Superconducting Qubits: A Short Review

1 SuperconductingQubits:A , y, andJ. yDepartment of AppliedPhysics,YaleUniversity, NewHaven,CT06520 Department of Physics,University of California,Santa Barbara,CA93106 October 11, theJosephsontunneljunction,theonlynon-di ssipative, stronglynon-linearcircuitelement availableat low as spinsor atoms,theytendto be wellcoupledto othercircuits,which make themappeallingfromthepoint of , newdesignsof supercon-ductingqubitsbasedonmulti-junct ioncircuitshave solvedtheproblemof thisreviewhow qubitdecoherenceis a ectedby theintrinsicnoiseof thejunctionandwhatcanbe doneto improve implementinga quantumcomputer..32 Basicfeaturesof dissipation:superconductivity .. noise:low temperature.. ,non-dissipative elements:tunneljunctions.

2 Quantumintegratedcircuits.. macroscopicquantummechanics..73 .. thevacuumimpedance:a usefulfeature,nota beingmacroscopic..914 equation.. theparameterdescribingtheJosephsonnon-li nearinductance..125 thehamiltonian.. theparametersof thehamiltonian..136 Why threebasictypes of Josephsonqubits? pairbox .. versussensitivity to noisein controlparameters.. versussensitivity to intrinsicnoise..227 Qubitrelaxationanddecoherence228 Readoutof thereadoutproblem.. andgeneralstrategies.. :tunnelingreadoutwitha DC-SQUIDon-chipam-pli .. withnon-linearinductive readout:the\Quantro-nium"circuit.. uxqubitwithbuilt-inreadout.. readoutschemes.. Couplingsuperconductingqubits3110 Cancoherencebe improvedwithbettermaterials?

3 3211 Concludingremarksandperspectives33A Quantumcircuittheory34B Eigenenergiesandeigenfunctionsof theCooper pairbox36C Relaxationanddecoherenceratesfora nitionof therates.. implementinga quantumcomputerThetheoryof informationhasbeenrevolutionizedby thediscoverythatquan-tumalgorithmscanrun exponentiallyfasterthantheirclassicalcou nterparts,andby theinventionof quantumerror-correctionprotocols[1].Thes efun-damentalbreakthroughshave leadscientistsandengineersto imaginebuildingentirelynovel types of ,theconstructionof acomputerexploitingquantum{ ratherthanclassical{ principlesrepresents aformidablescienti gatesto performquantumcomputation,theymustatthes ametimebe completelydecoupledfromexternalin uences,exceptduringthewrite,controlandre adoutphaseswheninformationmust ow freelyin andoutof culty does notexistfortheclassicalbitsof anordinarycomputer,which each follow stronglyirreversibledynamicsthatdampthen oiseof quantumcomputerhave beenbasedonqubitsconstructedfrommicrosco picdegreesof freedom.}}

4 Spinof eitherelectronsor nuclei,transitiondipolesof eitheratomsor ionsin freedomarenaturallyverywellisolatedfromt heirenvironment, andhencedecohereveryslowly. Themainchallengeof theseimplementationsis enhancingtheinter-qubitcouplingto thelevel ,we willdiscussa radicallydi erent experimentalapproachbasedon\quantuminteg ratedcircuits."Here,qubitsareconstructed fromcol-lectiveelectrodynamicmodesof macroscopicelectricalelements,ratherthan microscopicdegreesof thisapproach is thatthesequbitshave intrinsicallylargeelectromagneticcross-s ections,which impliestheymay be easilycoupledtogetherin complextopologiesviasimplelinearelectric alelements like capacitors,inductors, ,strongcouplingalsopresents a relatedchallenge:is it possibleto isolatetheseelectro-dynamicqubitsfromamb ient parasiticnoisewhileretaininge cient communi-cationchannelsforthewrite,contro l,andreadoperations?

5 Themainpurposeof thisarticleis to reviewtheconsiderableprogressthathasbeen madein thepastfewyearstowardsthisgoal,andto explainhow newideasaboutmethod-ologyandmaterialsare likelyto improve coherenceto ,we mustwarnthereaderthatthisreviewisatypica lin thatit is neitherhistoricalnorexhaustive. Someimportant workshave amplyciteworkby ourown,attheriskof irritatingthereader,butwe wantedtobaseourspeculationsonexperiments whosedetailswe have on3purposenarrowed ourfocus:we adoptthepoint of viewof anengineertryingtodeterminethebeststrate gyforbuildinga reliablemachinegiven obviouslyrunstheriskof presentinga biasedandevenincorrectaccount of recent scienti cresults,sincetheoptimizationof a complexsystemis always anintricateprocesswithbothhiddenpassagew ays hope neverthelessthatthefollowingsectionswill at leaststimulatediscussionsonhow to harnessthephysicsof quantumintegratedcircuitsinto dissipation.

6 SuperconductivityFor anintegratedcircuitto behave quantummechanically, the rstrequirementis theabsenceof cally, allmetallicpartsneedto bemadeoutof a materialthathaszeroresistanceatthequbito peratingtem-peratureandat thequbittransitionfrequency. Thisis essentialin orderforelectronicsignalsto be carriedfromonepartof thechipto anotherwithoutenergyloss{ a necessary(butnotsu cient) temperaturesuperconductorssuch as aluminiumorniobiumareidealforthistask[2] .For thisreason,quantumintegratedcircuitimple mentationshave beennicknamed\superconductingqubits" noise:low temperatureThedegreesof freedomof thequantumintegratedcircuitmustbe cooledtotemperatureswherethetypicalenerg ykTof thermal uctuationsis much lessthattheenergyquantum~!}

7 01associatedwiththetransitionbetweenthes tatesjqubit=0>andjqubit=1>. For reasonswhich willbecomeclearin subsequentsections,thisfrequencyforsuper conductingqubitsis in the5-20 GHzrangeandtherefore,theoperatingtempera turetemperatureTmustbe around20mK(Recallthat1 K correspondsto about20 GHz).Thesetemperaturesmaybe readilyobtainedby coolingthechipwitha ,the\electromagnetictemperature"of thewiresof thecontrolandreadoutportsconnectedto thechipmustalsobe cooledto theselow temperatures,which requirescarefulelectromagnetic low [3] andrequirements [4] forultra-low noise lteringhave beenknownforabout20 ~!01and~!01 ,where is theenergygap1In principle,othercondensedphasesof electrons,such as high-Tcsuperconductivity orthequantumHalle ect,bothintegerandfractional,arepossible andwouldalsoleadtoquantumintegratedcircu itsof thegeneraltype donotpursuethissubjectfurtherthanthisnot e,however,becausedissipationin thesenewphasesis, by far,notas wellunderstood as in thesuperconductingmaterial,onemustusesup erconductingmaterialswitha ,non-dissipative elements.

8 TunneljunctionsQuantumsignalprocessingca nnotbe quantumcircuits,however,thenon-linearele ments mustobey theadditionalrequirement of like PINdiodesorCMOS transistorsarethus forbidden,even if theycouldbe operatedat onlyoneelectronicelement thatis bothnon-linearandnon-dissipativeatarbitr arilylow temperature:thesuperconductingtunneljunc tion2(alsoknownas a Josephsontunneljunction[5]).Asillustrate din , thiscir-cuitelement consistsof a sandwich of two superconductingthin lmsseparatedby an insulatinglayer thatis thinenough(typically 1 nm)to allow latersectionswe willdescribe how thetunnelingof Cooper pairscreatesaninductive pathwithstrongnon-linearity,thus creatingenergylevelssuitablefora typicallyfabricatedfromoxidationof a reli-ablebarriersincetheoxidationprocess is self-terminating[7].

9 Thematerialspropertiesof amorphousaluminumoxide,alumina,make it anattractive partbecauseof itswell-behavedoxide,aluminumis thematerialfromwhich good quality tunneljunctionsaremosteasilyfabricated,a ndit is oftensaidthataluminiumis to superconductingquantumcircuitswhatsilico nis to ectis asubtlephysicale ectinvolvinga combinationof tunnelingandsuperconductiv-ity, thejunctionfabricationprocessis quantumintegratedcircuitsSuperconducting junctionsandwiresarefabricatedusingtechn iquesborrowedfromconventionalintegratedc ircuits3. Quantumcircuitsaretypicallymadeonsilicon wafersusingopticalor electron-beamlithography andthin themselves as a setof micron-sizeor sub-micron-sizecircuitelements (tunneljunctions,capacitors,andinductors )connectedby thechipandelements aresuch that,to a largeextent, theelectrodynamicsof thecircuitcanbe analyzedusingsimpletrans-missionlineequa tionsor even a lumped element thechipis madeby wiresbondedto veryshortsuperconductingweaklink(seefori nstanceRef.)

10 [6])is a alsoa possiblecandidate,providedtheAndreevleve lswouldbe su have toofewexperimentalevidenceforquantume ectsinvolvingthisdevice,we donotdiscussthisotherwiseimportant is worthmentioningthatchipswithtensof thousandsof junctionshave beensuccessfullyfabricatedforthevoltages tandardandfortheJosephsonsignalprocessor s,which areonlyexploitingthespeedof Josephsonelements, ! a)b)Figure1:a) Josephsontunneljunctionmadewithtwo superconductingthin lms;b)Schematicrepresentationof a is representedby a ,many of thedesignconceptsandtoolsof conventionalsemiconductorelectronicscanb e directlyappliedto ,therearestillimportant di theconceptuallevel, conventionalandquantumcircuitsdi erin that,in theformer,thecollective electronicdegreesof freedomsuch as currents andvoltagesareclassicalvariables,whereas in thelatter,thesedegreesof freedommustbe treatedby quantumoperatorswhich moreconcreteway of presentingthisratherabstractdi erenceis to say thata typicalelectricalquantity, such as thechargeon theplatesof a capacitor,canbe thoughtof as a simplenumber in conventionalcircuits,whereasin quantumcircuits.