteentanglementbetweeneachpairofneighbouringnodes.Inaddition,BobandCharlieeachuseanearby13Cnuclearspinasamemoryqubit.ThestepsoftheteleportationprotocolareshowninFig.1b.Topreparetheteleporter,weuseanentanglementswappingprotocolmediatedbyBob,similartoaquantumrepeaterprotocol15,toestablishentanglementbetweenAliceandCharlie.Oncesuccessfulpreparationoftheteleporterisheralded,theinputqubitstateispreparedonCharlieandfinallyteleportedtoAlice.Fig.1:Teleportingaqubitbetweennon-neighbouringnodesofaquantumnetwork.a,Threenetworknodes,Alice(A),Bob(B)andCharlie(C),areconnectedbymeansofopticalfibrelinks(lines)inalineconfiguration.Eachsetuphasacommunicationqubit(purple)thatenablesentanglementgenerationwithitsneighbouringnode.Furthermore,BobandCharliecontainamemoryqubit(yellow).b,Thestepsoftheteleportationprotocol.(1)WepreparetheteleporterbyestablishingentanglementbetweenAliceandCharlieusinganentanglementswappingprotocolonBob,followedbyswappingthestateatCharlietothememoryqubit.(2)ThequbitstatetobeteleportedispreparedonthecommunicationqubitonCharlie.(3)ABSMisperformedonCharlie’squbitsandtheoutcomeiscommunicatedtoAliceoveraclassicalchannel.Dependentonthisoutcome,Aliceappliesaquantumgatetoobtaintheteleportedqubitstate.FullsizeimageEntanglementfidelityofthenetworklinksAkeyparameterforquantumteleportationisthefidelityofthepre-sharedentangledstatebetweenAliceandCharlie.Aswegeneratethisstatebyentanglementswapping,itsfidelitycanbeincreasedbymitigatingerrorsontheindividuallinks.Ournetworkgeneratesentanglementbetweenneighbouringnodesusingasingle-photonprotocol16,17inanoptical-phase-stabilizedarchitecture18.Thebuildingblockofthisprotocolisaqubit–photonentangledstatecreatedateachnode.Togeneratethisentangledstate,weinitializethecommunicationqubitinasuperpositionstate\(|\psi\rangle=\sqrt{\alpha}|0\rangle+\sqrt{1-\alpha}|1\rangle\)andapplyastate-selectiveopticalpulsethattransfersthepopulationfrom\(|0\rangle\)toanopticallyexcitedstate.Followingspontaneousemission,thequbitstateisentangledwiththephotonnumber(0or1photon).Weperformthisprotocolonbothnodesandinterferetheresonantphotonicstatesonabeamsplitter(Fig.2a).Detectionofasinglephotoninoneoftheoutputportsideallyheraldsthegenerationofanentangledstate\(|\psi\rangle=(|01\rangle\pm|10\rangle)/\sqrt{2},\)inwhichthe±phaseissetbythedetectorthatclicked.Figure2bshowsthejointoutcomesofqubitmeasurementsinthecomputationalbasisafterentanglementisheralded,showingtheexpectedcorrelations.Fig.2:High-fidelityentanglednetworklinks.a,Simplifiedschematicoftheopticallinkusedforgeneratingentanglementbetweenneighbouringnodes.Photonsemittedbythecommunicationqubitsarefilteredbyadichroicmirror(DM)toseparatetheresonant(zero-phononline,ZPL)photons(3%ofemission)fromtheoff-resonant(phonon-sideband,PSB)photons(97%ofemission).Theresonantphotonsaresenttothebeamsplitter(BS);detectionofasinglephotonatoneoftheZPLdetectorsheraldssuccessfulgenerationofanentangledstatebetweenthetwonodes.b,Measuredcorrelationsofthecommunicationqubitsinthecomputationalbasis,conditionedonaheraldingeventontheZPLdetectors.c,Left,histogramsofthePSBphotondetectiontimesonAlice(top)orBob(bottom),conditionedonasimultaneousZPLdetectioninthesameentanglementgenerationattempt.Greylinesshowexpectedcorrelationsonthebasisofaquantum-opticalmodel(see SupplementaryInformation).ThecorrelationsmeasuredintheothermeasurementbasescanbefoundinExtendedDataFig.1.d,Measuredfidelityofthenetworklinks,withoutPSBrejection(left),withPSBrejection(middle)andwithPSBrejectionplusshorteneddetectionwindow(right).ThedarkbluebarsindicatethecorrespondingexpectedfidelityonAlice–Charlieafterentanglementswappingforeachcase(seeMethods).Allerrorbarsrepresentonestandarddeviation.FullsizeimageTheinfidelityofthegeneratedstatehasthreemaincontributions:double\(|0\rangle\)stateoccupancy,doubleopticalexcitationandfinitedistinguishabilityofthephotons18,19.Inthecaseofdouble\(|0\rangle\)stateoccupancy(whichoccurswithprobabilityα),bothcommunicationqubitsareinthe\(|0\rangle\)stateandhaveemittedaphoton.Detectionofoneofthesephotonsleadstofalseheraldingofanentangledstate.Thesecondeffect,doubleexcitation,isdue tothefinitelengthoftheopticalpulsecomparedwiththeopticallifetimeoftheemitter.Thereisafinitechancethatthecommunicationqubitemitsaphotonduringthispulse,issubsequentlyre-excitedandthenemitsanotherphoton,resultinginthequbitstatebeingentangledwithtwophotons.Detectionorlossofthefirstphotondestroysthecoherenceofthequbit–photonentangledstateanddetectionofthesecondphotoncanthenfalselyheraldthegenerationofanentangledstate.Crucially,falseheraldingeventscausedby double\(|0\rangle\)stateoccupancyanddoubleexcitationarebothaccompaniedbyanextraemittedphoton.Therefore,detectionofthisextraphotonallowsforunambiguousidentificationofsucheventsandthusforreal-timerejectionoffalseheraldingsignals.Weimplementthisrejectionschemebymonitoringtheoff-resonantphonon-sideband(PSB)detectionpathonbothsetupsduringandaftertheopticalexcitation(seeFig.2a).Toinvestigatetheeffectofthisscheme,wegenerateentanglementontheindividuallinksandextracttheentanglementheraldingeventsforwhichthePSBmonitoringflaggedthepresenceofanextraphoton.Fortheseevents,weanalysethecorrespondingqubitmeasurementsinthecomputationalbasis(Fig.2c).Weidentifytwoseparateregimes:oneduringtheopticalpulse(purple)andoneaftertheopticalpulse(yellow).WhenaphotonisdetectedonAlice’s(Bob’s)PSBdetectorduringtheopticalpulse,weseethattheoutcome01(10)ismostprobable(purpledatainFig.2c),showingthatonlyonesetupwasinthe\(|0\rangle\)stateandthusthatbothdetectedphotonsoriginatedfromAlice(Bob).ThedetectionofPSBphotonsduringtheopticalpulsethusprimarilyflagsdoubleexcitationerrors.Bycontrast,whenaphotonisdetectedaftertheopticalpulseineitherAlice’sorBob’sPSBdetector,theoutcome00ismostprobable(yellowdatainFig.2c),indicatingthatbothsetupswereinthe\(|0\rangle\)stateandemittedaphoton.PSBphotondetectionaftertheopticalpulsethusflagsthedouble\(|0\rangle\)stateoccupancyerror.WefindsimilarresultstoFig.2cfortheentangledstatesgeneratedontheBob–Charlielink;seeExtendedDataFig.2.Theimprovementinfidelityfromrejectingthesefalseheraldingeventsinourexperimentissetbythecombinedprobabilityofoccurrence(≈9%;see SupplementaryInformation)multipliedbytheprobabilitytoflagthem(givenherebythetotalPSBphotondetectionefficiencyof≈10%).Thethirdmainsourceofinfidelity,thefinitedistinguishability,canarisefromfrequencydetuningsbetweentheemittedphotons20.Whereasmostofthesedetuningsareeliminatedupfrontbythecharge-resonance(CR)checkbeforethestartoftheprotocol(see SupplementaryInformation),thecommunicationqubitsmaystillbesubjecttoasmallamountofspectraldiffusion.Inoursingle-photonprotocol,thisleadstodephasingthatisstrongerforphotonsthataredetectedlaterrelativetotheopticalpulse.Byshorteningourdetectionwindow,wecanincreasethefidelityoftheentangledstateattheexpenseofalowerentanglingrate.Fortheexperimentsbelow(unlessmentionedotherwise),weuseadetectionwindowlengthof15 ns.Figure2dsummarizesthemeasuredimprovementsontheindividuallinksandtheestimatedeffectontheAlice–Charlieentangledstatefidelity.Theincreaseof≈3%isinstrumentalinpushingtheteleportationfidelityabovetheclassicalbound.MemoryqubitcoherenceInthepreparationoftheteleporter,wereliablypreservetheAlice–Bobentangledlinkonthememoryqubit,byabortingthesequenceandstartingoverwhentheBob–Charlieentangledstateisnotheraldedwithinafixednumberofattempts,thetimeout.The13Cmemoryqubitscanbecontrolledwithhighfidelitybymeansofthecommunicationqubit,althoughtheycanbeefficientlydecoupledwhennointeractionisdesired.Recentworkshowedthat,inamagneticfieldof189 mT,entanglementgenerationattemptswiththecommunicationqubitdonotlimitthememorydephasingtime\({T}_{2}^{\ast}\)(ref. 18),openingthedoortosubstantially extendingthememorypreservationtimewithactivecoherenceprotectionfromthespinbath21.Werealizethisprotectionbyintegratingadecouplingπ-pulseonthememoryqubitintotheexperimentalsequencethatfollowsaheraldingevent,whileensuringthatallphasesthatarepickedupowingtotheprobabilisticnatureoftheremoteentanglingprocessarecompensatedinrealtime(Fig.3a).Fig.3:Memoryqubitcoherenceandreadout.a,GatesequenceonBobforentanglementgenerationwiththecommunicationqubitwhilepreservingstatesstoredonthememoryqubit.Entanglementgenerationattemptsarerepeateduntilsuccessorapredeterminedtimeout.Onsuccessinthenthattempt,aphasefeed-forwardisappliedtomaintainthecorrectreferenceframeofthememoryqubit18,followedbyadecouplingpulseonthememoryqubit.ThedecouplingπMpulsecausesaZrotationonthecommunicationqubit.Afterwards,werephasethememoryqubitforthesameamountoftimeasittooktoheraldentanglement(byapplyingqblocksofXY8decouplingsequencesonthecommunicationqubit,inwhichqdependsonthenumberofentanglementattemptsneededn)andweendwithanotherphasefeed-forwardonthememoryqubit,tocompensateforanyphasepickedupduringthisdecoupling.b,Blochvectorlengthofasuperpositionstatestoredonthememoryqubitfordifferentnumberofentanglementattemptsoratime-equivalentwaitelement.Inthecaseofnodecoupling(noπM)onthememoryqubit,thegatesintheyellowshadedboxinaareleftout.Thegreydashedlineindicatesthechosentimeoutof1,000entanglementattempts.c,Gatesequenceforthebasis-alternatingrepetitivereadoutofthememoryqubit.d,Readoutfidelityforeachreadoutrepetition,forstates\(|0\rangle\)and\(|1\rangle\)e,Readoutfidelityofthebasis-alternatingrepetitivereadoutschemefordifferentnumberofreadoutrepetitions.f,Fractionofinconsistentreadoutpatternsfordifferentnumberofreadoutrepetitions.Ind–f,thedashedlinesshowanumericalmodelusingmeasuredparameters.Allerrorbarsrepresentonestandarddeviation.FullsizeimageInFig.3b,wechecktheperformanceofthissequencebystoringasuperpositionstateonthememoryqubitandmeasuringtheBlochvectorlength.Weobservethat,withoutthedecouplingpulse,thedecayoftheBlochvectorlengthisnotalteredbytheentanglementattempts,inlinewithpreviousfindings18.Bycontrast,whenweapplythedecouplingpulse,thedecayissloweddownbymorethanafactorof6,yieldingaN1/edecayconstantof≈5,300entanglementattempts,thehighestnumberreportedsofarfordiamonddevices.Thedifferenceintheshapeofthedecayindicatesthatintrinsicdecoherenceisnolongertheonlylimitingfactor.Theimprovedmemorycoherenceenablesustouseatimeoutof1,000entanglingattempts,morethandoublethatofref. 18,whichdoublestheentanglementswappingrate.MemoryqubitreadoutHigh-fidelitymemoryqubitreadoutisrequiredbothinthepreparationoftheteleporter(atBob)andduringtheteleportationprotocolitself(atCharlie).Thememoryqubitisreadoutbymappingitsstateontothecommunicationqubitusingquantumlogicfollowedbysingle-shotreadoutofthecommunicationqubitusingstate-dependentopticalexcitationanddetection22.Owingtolimitedphotoncollectionefficiency(≈10%)andfinitecyclicityoftheopticaltransition(≈99%),thecommunicationqubitreadoutfidelityisdifferentfor\(|0\rangle\)and\(|1\rangle\)andtheprobabilitythatthecorrectstatewasassignedismuchlargerifoneormorephotonsweredetected(assignedoutcome0)thanifnophotonsweredetected(assignedoutcome1)23.Inpreviouswork,wecircumventedthisissuebyconditioningonobtainingtheoutcome0(ref. 18).However,thisapproachscalesunfavourably,asitforcestheprotocoltoprematurelyabortwithprobability>50%ateachmemoryqubitreadout.Weresolvethischallengebyintroducingabasis-alternatingrepetitivereadoutforthememoryqubit(seeFig.3c).Thekeypointofthisreadoutstrategyis,incontrasttoearlierwork24,toalternatinglymapthecomputationalbasisstatesofthememoryqubittothecommunicationqubitstate\(|0\rangle\)Figure3dshowsthereadoutfidelitiesofthenthreadoutrepetitionforthetwoinitialstatesforthememoryqubitonBob(forCharlie,seeExtendedDataFig.3).Weclearlyobservetheexpectedalternatingpatternowingtotheasymmetryofthecommunicationqubitreadoutfidelities.Notably,thereadoutfidelitydecaysonlyby≈1%perreadout,showingthatthereadoutismostlynon-demolitionandseveralreadoutsarepossiblewithoutlosingthestate.Next,weassignthestateusingthefirstreadoutandcontinuethesequenceonlywhentheconsecutivereadoutsareconsistentwiththefirstreadout.Thesubsequentreadoutsthereforeaddconfidencetotheassignmentinthecaseofconsistentoutcomes,whereascasesofinconsistentoutcomes(whichhaveahigherchanceofindicatinganincorrectassignment)arefilteredout.InFig.3e,weplotthereadoutfidelityresultingfromthisstrategyforuptofivereadouts,withthecorrespondingrejectedfractiondue toinconsistentoutcomesplottedinFig.3f.Weobservethatusingtworeadoutsalreadyeliminatesmostoftheasymmetry,reducingtheaverageinfidelityfrom≈6%tobelow1%.Atthispoint,theremainingobservedinfidelitymainlyresultsfromcasesinwhichthememoryqubitwasflippedduringthefirstreadoutblockbecauseof imperfectmemoryqubitgates.Fortheexperimentsreportedbelow(unlessmentionedotherwise),weusetworeadoutrepetitionstobenefitfromahighaveragereadoutfidelity(Bob:99.2(4)%,Charlie:98.1(4)%)andahighprobabilitytocontinuethesequence(BobandCharlie:≈88%).TeleportingqubitstatesfromCharlietoAliceWithallinnovationsdescribedaboveimplemented,weperformtheprotocolasshowninFig.4a.First,wegenerateentanglementbetweenAliceandBobandstoreBob’spartoftheentangledstateonthememoryqubitusingacompiledSWAPoperation.Second,wegenerateentanglementbetweenBobandCharlie,whilepreservingthefirstentangledstateonthememoryqubitwiththepulsesequenceasdescribedinFig.3a.Next,weperformaBSMonBobfollowedbyaCRcheck.Wecontinuethesequenceifthecommunicationqubitreadoutyieldsoutcome0,thememoryqubitreadoutgivesaconsistentoutcomepatternandtheCRcheckispassed.AtCharlie,weperformaquantumgatethatdependsontheoutcomeoftheBSMandonwhichdetectorsclickedduringthetwo-nodeentanglementgeneration.Next,weswaptheentangledstatetothememoryqubit.Atthispoint,theteleporterisreadyandAliceandCharlieshareanentangledstatewithanestimatedfidelityof0.61.Fig.4:Qubitteleportationbetweennon-neighbouringnetworknodes.a,CircuitdiagramoftheteleportationprotocolusingnotationdefinedinFig.3.m(n)isthenumberofattemptsneededtoheraldentanglementfortheAB(BC)entangledlink.Seethe SupplementaryInformationforthefullcircuitdiagram.b,Teleportedstatefidelitiesforthesixcardinalstatesandtheiraverage(Avg.).Thegreylinesshowtheexpectedfidelitiesfromsimulations.Thedashedlinesinb–drepresenttheclassicalboundof2/3.c,AverageteleportedstatefidelityforthedifferentoutcomesoftheBSMonCharlie.Theright-mostbarshowstheresultingfidelitywhennofeed-forwardoperationonAlicewouldbeapplied.ThenumericalvaluesofthebarplotsshowninbandccanbefoundinExtendedDataTables1and2.d,Averagestatefidelityforaconditionalandanunconditionalteleportation,fordifferentdetectionwindowlengthsofthetwo-nodeentanglementgenerationprocesses.Theblue-bordereddatapointisthesamepointasshowninb.Allerrorbarsrepresentonestandarddeviation.FullsizeimageSubsequently,wegeneratethequbitstatetobeteleported,\(|\psi\rangle,\)onCharlie’scommunicationqubitandruntheteleportationprotocol.First,aBSMisperformedonthecommunicationandmemoryqubitsatCharlie.Withtheexceptionofunconditionalteleportation(discussedbelow),weonlycontinuethesequencewhenweobtaina0outcomeonthecommunicationqubit,whenwehaveaconsistentreadoutpatternonthememoryqubitandwhenCharliepassestheCRcheck.TheoutcomesoftheBSMaresenttoAliceand,byapplyingthecorrespondinggateoperation,weobtain\(|\psi\rangle\)onAlice’sside.Weteleportthesixcardinalstates\((\pm{\rm{X}},\pm{\rm{Y}},\pm{\rm{Z}}),\)whichformanunbiasedset25,andmeasurethefidelityoftheteleportedstatestotheideallypreparedstate(Fig.4b).WefindanaverageteleportedstatefidelityofF = 0.702(11)atanexperimentalrateof1/(117 s).Thisvalueexceedstheclassicalboundof2/3bymorethanthreestandarddeviations,therebyprovingthequantumnatureoftheprotocol.Wenotethatthisvalueprovidesalowerboundtothetrueteleportationfidelity,asthemeasuredfidelityisdecreasedbyerrorsinthepreparationofthequbitstatesatCharlie(estimatedtobe0.5%;see SupplementaryInformation).Thedifferencesinfidelitybetweentheteleportedstatesarisefromaninterplayoferrorsindifferentpartsoftheprotocolthateitheraffectallthreeaxes(depolarizingerrors)oraffectonlytwoaxes(dephasingerrors).Thesedifferencesarequalitativelyreproducedbyourmodel(greybarsinFig.4b).InFig.4c,weplottheteleportationfidelityforeachpossibleoutcomeoftheBSM.Owingtothebasis-alternatingrepetitivereadout,thedependenceonthesecondbit(fromthememoryqubitreadout)issmall,whereasforthefirstbit(communicationqubitreadout),thebestteleportedstatefidelityisachievedforoutcome0,due totheasymmetricreadoutfidelities.Wealsoanalysethecaseinwhichnofeed-forwardisappliedatAlice(see Methods);asexpected,theaveragestatefidelityreducestoavalueconsistentwithafullymixedstate(fidelityF = 0.501(7)),emphasizingthecriticalroleofthefeed-forwardintheteleportationprotocol.Finally,wedemonstratethatthenetworkcanachieveunconditionalteleportationbetweenAliceandCharliebyusingtheBSMinadeterministicfashion.Tothisend,werevisetheprotocolatCharlietoacceptbothcommunicationqubitoutcomes,useallmemoryqubitreadoutpatterns,includingtheinconsistentones,anddisregardtheoutcomeoftheCRcheckaftertheBSM.UsingthisfullydeterministicBSMlowerstheaverageteleportationfidelitybyafewpercent(Fig.4d).Atthesametime,shorteningthedetectionwindowsofthetwo-nodeentanglementgenerationisexpectedtoyieldanimprovementinthefidelity,asdiscussedabove.Wefindthat,indeed,theaverageunconditionalteleportationfidelityincreaseswithshorterwindowlengths,reachingF = 0.688(10)foralengthof7.5 nsandarateof1/(100 s);seeExtendedDataFig.4.Thecurrentquantumnetworkisthusabletoperformteleportationbeyondtheclassicalbound,evenunderthestrictconditionthateverystateinsertedintotheteleporterbetransferred.OutlookInthiswork,wehaverealizedunconditionalqubitteleportationbetweennon-neighbouringnodesinaquantumnetwork.Theinnovationsintroducedhereonmemoryqubitreadoutandprotectionduringentanglementgeneration,aswellasthereal-timerejectionoffalseheraldingsignals,willbeinstrumentalinexploringmorecomplexprotocols2,11,12,13,26.Also,thesemethodscanbereadilytransferredtootherplatforms,suchasthegroupIVcolourcentresindiamond,thevacancy-relatedqubitsinSiCandsinglerare-earthionsinsolids27,28,29,30,31,32,33.Thedevelopmentofanimprovedopticalinterfaceforthecommunicationqubit34willincreaseboththeteleportationprotocolrateandfidelity.Becauseoftheimprovedmemoryqubitperformancereportedhere,thenetworkalreadyoperatesclosetothethresholdatwhichnodescanreliablydeliveraremoteentangledstatewhilepreservingpreviouslystoredquantumstatesintheirmemoryqubits.Withfurtherimprovements,forinstance,byintegratingmulti-pulsememorydecouplingsequences21intotheentanglementgeneration,demonstrationofdeterministicqubitteleportation(withnopre-sharedentangledstate)maycomewithinreach,whichopensthedoortoexploringapplicationsthatcalltheteleportationroutineseveraltimes.Inaddition,futureworkwillfocusonfurtherimprovingthephasestabilizationandextendingthecurrentschemesforuseindeployedfibre35.Finally,byimplementingarecentlyproposedlinklayerprotocol36,qubitteleportationandapplicationsmakinguseoftheteleportationprimitivemaybeexecutedandtestedonthenetworkthroughplatform-independentcontrolsoftware,animportantprerequisiteforalarge-scalefuturenetwork.MethodsExperimentalsetupThebasicsoftheexperimentalsetuparedescribedinref. 18.Inthecurrentexperiment,Charliehasaccesstoacarbon-13nuclearspinthatactsasamemoryqubit.TheparametersusedforthememoryqubitsofBobandCharliecanbefoundinExtendedDataTable3.Furthermore,wehavesetupaclassicalcommunicationchannelbetweenCharlieandAlice,suchthatCharliecandirectlysendtheresultsoftheBSMtoAlice.TemporalselectionofheraldingphotonsToeliminateanyreflectedexcitationlightintheheraldingdetectors,wemakeuseofacross-polarizationschemeandperformtemporalselectionofthedetectedphotonsasdescribedinref. 37.Westartthedetectionwindows4 ns(5 ns)afterthehighestintensitypointoftheexcitationpulse,fortheAB(BC)entangledlink,toensuresufficientsuppressionofexcitationlaserlightinthedetectionwindow.MemoryqubitcoherenceBobWeusethesequencedescribedinFig.3atopreservethestateofthememoryqubitduringentanglementattempts.Tocharacterizethedecouplingsequence,wecompareittothesequenceinwhichwedonotapplythedecouplingpulseonthememoryqubitand/orthesequenceinwhichweidleinsteadofperformingentanglementattempts.Wecharacterizethecoherenceofthememoryqubitbystoringthesixcardinalstates.Weaveragetheresultsfortheeigenstates\((|0\rangle,|1\rangle)\)andsuperpositionstates\((|\pm{\rm{X}}\rangle{\rm{and}}|\pm{\rm{Y}}\rangle).\)InExtendedDataFig.5a,weplottheBlochvectorlength\(b=\sqrt{{b}_{x}^{2}+{b}_{y}^{2}+{b}_{z}^{2}},\)withbitheBlochvectorcomponentindirectioni.Overthemeasuredrange,theeigenstatesshowlittledecay.Thedecayofthesuperpositionstatesisfittedwiththefunction\(f(x)=A{{\rm{e}}}^{-{(x/{N}_{1/{\rm{e}}})}^{n}}.\)ThefittedparameterscanbefoundinExtendedDataFig.5b.TheuseofthedecouplingpulseπMonthememoryqubitincreasestheN1/ebymorethanafactorof6.Moreover,theinitialBlochvectorlengthAishigherwiththeπMpulse.Thisismainlyexplainedbythesecondroundofphasestabilization18inbetweenswappingthestateontothememoryqubitandstartingtheentanglementgenerationprocess.Thephasestabilizationtakes≈350 μsand,duringthistime,thememoryqubitissubjecttointrinsic\({T}_{2}^{\ast}\)dephasing,whichcanbeefficientlydecoupledusingtheπMpulse.CommunicationqubitcoherenceInvariouspartsoftheprotocol,wedecouplethecommunicationqubitsfromthespinbathenvironmenttoextendtheircoherencetime.OnAlice,westartthedecouplingwhenthefirstentangledlinkisestablishedandstopwhentheresultsoftheBSMtoteleportthestatearesentbyCharlie.OnBob,wedecouplethecommunicationqubitwhenthememoryqubitisbeingrephased.OnCharlie,thecommunicationqubitisdecoupledfromthepointthatentanglementwithBobisheraldeduptothepointatwhichBobhasfinishedtheBSM,performedtheCRcheckandhascommunicatedtheresults.AllthesedecouplingtimesaredependentonhowmanyentanglementattemptsareneededtogeneratetheentangledlinkbetweenBobandCharlie.Wecharacterizetheaveragestatefidelitiesfordifferentdecouplingtimes;seeExtendedDataFig.6a.Weinvestigateeigenstatesandsuperpositionstatesseparately.Wefitthefidelitywiththefunction\(f(t)=A{{\rm{e}}}^{-{(t/{\tau}_{{\rm{coh}}})}^{n}}+0.5.\)ThefittedparametersaresummarizedinExtendedDataFig.6b.Foreachsetup,theminimumandmaximumdecouplingtimesusedareindicatedbytheshadedregionsinExtendedDataFig.6a.Theleft-mostborderisthedecouplingtimewhenthefirstentanglementattemptonBobandCharliewouldbesuccessfulandtheright-mostborderiswhenthelastattemptbeforethetimeoutof1,000attemptswouldheraldtheentangledstate.ModeloftheteleportedstateAdetailedmodeloftheteleportedstatecanbefoundathttps://doi.org/10.4121/16645969.Themodelcompriseselementsfromref. 18andisfurtherextendedfortheteleportationprotocol.Wetakethefollowingnoisesourcesintoaccount: ImperfectBellstatesbetweenAliceandBob,andbetweenBobandCharlie. DephasingofthememoryqubitofBobduringentanglementgenerationbetweenBobandCharlie. DepolarizingnoiseonthememoryqubitsofBobandCharlie,owingtoimperfectinitializationandswapgates. ReadouterrorsonthecommunicationqubitsofBobandCharlieandreadouterrorsonthememoryqubitsofBobandCharliewhenusingthebasis-alternatingreadoutscheme,whichresultinincorrectfeed-forwardgateoperationsaftertheBSMs. DepolarizingnoiseonAliceduringthedecouplingsequence. IonizationprobabilityonAlice. AnoverviewoftheinputparametersandtheeffectofthedifferenterrorsourcesaregiveninExtendedDataTable4.Calculationofteleportedstatefidelitywithoutfeed-forwardoperationInFig.4c,weshowthefidelityoftheteleportedstateincasenofeed-forwardoperationswouldhavebeenappliedonAlice.Toextractthisdata,wefollowthesamemethodasinref. 9.Weperformclassicalbitflipsonthemeasurementoutcomestocounteracttheeffectofthefeed-forwardgateoperations(asifthegatewasnotapplied)foreachBSMoutcome.Wedothisforallsixcardinalstatesandcomputetheaveragefidelity.Weassumetheerrorsofthegateinthefeed-forwardoperationstobesmall. Dataavailability Thedatasetsthatsupportthismanuscriptandthesoftwaretoanalysethemareavailableathttps://doi.org/10.4121/16645969. ReferencesKimble,H.J.Thequantuminternet.Nature453,1023–1030(2008).ADS  CAS  Article  GoogleScholar  Wehner,S.,Elkouss,D.&Hanson,R.Quantuminternet:avisionfortheroadahead.Science362,eaam9288(2018).ADS  MathSciNet  Article  GoogleScholar  Bennett,C.H.etal.TeleportinganunknownquantumstateviadualclassicalandEinstein-Podolsky-Rosenchannels.Phys.Rev.Lett.70,1895–1899(1993).ADS  MathSciNet  CAS  Article  GoogleScholar  Bouwmeester,D.etal.Experimentalquantumteleportation.Nature390,575–579(1997).ADS  CAS  Article  GoogleScholar  Boschi,D.,Branca,S.,DeMartini,F.,Hardy,L.&Popescu,S.ExperimentalrealizationofteleportinganunknownpurequantumstateviadualclassicalandEinstein-Podolsky-Rosenchannels.Phys.Rev.Lett.80,1121–1125(1998).ADS  MathSciNet  CAS  Article  GoogleScholar  Furusawa,A.etal.Unconditionalquantumteleportation.Science282,706–709(1998).ADS  CAS  Article  GoogleScholar  Olmschenk,S.etal.Quantumteleportationbetweendistantmatterqubits.Science323,486–489(2009).ADS  CAS  Article  GoogleScholar  Nölleke,C.etal.Efficientteleportationbetweenremotesingle-atomquantummemories.Phys.Rev.Lett.110,140403(2013).ADS  Article  GoogleScholar  Pfaff,W.etal.Unconditionalquantumteleportationbetweendistantsolid-statequantumbits.Science345,532–535(2014).ADS  MathSciNet  CAS  Article  GoogleScholar  Langenfeld,S.etal.Quantumteleportationbetweenremotequbitmemorieswithonlyasinglephotonasaresource.Phys.Rev.Lett.126,130502(2021).ADS  CAS  Article  GoogleScholar  Ben-Or,M.,Crépeau,C.,Gottesman,D.,Hassidim,A.&Smith,A.Securemultipartyquantumcomputationwith(only)astricthonestmajority.InProc.200647thAnnualIEEESymposiumonFoundationsofComputerScience(FOCS’06)249–258(IEEE,2006).Arora,A.S.,Roland,J.&Weis,S.Quantumweakcoinflipping.InProc.51stAnnualACMSymposiumonTheoryofComputing(STOC2019)205–216(ACM,2019).VanMeter,R.QuantumNetworking(Wiley,2014).Bao,X.-H.etal.Quantumteleportationbetweenremoteatomic-ensemblequantummemories.Proc.NatlAcad.Sci.109,20347–20351(2012).ADS  CAS  Article  GoogleScholar  Briegel,H.-J.,Dür,W.,Cirac,J.I.&Zoller,P.Quantumrepeaters:theroleofimperfectlocaloperationsinquantumcommunication.Phys.Rev.Lett.81,5932–5935(1998).ADS  CAS  Article  GoogleScholar  Cabrillo,C.,Cirac,J.I.,García-Fernández,P.&Zoller,P.Creationofentangledstatesofdistantatomsbyinterference.Phys.Rev.A59,1025–1033(1999).ADS  CAS  Article  GoogleScholar  Bose,S.,Knight,P.L.,Plenio,M.B.&Vedral,V.Proposalforteleportationofanatomicstateviacavitydecay.Phys.Rev.Lett.83,5158–5161(1999).ADS  CAS  Article  GoogleScholar  Pompili,M.etal.Realizationofamultinodequantumnetworkofremotesolid-statequbits.Science372,259–264(2021).ADS  CAS  Article  GoogleScholar  Humphreys,P.C.etal.Deterministicdeliveryofremoteentanglementonaquantumnetwork.Nature558,268–273(2018).ADS  CAS  Article  GoogleScholar  Legero,T.,Wilk,T.,Kuhn,A.&Rempe,G.Time-resolvedtwo-photonquantuminterference.Appl.Phys.B77,797–802(2003).ADS  CAS  Article  GoogleScholar  Bradley,C.etal.Aten-qubitsolid-statespinregisterwithquantummemoryuptooneminute.Phys.Rev.X9,031045(2019).CAS  GoogleScholar  Cramer,J.etal.Repeatedquantumerrorcorrectiononacontinuouslyencodedqubitbyreal-timefeedback.Nat.Commun.7,11526(2016).ADS  CAS  Article  GoogleScholar  Robledo,L.etal.High-fidelityprojectiveread-outofasolid-statespinquantumregister.Nature477,574–578(2011).ADS  CAS  Article  GoogleScholar  Jiang,L.etal.Repetitivereadoutofasingleelectronicspinviaquantumlogicwithnuclearspinancillae.Science326,267–272(2009).ADS  CAS  Article  GoogleScholar  vanEnk,S.J.,Lütkenhaus,N.&Kimble,H.J.Experimentalproceduresforentanglementverification.Phys.Rev.A75,052318(2007).ADS  Article  GoogleScholar  Broadbent,A.,Fitzsimons,J.&Kashefi,E.Universalblindquantumcomputation.InProc.200950thAnnualIEEESymposiumonFoundationsofComputerScience517–526(IEEE,2009).Rose,B.C.etal.Observationofanenvironmentallyinsensitivesolid-statespindefectindiamond.Science361,60–63(2018).ADS  CAS  Article  GoogleScholar  Nguyen,C.etal.Quantumnetworknodesbasedondiamondqubitswithanefficientnanophotonicinterface.Phys.Rev.Lett.123,183602(2019).ADS  CAS  Article  GoogleScholar  Trusheim,M.E.etal.Transform-limitedphotonsfromacoherenttin-vacancyspinindiamond.Phys.Rev.Lett.124,023602(2020).ADS  CAS  Article  GoogleScholar  Son,N.T.etal.Developingsiliconcarbideforquantumspintronics.Appl.Phys.Lett.116,190501(2020).ADS  CAS  Article  GoogleScholar  Lukin,D.M.,Guidry,M.A.&Vučković,J.Integratedquantumphotonicswithsiliconcarbide:challengesandprospects.PRXQuantum1,020102(2020).Article  GoogleScholar  Kindem,J.M.etal.Controlandsingle-shotreadoutofanionembeddedinananophotoniccavity.Nature580,201–204(2020).ADS  CAS  Article  GoogleScholar  Chen,S.,Raha,M.,Phenicie,C.M.,Ourari,S.&Thompson,J.D.Parallelsingle-shotmeasurementandcoherentcontrolofsolid-statespinsbelowthediffractionlimit.Science370,592–595(2020).CAS  Article  GoogleScholar  Ruf,M.,Wan,N.H.,Choi,H.,Englund,D.&Hanson,R.Quantumnetworksbasedoncolorcentersindiamond.J.Appl.Phys.130,070901(2021).ADS  CAS  Article  GoogleScholar  Grein,M.E.,Stevens,M.L.,Hardy,N.D.&BenjaminDixon,P.Stabilizationoflong,deployedopticalfiberlinksforquantumnetworks.InProc.2017ConferenceonLasersandElectro-Optics(CLEO2017)1–2(IEEE,2017).Dahlberg,A.etal.Alinklayerprotocolforquantumnetworks.InProc.ACMSpecialInterestGrouponDataCommunication(SIGCOMM’19)159–173,(ACM,2019).Hensen,B.etal.Loophole-freeBellinequalityviolationusingelectronspinsseparatedby1.3kilometres.Nature526,682–686(2015).ADS  CAS  Article  GoogleScholar  DownloadreferencesAcknowledgementsWethankS.Wehner,T.Taminiau,C.BradleyandH.deRiedmattenfordiscussions.WeacknowledgefinancialsupportfromtheEUFlagshiponQuantumTechnologiesthroughtheprojectQuantumInternetAlliance(EUHorizon2020,grantagreementno.820445);fromtheEuropeanResearchCouncil(ERC)throughanERCConsolidatorGrant(grantagreementno.772627toR.H.);fromtheNetherlandsOrganisationforScientificResearch(NWO)throughaVICIgrant(projectno.680-47-624)andtheZwaartekrachtprogramQuantumSoftwareConsortium(projectno.024.003.037/3368).S.B.acknowledgessupportfromanErwin-Schrödingerfellowship(QuantNet,no.J4229-N27)oftheAustrianNationalScienceFoundation(FWF).AuthorinformationAuthornotesS.BaierPresentaddress:InstitutfürExperimentalphysik,UniversitätInnsbruck,Innsbruck,AustriaTheseauthorscontributedequally:S.L.N.Hermans,M.PompiliAuthorsandAffiliationsQuTechandKavliInstituteofNanoscience,DelftUniversityofTechnology,Delft,TheNetherlandsS.L.N.Hermans, M.Pompili, H.K.C.Beukers, S.Baier, J.Borregaard & R.HansonAuthorsS.L.N.HermansViewauthorpublicationsYoucanalsosearchforthisauthorin PubMed GoogleScholarM.PompiliViewauthorpublicationsYoucanalsosearchforthisauthorin PubMed GoogleScholarH.K.C.BeukersViewauthorpublicationsYoucanalsosearchforthisauthorin PubMed GoogleScholarS.BaierViewauthorpublicationsYoucanalsosearchforthisauthorin PubMed GoogleScholarJ.BorregaardViewauthorpublicationsYoucanalsosearchforthisauthorin PubMed GoogleScholarR.HansonViewauthorpublicationsYoucanalsosearchforthisauthorin PubMed GoogleScholarContributionsS.L.N.H.,M.P.andR.H.devisedtheexperiment.S.L.N.H.,M.P.andH.K.C.B.carriedouttheexperimentsandcollectedthedata.S.L.N.H.,M.P.,H.K.C.B.andS.B.preparedtheexperimentalapparatus.J.B.developedthequantum-opticalmodel.S.L.N.H.andR.H.wrotethemainmanuscript,withinputfromallauthors.S.L.N.H.,M.P.andJ.B.wrotethesupplementarymaterials,withinputfromallauthors.S.L.N.H.andM.P.analysedthedataanddiscussedwithallauthors.R.H.supervisedtheresearch.CorrespondingauthorCorrespondenceto R.Hanson.Ethicsdeclarations Competinginterests Theauthorsdeclarenocompetinginterests. Peerreview Peerreviewinformation NaturethanksFlorianKaiserandtheother,anonymous,reviewer(s)fortheircontributiontothepeerreviewofthiswork. Peerreviewerreportsareavailable. AdditionalinformationPublisher’snoteSpringerNatureremainsneutralwithregardtojurisdictionalclaimsinpublishedmapsandinstitutionalaffiliations.ExtendeddatafiguresandtablesExtendedDataFig.1PSB-flaggedcorrelationsAlice–Bob.Top,histogramsofthedetectedPSBphotonsconditionedonasimultaneousZPLdetectionintheentanglementgenerationattempt,forAlice(left)andBob(right).Bottom,correspondingmeasuredcorrelationsinallbases.ThegreybarsintheZbasisrepresentthesimulatedvalues.FortheXandYbases,onewouldexpectaprobabilityof0.25foralloutcomes.Allerrorbarsrepresentonestandarddeviation.ExtendedDataFig.2PSB-flaggedcorrelationsBob–Charlie.Top,histogramsofthedetectedPSBphotonsconditionedonasimultaneousZPLdetectionintheentanglementgenerationattempt,forBob(left)andCharlie(right).Bottom,correspondingmeasuredcorrelationsinallbases.ThegreybarsintheZbasisrepresentthesimulatedvalues.FortheXandYbases,onewouldexpectaprobabilityof0.25foralloutcomes.Allerrorbarsrepresentonestandarddeviation.ExtendedDataFig.3Basis-alternatingrepetitivereadout.Basis-alternatingrepetitive(BAR)readoutresultsforCharlie’smemoryqubit.a,Readoutfidelityforeachreadoutrepetition, forstates \(|0\rangle\)and\(|1\rangle.\)b,ReadoutfidelityoftheBARreadoutschemefordifferentnumberofreadoutrepetitions.c,Fractionofinconsistentreadoutpatternsfordifferentnumberofreadoutrepetitions.Thedashedlinesrepresentanumericalmodelusingmeasuredparameters,whichcanbefoundathttps://doi.org/10.4121/16645969.Allerrorbarsrepresentonestandarddeviation.ExtendedDataFig.4Experimentalrates.Experimentalratesoftheconditionalandunconditionalteleportationprotocolfordifferentdetectionwindowlengthsinthetwo-nodeentanglementgeneration.ExtendedDataFig.5Memoryqubitcoherence.a,CoherenceofBob’smemoryqubitforsuperpositionstates(trianglesandcircles)andeigenstates(squaresanddiamonds).WeperformthesequenceasdescribedinthemaintextwithandwithoutthedecouplingpulseπMonthememoryqubit,thedarkblueandpurplepoints,respectively.Furthermore,weperformthesequencewithawaittimeinsteadofentanglementattemptswith(pinkpoints)andwithout(yellowpoints)thedecouplingpulse.Thegreydashedlineindicatesthetimeoutoftheentanglementgenerationprocessusedintheteleportationprotocol.b,Fittedparametersforthememorycoherencedecayofthesuperpositionstates.Allerrorbarsrepresentonestandarddeviation.ExtendedDataFig.6Communicationqubitcoherence.a,Decouplingofthecommunicationqubits.Theaveragestatefidelityisplottedfordifferentdecouplingtimesforeachsetup.Theshadedarearepresentsthedecouplingtimesusedintheteleportationprotocol.b,Fittedparametersforaveragestatefidelityduringcommunicationqubitdecoupling.Allerrorbarsrepresentonestandarddeviation.ExtendedDataTable1TeleportedstatefidelitiesFullsizetableExtendedDataTable2AverageteleportedstatefidelitiesperBSMoutcomeFullsizetableExtendedDataTable3MemoryqubitcharacteristicsFullsizetableExtendedDataTable4Two-nodeandteleportationsimulationparametersFullsizetableSupplementaryinformation SupplementaryInformationThisfilecontainsSupplementaryText;equations,figures,tablesandreferences.PeerReviewFileRightsandpermissions OpenAccessThisarticleislicensedunderaCreativeCommonsAttribution4.0InternationalLicense,whichpermitsuse,sharing,adaptation,distributionandreproductioninanymediumorformat,aslongasyougiveappropriatecredittotheoriginalauthor(s)andthesource,providealinktotheCreativeCommonslicense,andindicateifchangesweremade.Theimagesorotherthirdpartymaterialinthisarticleareincludedinthearticle’sCreativeCommonslicense,unlessindicatedotherwiseinacreditlinetothematerial.Ifmaterialisnotincludedinthearticle’sCreativeCommonslicenseandyourintendeduseisnotpermittedbystatutoryregulationorexceedsthepermitteduse,youwillneedtoobtainpermissiondirectlyfromthecopyrightholder.Toviewacopyofthislicense,visithttp://creativecommons.org/licenses/by/4.0/. ReprintsandPermissionsAboutthisarticleCitethisarticleHermans,S.L.N.,Pompili,M.,Beukers,H.K.C.etal.Qubitteleportationbetweennon-neighbouringnodesinaquantumnetwork. Nature605,663–668(2022).https://doi.org/10.1038/s41586-022-04697-yDownloadcitationReceived:05October2021Accepted:29March2022Published:25May2022IssueDate:26May2022DOI:https://doi.org/10.1038/s41586-022-04697-ySharethisarticleAnyoneyousharethefollowinglinkwithwillbeabletoreadthiscontent:GetshareablelinkSorry,ashareablelinkisnotcurrentlyavailableforthisarticle.Copytoclipboard ProvidedbytheSpringerNatureSharedItcontent-sharinginitiative CommentsBysubmittingacommentyouagreetoabidebyourTermsandCommunityGuidelines.Ifyoufindsomethingabusiveorthatdoesnotcomplywithourtermsorguidelinespleaseflagitasinappropriate. DownloadPDF AssociatedContent Breakthroughinteleportationfurthersquantumnetworkdevelopment OliverSlatteryYong-SuKim Nature News&Views 25May2022 Advertisement Explorecontent Researcharticles News Opinion ResearchAnalysis Careers Books&Culture Podcasts Videos Currentissue Browseissues Collections Subjects FollowusonFacebook FollowusonTwitter Signupforalerts RSSfeed Aboutthejournal JournalStaff AbouttheEditors JournalInformation Ourpublishingmodels EditorialValuesStatement JournalMetrics Awards Contact Editorialpolicies HistoryofNature Sendanewstip Publishwithus ForAuthors ForReferees Submitmanuscript Search Searcharticlesbysubject,keywordorauthor Showresultsfrom Alljournals Thisjournal Search Advancedsearch Quicklinks Explorearticlesbysubject Findajob Guidetoauthors Editorialpolicies Closebanner Close SignupfortheNatureBriefingnewsletter—whatmattersinscience,freetoyourinboxdaily. Emailaddress Signup IagreemyinformationwillbeprocessedinaccordancewiththeNatureandSpringerNatureLimitedPrivacyPolicy. Closebanner Close Getthemostimportantsciencestoriesoftheday,freeinyourinbox. SignupforNatureBriefing