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4airmoleculesbouncingoffthephysicalsystemsthatrepres ainducedbyuthrough ahadamardiscase1 ahadamard agladz aggreg 2ageneralstateofnqubitswillbeasuperpositionofthese2 aftersuchameasure affect aerogel Jadvantageofthepossibilitiesmadeavailablebythequantummechanicalbehaviorofit advantage adsorb adoptinganotationmuchbelov :adjointofaproductistheproductoftheadjointsintheoppositeord Nadifferentstatisticaldistributionofreadingsdependingonwhetheralicedidordidnot actsonlyonthesecondandfourth actsonlyonalice actsastheswapoper actsasaqubitnumberoperatoron 6actionatadistancewhichcouldactuallybeusedtosendmessage actingonthevect -actedonbyacorrespondinglyrichersetoftransform &aclassicalcomputeroperatesonstringsof0 *aclassicalbitisaprettysicklyspecimenofatwo acid achievethesameresult achiev Aaccumulateenoughstatisticstogetagoodestimateofatleastthemagnitude Eaccountforthebehaviorofmatterandradiationattheatomicandsubatomiclevel account accordingtothefollowingrule 'accordingtoourhypothesisbobwouldhavegot accordingto abouttheaxisnthroughtheangleθ ableon abhai aashish aaron aandbbetwothre a15 =1wecanuse !=1foranyunitvectornasaspecialcase =1andu =1andthest =1a1 =0then =0soalice =0caseofthemoregeneralrule 525 508 45-6 311 30-6 30,31 224 2004 1⊗1 1|1 1|0 1st 51andthethreepaulimatricesarelinearlyindependentinthe4 1andthescalar1 154 126 125 124 123 122 121 120 12-1 119 118 117 116 115 114 113 112 111 11001 110 109 108 107 106 105 104 103 102 101thatthethre 101 0|1 0|0 0or1 00-2 00-12 00-1 .1ofquantumcomputationand -681 -4he -1homomorphismisfromthegroupsu -1homomorph -10 ,2002 ,2000 ,01100 flipsy⇐ficientlyshorttime⊗1⊗1−2|11,−1valuesofylinearlyindependentunderbitwise→0,1|001101zallanticommutexonlytermwithiwithuunitaryandthereforelinearwithprobwithhighprobabilityone witheachx willfindnwherexˆisqubitnumberoperwhereeachunitarygatev$whatyoucoulddoifyouhadaquantumcomputwhatisthevalueofa weneedtoimplweek2week1valuesofxfornonuusuallytaketobeoftheformunitarytransformationsu umparallelumfouriertransformissuperfastumfouriertransformumcomput umcircuit ufromcontroluandcctwocture3ture2ture1toffolitowithinaconstantphase tothesuperfastquantumfouriertran tonearlyalong toinputregisttoharmlessphasetogetctogetaresultthatitochangeunknownshiftxthqubitthesineoftheangleθbetweenthesimplestpossiblefunctionfthesi.thesetofsimultaneouseigenstatesofallthequbitσthesetofeigenv?thesetofallnumberslessthannthatarecoprimetonformsafinitegroupg!thesestatesarealsoeigenstatesofσ%theseoperatorsactingonthesinglestate9theseobservablescharItheseelementaryfactsprovidethebasisforasecureencryptionschemethatreliesonthesamesubgroupofg thesameattherightsideof0therewasnothingterriblyinterestingtocalculateand8therewasnoconceivablepracticalwaytogoaboutdoingthecalcul.theresultwillbeasuperpositionofstatesoftheformtheresultofapplyingv0theresultofaddingcorrespondingbitsofxandymodulo2theremaining21corruptedtermsin#thereisnowayforaclassicalcomputertothereisatleastonesolutiondto9thereisasimpleextensionoftheeuclideanalgorithmthatdoesthithereisanintegermforwhich0thereisaclashinwhatfollowsbetweenstandardangular thereisa5thereitheuseofquantumcomputationincodebreakingisdescribedinsectioniv#theunitarytransformationuisgivenbyu)theunitarytransformationthattakesthetwost&theunitarytransformationsvandwsatisfyvtheunitaryextensionofuthetwostatesareorthogonthetwoorthonormalstthetwo thetrickisthiAthetransformationoftheinitialstatebroughtaboutbyaquantumcomputerithetotalangularmomentumthethreeoperatorszthethrethethirdisequivalentto%thetelephonecompanyinitsmanymanifesta thetargetqubBthesymmetryisreflectedinthefactthatthealternativechoiceoffieldsh thesuperposit thesumoverxinthesubjectofsectioniv9thesubgroupgeneratedbyanelementisthesetofallpowersoftheel thesubgroupgeneratedbyacontainsathestructureofsh.thestructureofeachencodedqubitremainsoftheformthestructureofthestatethespellingisabominthespecialonezbeingsoughtweshallneedthisingrov(weshallcomeuponanotherexampleinourexamin wenowletuwellwelcomewelcom weinstockweinitiallypreparewehavefoundamultiplwehaveweekliweekendweek wednesdai'wecouldestablishthisintwoevaluationsoffwecouldcorrecttheerr7wecanuseourquantumcomputerintheusualwaytoconstructthest(wecanrealizethisbysandwichingbetweentwonwecanlearnasingl6wecanexploitthelinearityofthequantumimplementationofthwebmastweb7weassumetheinteractionsofeachqubitwithitsenvironmentare*weassumethatahasnofactorsincommonwithnsinc!wearetoldthatthefunctionistwotoonweapplyhway wavefunctwatson wastodiscovwasn washingtonwasanothereminentskeptwarwantwallvorticevoicvividlivitavisitvisionvinaiviiviewofthepracticalfeasibilityofviewvictoriviavgateverynearlyequaltoverticversionveritiveriverbalvenervectorysatisfi#vectorwhosecomponentsare0or1andletf.vectorsxtoacquireenoughinformationtodetermineavectorintotheinnerproductavaziraniproblemvariouvarietivariablvapvanishunlessyisamultipleof2vanish valuesoffvalueofysatisfivaluvaliduvvww uuttuummnnuutttuuummmnnn uuaannttuummutterliusual;useofancillaryqubitstostoretheresultsofthediagnosticmeasure*usefulinformationhastobeextractedfromthestusurgentupwardupsideupperuponupisnotexponentiupdateuoutof3unusuuntitluntilitproducestheoutput1until unrestrict unreservedliunlikeshunlikeunlessunknownunixunivers7unitarytransformationvhasaunitarysquarerootusatisfyingu-unitarytransformationthatconstitutesthecomput*unitarytransformationsappliedtosinglequbitunitarytransformunitarygunitariuniquuninformuniformunifiunhotunfortununexpl undisturb undifferenti undertheact understood understandundermultiplicationmodulon undermine undergraduunderuncorrelatedstateoftheenviron unconvent uncontroversi uncommonliuncharacteristunchangedandtakunchang uncertaintiunambiguunabl ultrasound ultrafastugateoutofthreeon ugatecanbebuiltupoutoftwocontrolugateuesrevealswhichtermucla uappliesutoonequbitifandonlyifthuaaannntttuuummmtypictypetwothingsdistinguishquantumbit twoprimefact2twonumbersaresaidtobecoprimeiftheyhavenocommonfact twocontroltwotwentiturn+tumcomputationdemonstratesthetruthofthemani tumcomputtumttuueessddaayitttuuueeesssdddaaayyitrytruthtrulitruetroubltrophiItroglodytejournalslikesciencewhichhindertheadvancementofsciencebyprohibittrivial triumphantlitriptriggertrickitricktritreachertraveltrap transmiss translation translate transitionJtransformovertheclassicalfastfouriertransformcomeswiththeusualcatchthatonetransformedintotheorthogonalsttransformationxHtransformationwhatevercanbeapproximatedtoarbitrarilyhighprecisionbyasmal8transformationvtothethirdqubitonlyifthefirsttwoareboth1JtransformationsarelinearandthereforewhenappliedtoaqubitorcollectionofqubitMtransformationsareinvertibleitisnecessarythatthefunctionfactuallytaketheintegtransformations1transformationonn7transformationappliedtothecorruptedqubitwilluncorruptit transformtransduc traditiontracktracetofindasolutionto towithinround!towithinarotationonthesecondqubit$towithinaconstantmultiplicativephasetoward touseindecodtoughtotheregisterontheleft totheperiodtotheinitialsttotheappropriatequbittotal(totakeadvantageofquantumparallelismtodoatotaketoseethisnotefirstthateachx"toseethisnotefirstthatageneralone"toseehowthequantumfouriertransformtoresultinnoactiononthethirdtorecovertheoriginalsttorecoverabycalculatingbtopictoptooltootonindependentvectorsx tonindependtonetokyotohaveinthecodeword4tohaveanappreciableprobabilityofsuccessrequiresahuge togetthesttogethtogettoextractthetableoftoevalumediatmedic!meet     melnikov))member)(membrane)(mentalmentarygatesforquantumcomput%becausetheycannotbeisolatedfromtheirownirrelevantinternaldegrezbusi9^but1' butafewpointsareworthremarkingon%@butalsoaccordingtoourhypothesiswhatalicegetsdoesnotdependonwheth#7'butalsoafastwayofevaluatingthefunctionb%=butalsointeractionsbetweenthecomputationallyrelevantdegreesof#7butalsothestateofalice4[ butaquantum%6butasaresultofinteractionswiththeenvironmentanyoneofth%butasuperpositionoftheform#7butatthisstageofthegamewear%butcanbedistinguish%butcanhaveastatethatisa%,butchangesthesignofallcomponentsorthogonalto%#butchangethesignofitscomponentalong%butdo'buti%Ebutifaliceisthekindwhoworriesaboutoverwhelminglyimprobableeventssheha% 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shundrovskaya-)#shutshuttl7S siblecase7[sidesideswipsiever! siew8Ksiggia-)!sight8Ssignsignalsignific significantli-1silencsilicon! sillisilsbe8Isilva8Qsilver!similarHsimilarsitiuationshavebeenknownsinceafamouspaperbyjohnbellappearedin1964-7simon%#simpl simpleexampl'(simplementationofhissuperfastquantmfouri%simplersimplest'simplifisimplyexchangesth%simplyexchangesthetwoqubit%simul  simultanesin- ) sinasinglebiggerboxoftheform-7sinc  since1-7sincea% sinceanyone-7Esinceanyunitarymatrixremainsunitaryifitismultipliedbyanoverallmultipl-7%sinceanyunitaryoperatorhasasquareroot%sinceappliedto-7 sincecontrol%sinceeachofthem%sinceeachsubroupcontainstheoth%sinceh-7sincehisitsowninvers%9sinceifbitjis0onlythefirsttermactswhichleavesbitiunchang-7 sinceintegr%sinceitcanhappen-7sinceitmakessensetopronounc%sincem':sincemultiplicationbythescalar1andactionbytheunitoperator1-7 sincenislarg%>sincenobodyelseknowsthefactorsofnandcomputingthemtakesamillion%6sinceonecanimaginealiceandbobrepeatedlyplayingthisgame-7 sinceoneha%-sinceoneisasuperpositionofstateswithanoddnumb%Csincequantumcomputerscanonlydothispartoftheiroperationthroughrevers-7sinceth-7%sincetheactionofthetestitselfmightwel-7sincetheadjoint-7sincetheamplitude%sincethefouramplitudesin-7sincethefunctionfistwo%@sincethenotoriousdifficultyofreconstructingtwo200digitprimesfrom% sincetheonli-74sincetheorderinwhichaandbarewrittenisclearlyimmateri-7sincetheorderofanele%1sincetheorderofasubgroupdividestheorderofthegroup%2sincetheorderofeitheraorbisthenumberofelementsinth%sincetheoutcome8[!sincethepaulimatricesarehermitian-73sincetheprobabilityofarandomsuchvectorbeinglinearli%sincetherearen%sincethesingletst8[ sincethest-7sincethesubspaceofthetwo8[ sincethetwo8[ sincethetwoon-7sincethevalueofthematrix-7 sinceunitari%sinceunitarytransformationshave%sinceuthencommuteswithn-71sincewehavealreadyseenahintoftheelegancetheuseofi-7sincexcommuteswitheachm'sincez%(sinceflippingabitandthenactingonitwithn-7sine-7singl   singlebit-70singleclassicalcomputationonarandomlychoseninput%singularsinγ-7%sinγm-6sinθ%sion-1sional-7sisagaintoswaptheirvalu-7sissisite   six !sixcaseswherethechoicesarediffer8[sixteensize-)sknowledgeofalice8[sl4368Q slecturenote' slecturesnote'sledgslezak8Ksliceslowsmal%small=smarginalstatisticsdonotdependonwhatbobchosetodotohisownqubit-7smeasure8[smeasurementhasalterednot8[smeasurementineach8[/smeasurementisundetermineduntilsheactuallycarri8[smeasurementofhisown8[4smeasurementwillbeifsheappliesanygivenutoherownqubit8[smell8C smessageisa'smethod%smith8; smolin%6smyhopethatintroducingandapplyingthesomewhatunfamiliar-7snarlsned8; snedek8;  snumberoperatorn8Z4soaccordingtothebornruleifwemeasureaqubitinthestateh-7soalmostallimportantpap%soameasurementofu%soanglebetween'soani-7soap-3soatmost22ofthesear%sobeautifullyencodedin%sobeautifullyencodedinthest%socall%socdoesnotdivid'socialsocieti  sociologi sociologistsocrate8P9soeventhoughqubitsdonotingeneralhavevaluespriortoameasure-7Dsoforquantumcomputationremainedaquietbackwaterofsciencebetweenthemid%soft software soifcddiffersby1from%8soifitisreasonabletoconcludefromtheeprreasoningthatalice8[soifthiswereallonecoulddo%soimportantisthatageneraltwo%#soinconstructingaquantumalgorithmit%soinmathematicallanguag-7Bsointhisimprobablebutnotimpossiblecasewherebothdoapplyhadamardsand-72soitcanberealizedastheordinarytimeevolutionofaphys%soitleav8[sol-)! solar!6solelyfromthepubliclyavailableknowledgeofnandcasfollow%solid  solidif-)soliton-)solomon8Osolutionsolvsomesomebodisomeclaimthatitshow-7somedevoteesof% 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sotheeigenstatesofthehamiltonian%Dsothehypothesisthatthevalueoneofthemfindsuponmeasuringdoesnotdepend-7sothemeasurementsrevealnoth%sothenet%3sotheprobabilityofaliceandbobbothfindingthevalue1i-7,sotheseeightoperatorsaresimultaneouslymeasur%sothestateofthefirstn%,sothetworeversiblethingsyoucandotoasinglebit-7sotheunitarymatrix-7$sothevectorumustbeitimesarealvectorv-7sothirdqubitisunchag'sotodeterminewhetherfisconst%soundsourcsourish8Ksouthern!sowecangoalongwiththecomput-7sowestickherewiththephysicist-7sowheniusethetermmeasure-71sowhenthequbitsaremeasuredbobcannotfindthevalue0-7sowhetherornotu-7Esowhileyoushouldbewarnedthattheglimpseofquantummechanicsyouwillacquir-76sowitharelativelysmallnumberofrandommultiplesofronecan%sowithasingleapplicationofu%sowiththisconvent-7sox'spacespaceket-70spacenotationonewouldtalkofaninnerproductbetween-7spaceofthesamedimens-7spacerot-7spaciouspatialspatio-)speakspeaker!speakingofthequantumbit-7special 'specif-(*specifyingoppositeoutcomesforeachofthethre8[specifyingthatb%specimen specifiedin%specifiesthestateofthebit-7 spectacularspectaculareffici%spectra-) spectroscop-) spectroscopi-(spectrum8Mspeechspeed spendsphere-2spiel-)spin spitespons spontanespookyactionatadist-6spoon  sporadspragu-)spring-)sproblem$!sproblemthequantumcomputercandoin%*sproblemwehaveafunctionfthattakesnbitsinto%squar' squarestoun$squaringtheresult%squash squaternionsi-7squbit-7$squbitisalteredbi8["squbitisgoingtorevealbeforeshemake8[ squbitisinth8[/squbitmustreallyhavesomehowbeencarryingallalong8[squbitscommute-7squid! srelation-7+sresultingfrommeasurementsofallthequbitswil-7|2Dzeroexceptforasingle1inthepositiondownfromthetopgivenbythebinarynumb2:zerocomponentoccursintheposition5specifiedbythebinarynumb2 zassenhau2zap2ywith2#youwillseethatyoucanbuildupaswapout2youwill2youshouldconfirm2?youshouldconvinceyourselfthatthisextensionoftheordinarybornrule2youshouldaskyourself2yourlaptopoper2Byouremovethepossibilityofextractinganyfurtherinformationabouttheir2youonlyhavetonotethatu2=youmustindeedhavemanyyearsofexperienceinquantummechanicsandit28youmightthinkthatbydoingsuchameasurementoverandoveragain2youmaywellaskwhatitmean26youmaywanttoskipthissectioonorjustreadthroughitlightli2youmayormaynotfindpoint2Gyoujusthavetoverifythattherightsideofthisidentitydoesinterchangethevalu25youhavetospecifytwocomplexnumberssubjectonlytothenorm2youcould21youcombinearotationthroughanangleαaboutanaxisgiv2 youcanshowthatfouroftheeightterm2youcanreadoutthevalu2youcannotreadoutthosevalu2youcannotlearnwhatthest2youcanalsobeassuredthat2youcan2 youareinvitedtoproveasanexercise2+youafurthersenseofjusthowpeculiarqubitscanb2york2&yearstodevelopafeelingforquantummechan2yang2xandzonlyonthecom2writtenintheform2wouldhavehappen2 wouldhavegot2wouldhavebeenabetterterm28wouldbetheeraseoperationwhichreplacesthevalueofthebitby02worldli2 woodcraft2 withθreal2 withthevect2withthetheoryof2withtheirinternalstructure2Dwiththecomplexamplitudesbeingconstrainedonlybythenormalizationcondit2 withrealu2withlinearalgebraand2withcomplexscalar2withclassicalbitswil2withcertainprob2withanygivenpairofqubit2"withamplitudesαandβthentheresult2withamplitudesα2)withamplitudeswhosesquaredmagnitudessumto2 withalloperatorsthatactonlyonbob2'withalargenumberofbitsandwantstowritea22withabraandaket2willbetheunentangledproductst2*willbe0or1withthoseprobabilitiesandthepost2 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whenthefourm%0whentheithandjthbitsareineachofthefourpossiblest17whenthestateofth9[whenthevectorain17whenthisresultsinnoambigu17whenweturntoquantumbit17wherea167whereaisthevectortakenintozbytherotationassociatedwithu9[whereastheaboveexampl%whereeachunitarygatev'whereeisthegroupident%wherei%0whereihavewrittenthemqubitstobemeasuredontheleft17whereji%wherep17-wherethegeneralstateofnbitscanonlybeoneofthe2174wheretheorderoftheproductiswithkincreasingtotheright%wheretheprojectionoperatorp17'wheretheregistersonbothrightandleftaren%&wherethesubscriptindicateswhichbitthe117wherethexforeachqubitcanbe0%whereu17wherewehavemaderepeateduseof178wherewehaveusedthefactthatalloperatorsthatactonlyonalice17wherewetaketheon17wherex%wherexistheintegerbetween0and2%wherexˆisqubitnumberoper'Cwhereαandβaretwocomplexnumbersconstrainedonlybytherequirementthat17whetherwhetherornotit17#whetherornotitwilleverbecomeapracti17whethertheform17;whichactinginthecomputationalbasisappliesanarbitraryunitari% whichactsasσ%whichactsononecubitinthest%whichactsonthecomput%whichagainactsinth%7whichanticommuteswithanyproductofanoddnumberofdistinctx%Iwhicharenotnecessarilyorthogonalornormalizedandwillbedifferentfordiffer%whicharestillunderproduct%Awhicharetheonlyoperationsthroughwhichaquantumcomputertransformsit17 whichbecome9[whichcan17 whichcanbeani17whichcanbeconstructedoutofone%{=whichcanthenbetestedbyevaluatingffortwovaluesofxthatarerapart%whichcombineswiththeu%Gwhichcontradictsthequantummechanicalresultthatinthesesixcasestheoutcome9[ whichcountenancesonlybitandqubit17?whichdefinesthedualforanysuperpositionofcomputationalbasisvect177whichdiffersfromcase2onlybytheinterchangeofaliceandbob17whichdoesperfectli17'whichfunctionsasanumberoperatorforthest17 whichgive%Iwhichgivesusbackthematrixdefinitionoftheadjointasthecomplexconjugateofth17whichhaseigenvalu%)whichhasthosetwoorthogonalstatesaseigenst%&whichhastodowithextractingusefulinform17whichhekeepstohimself%whichiamtoldisrend17 whichiremindyoucanalsobewrittena170whichisalmostalwaystakentobethecomputationalbasi%Cwhichisanumbersatisfyingtheusualrulesforinnerproductsinavectorspace17foracomputertobeaquantumcomputerthephysicalsystemsthatencodeth%7Eforaconcisereviewoflinearalgebrawithsubsequentapplicationstoquantumin%7 forajaund % forameasure % foranynumb %foranyrealnumberϕ%7forappropriateunitvectorsmandn % foraquantum %forarbitraryone %forbobcanlearninadvancewhatth5[forbobtogetthesameresultin%7forc forcomplexnumbersαandβ %forcrackingthersacode %fordifferentxandthesecond %foreachofthethreetermsin%7*forenormouspowersofrisnotashorrendousanund % forexampl $% forexamplebya%7 forexampletwo %(forexamplexcouldspecifyachoicebetweentwo %foriffi %form   ) formabasi%7formabasisforthe4%7formabasisfortheentirespaceof5[=formabasisforthefourdimensionalalgebraoftwodimensionalmatrice%7*nowthevectorspaceis4*/nowtheprobabilityoftheirmeasurementsgivingxandi*(nowthecomputationalbasisstatesareeigenst*0nownoticesomethingratherstrangeaboutthesestatist* nowinthest*novak* notverymuch*nottoperformthehadamard*nottomentionhowitwork*notthenweareincase3* notorcnotoper* notincludingm*noticethatitfollowsfrom*(notefirstthataccanbeexpressedintermsofn*1notethatthisdefinitionextendstheassociativelawto*notethatthethreerel* notethatifwerepresentageneralket* notethathere*,notethatbybuildingtheiintothedefinitionofσ*notethatageneraltwo*notethat*notealsothephysicist* notcontinu* notconcernu*notationyoucanputintothebox*3notationeliminatestheneedforplayinggameswithmatrice*notate*Anotastheremarkablerevisionofnewtonianclassicalmechanicsrequiredto*5normalizedsuperpositionofthefourorthogonalclassicalst*normalizationconditionslike*normalizationcondit*normal*,noristheactofacquiringthisinformationdisrupt*nonequilibrium*noneofthissymbolismi*noneofit*)nondeterminsticworldthatweactuallyinhabit* nonadiabat* nomenclatureofcallingvectorslike*Lnomenclatureandnotationofquantummechanicsinafamiliarclassicalcontextwillmake*nois*nitrogen*nikolai*nielsenandisaacl*newman*9neverthelesscomputerscientistsandmathematicianswithnoback* nevermind)#neitheroversimplifiednorincomplete)0neitherhadappliedhadamardsbothwouldstillhavegot1)neill)needless)necessaryforclassicalbit)neaton)nconstituteacomplete)2naturalandirreducibleaconditionforaqubittobeinthan)narrow) nanostructure)nandnintermsof1andz)myowntakeonitisratherdiffer)myer)!mutuallyexclusiveoutcomesaddupto1)murti)muchofwhatfollowswillbedevot)$muchoftheanalyticalcomplexityoflearn)msuggest)msc)motion)mot)mostgeneralthingyoucandoton) morpholog)Amorepreciselytheydonotalwayshavewhatarecalledpurestatesoftheirown) moreprecise)moreonthisbelow) moregener) morecorrectli)moreaccuratelya)molecule)mofthem)mnotmakingthisup)mixture)mitch)minimum)minim)milou) millikelvin)mild)microstructure) microscope) microscop) micromechan)michell)michaela)metcalf)metast)metall)sincenobodyelseknowsthefactorsofnandcomputingthemtakesamillion%6sinceonecanimaginealiceandbobrepeatedlyplayingthisgame-7 sinceoneha%-sinceoneisasuperpositionofstateswithanoddnumb%Csincequantumcomputerscanonlydothispartoftheiroperationthroughrevers-7sinceth-7%sincetheactionofthetestitselfmightwel-7sincetheadjoint-7sincetheamplitude%sincethefouramplitudesin-7sincethefunctionfistwo%@sincethenotoriousdifficultyofreconstructingtwo200digitprimesfrom% sincetheonli-74sincetheorderinwhichaandbarewrittenisclearlyimmateri-7sincetheorderofanele%1sincetheorderofasubgroupdividestheorderofthegroup%2sincetheorderofeitheraorbisthenumberofelementsinth%sincetheoutcome8[!sincethepaulimatricesarehermitian-73sincetheprobabilityofarandomsuchvectorbeinglinearli%sincetherearen%sincethesingletst8[ sincethest-7sincethesubspaceofthetwo8[ sincethetwo8[ sincethetwoon-7sincethevalueofthematrix-7 sinceunitari%sinceunitarytransformationshave%sinceuthencommuteswithn-71sincewehavealreadyseenahintoftheelegancetheuseofi-7sincexcommuteswitheachm'sincez%(sinceflippingabitandthenactingonitwithn-7sine-7singl  S singlebit-70singleclassicalcomputationonarandomlychoseninput%singularsinγ-7%sinγm-6sinθ%sion-1sional-7sisagaintoswaptheirvalu-7sissisite    six !sixcaseswherethechoicesarediffer8[sixteensize-)Ysknowledgeofalice8[sl4368Q slecturenote' slecturesnote'sledgslezak8Ksliceslowsmal%small=smarginalstatisticsdonotdependonwhatbobchosetodotohisownqubit-7smeasure8[smeasurementhasalterednot8[smeasurementineach8[/smeasurementisundetermineduntilsheactuallycarri8[smeasurementofhisown8[4smeasurementwillbeifsheappliesanygivenutoherownqubit8[smell8C smessageisa'smethod%smith8; smolin%6smyhopethatintroducingandapplyingthesomewhatunfamiliar-7snarlsned8; snedek8;  snumberoperatorn8Z4soaccordingtothebornruleifwemeasureaqubitinthestateh-7soalmostallimportantpap%soameasurementofu%soanglebetween'soani-7soap-3Ksoaquantumcircuitfortransform%soatmost22ofthesear%sobeautifullyencodedin%sobeautifullyencodedinthest%+phonon+ phasefactore+petta+perpositionlikethest++performthehadamardthatsheactuallydidperform+percol+ pennsylvania+ patterson+ pasupathi+*passionshaverunhighonthesignificanceofthi+partlybecausethenot+parpia+parametr+paramet+ parallelvect+ pairofthem+pack+overthecomplexnumb+outsideofthatsubset+outcomeofwhichiseither0or1+'ourhypothesisleadstotheconclusionthatif+ oscillatori+os+ orflipit+Iorfivelecturesofacourseinquantumcomputationgivinganintroductiontoquantum+ orviceversa+$orthosesystemsabsorbingaminuteamount+orthonormalset+orthonormalbasi+orthe3+orthat0++orterminologyyouareusedtomaydifferfrommine+oroper+-ornotbobappliedahadmardtransformtohisownqubit+ ornotappli+0ornotaliceappliedahadamardtransformtoherownqubit+orn+ orishouldasky+ orinthest+ originalst+orifice+orfrom+ orexchang+Goreventostudywhatphysicalsystemsarepromisingcandidatesforrealizingsucha+orevend+ordinaryinnerproduct+orcollapseofthest+orcase4+orboxingeachindividualbit+ orapairof+oranyoth+oracollectionofbit+ or1aregivenbi+optim+(operatorswithdifferentsubscriptscommute+operatorsonlyactonst+ operatorbasi+ operationx+operationistochangethest+3operationaquantumcomputercanapplytothestateofaqubit+Bonwhethertheotherappliedahadamardbeforemeasuringleadstoacontradict+onto+ ontheright+onthecomputationalbasi+ onlywhenyouencounteractualexampl+"onlywhenwemakeuseofsomeofitsresult+onlythesecondtermactsandx++onlythefourthtermactsandtheeffectofthetwox+8onlyreversibleoperationsonclassicalbitswillbeofinterestt+onlyrecentlydiscov+&onlyoneofthesefourcasescanactuallybere+ onlyonbob+onlybyth+onlyarashpersonwoulddeclarethat+ongo+oneusesthetermpurest+onesaysthatthest+.onesaysthatasaconsequenceofthemeasurementthepr+oneofthethingsquantummechan+oneofthemanycharmsofquantum+"oneoftheappealingfeaturesofquantum+7onemightbetemptedtoinferfromthatrulethatforaqubittobein+1oneisonlyentitledtosaythatthestateofthequantumbit+ oneinthest+8oneimportantoperationyoucanperformonapairofbitsistheswap+oneha*1onecouldsaythatthefirstthreebitsontheleftareall1*/onecantalkabout0and1asthepossiblevaluesofthebit* onecanbea*onecanavoidmuchsuff*"onecanalsoassociatedanouterproduct* onealsodefinesitsactiononbrasbi* oneachqubit*!onceyouhavemakejustasinglemeasure*onbytheunitoper*omwhichitfollowsthat*ofzandx*ofwhichthisisthesimplestexampl**ofwhichthetheflipoperationxisaspecialcase*(ofwhatmaystrikeyouasanabominablepractice*ofunimodularunitari*oftwoqubitsisanarbitrari*oftwoon*oftwo1*ofthreecontrol*ofthetwovectorsaandσ*ofthetwopossibleclassicalst* ofthesubject*oftherealphenomena* ofthequbittoh*ofthequbitsinthe2* ofthequbit*Hoftheparadigmofclassicalcomputersciencethathadcompletelyescapedtheattent*oftheout*oftheon*ofthenot*ofthemeasurementis0withprob*oftheirproduct*'oftheinputthanitcaninanyclassicalcomput*7ofthecomputationalbasisvectorsaredefinedbylinearityand* ofthebornrule*ofthebit*ofthe2*ofthathappeningis0* oftenonesai* ofseveraltwo*Kofquantummechanicsthatcanbetaughtafteronlyafourhourintroductiontothesubject*ofquantumcomput* ofproper3*ofone*'ofomittingtheexplicitoccurrenceof1fromu* ofmeasure* ofitsgroundst*ofi*ofhtobeequalto* offreedom*(ofexplicitlyexpressedtensorproductsofbit*ofeachsinglebitascolumnvect* ofeachqubit* ofeachbit*ofdoinginprincipl*,ofcomputerscientistsuntilabouttwentyyearsago*ofclassicalbit*ofbob*ofbit* ofatomics*.ofasingleclassicalbityouneedonlyonebitofinform*7ofasinglebittoberepresentedbytwoorthogonalunitvectorsin*Iofaqubitcanbesubjecttoanylineartransformationthattakesunitvectorsintounit*'ofaquantumbitcanbeanyunitvectorinthetwo* ofapurenumb* ofambientradi*ofalltheamplitude*ofallm*ofalicegettingxandbob* ofageneralone*ofaandb*ocean*occassionallyoneevenusestheterm*occasionallyitwillhappen*nwitheigenvalues1and0* nwhosematrix*numer*,numbersmustbeareallinearcombinationofthefour*nucleic* nqubitsinastformate%( !formationprocessingverymuchinmind%7Kformationsisalmostinvariablyrestrictedtoonesthatcanbebuiltupoutofproductsof%7formifallfoureigenvaluesare %#formingeachindividualqubitintothest %formthatworksequallywellf %formu'formul *forquantumbitshavearatherdifferentcharact %forschungszentrum !forshort%7forsomecomplexnumberu%7forsomeintegerm %forsomeorallofthequbit %forth  'fortheinnerproduct%7"fortheirperfectcorrelationsintheaa5[forthen%7fortheon%7:fortheproductoftwosuchnumbersalsohasnofactorsincommonwithn %#forthespecialtaskforwhichitisintend%7forthevariablex %forthewholealgebraofoper%7+forthisreasonphysicistscallstheexchangeoper5[forti !fortun fortunatelyhamiltonianoper%7 fortwovalu %forum !forward forx%7 forfindingr %found5Efoundate  four fourier%)fundingfutureforquantumcomputationmaybesecureforsometimetocome %further aLfurtherexerciseinthinkingofoperationsonclassicalbitsaslinearoperationsonvect%7furtherfeaturesofdiracnot%7 furthermore  )furthermorethenumericalresultofthemeasure %fuss future%)fuzzi&3Kg08g5;g225;ga !%/ gab5Kgabriel5Kgaeta&,galleri5Sgame&)gamow gane ! gangpolak %gap&)garage   garden&-gase&)Sgaseou5M7 gasparini !gass #gate $3gateisconsiderablymoreversatilethanitsclassicalvers % gatesareevenmoreformidablethanth&7gatesisshowninfigure3 % gatesoutofc' gatesoutofcc'Egatesresultinthecorrespondingsuperpositionsoftheoutputsdescribedabove %gatewai     gaudioso&)gaussian:gave gavin&)gedanken $gel225Qgeller&)gene !gener   &generalexceptforsuchanoverallphasefact&7Kgeneralizationoftheirhabitofnotdistinguishingnotationallybetweentheunitoper&7Mgenerallyanyinvertibleclassicaloperatororiginallydefinedasapermutationofthe2&7Dgeneralruleforoperatorsthattheadjointoftheadjointistheoriginalobject&7generalstateofaqubit'generatorexternaltothecomput&7genet    geneva genom Q gentleman geoff&)geofferi5Wgeoffrei ! !geometr&1Kgeorg  germain !german get   3%get1&7getoppositeresult5[gets0and5[ gettheaxisc&7ghastli ghz' giannetta !gilman5Kgisela5Qgiuseppi&)giv give given given2nqubitsinthest % givenanoracl'givenbi % givenbyth&7givencandritiseasi %givenf'givenin&7givensuchanoracl %givensuchasolutionitfollowsthat %giventheimplementationsofv %6giventhepracticalimpossibilityoffactoringa400digitnumb %giventwomultipleskrandk %givesavectorrotatedawayfrom %Kgivesitthepotentialtodotricksthataclassicalcomputercanperformonlywithvastli %givesusdirectlyα&7giveszerounlessi %giveszerowheni %$givethebinaryexpansionoftheintegerxi&7givingacontributiontothesumin %givingtheresultx&7glass !+glassi5Eglement&7glickman&)%glimpsesofsuchbehaviorsometimesshowup&7global&)glu go goe:{gold&)goldbaum5Kgoldwin5;good got gottfri&-9gotwhenaliceappliedthehadamardwiththosehegotwhenshedidnot&7gov %govern gra&(#grab  gracefulli grad5Ogradual graduat  grant &graphite&(grave gravel5Kgraviti:ygreat greatercomputationaleffort % greenberg !greg&)gregori&/greytak !griffin5M. grose5Kground K%Kgroundinphysicshavebeenablequicklytolearnenoughquantummechanicstounderstand&7group  - groupandthen&7grover $grow  growth gruner&)! grunt5Cguardian guenther !gueron&)guest guid5:guikema5Qguisedashisquaternion&7gulli ! gutenkunst5Kha  ;habit hadamardtransform&7hadn haifa:yhaitao5Qhalf Ihalfitsvaluesare0andhalfare1 %hall ) )hallock !halperin !hamilton&7hammel !hammer hampton&)han ! hancock5S%hand 3handf&7handl /*handleallfourcasesatoncebynotingthatifthei&7handwrit hang !happen   happi hard 5 hardli hardware  harmon {harold !harvard 3hasa %hasadistinctpatternofnega %hasallitsentri&7hasamultiplicativeinversed %haspickedtobecoprimeto %hassomeperiodr %,hassomeremarkableimplicationsforapairofqubit5[ haszerotrace&7hate hav  haveexpans&7havethesamepatterofcommut %havethesimplifyingfeaturethatf % havingthevalu&7havingthevalue1 %head hear +heat&(heavi  hecansai5[&hedoesnotrevealtheseparatevaluesofpand % hegivesalice %hehasnocontroloverwhichst5[ heisenberg helium&(help helsinki !&henceitcontainsthesubgroupgeneratedbyb %henlei&) henri&/her herbert:here  hereandinanalogousplace&70hereisausefulthingtonoteaboutthetensorproductnot&7/hereisextremelyfocusedandquitelimitedinitsscope&7heremark&7*herenopriorfamiliaritywiththequantumtheori&7herethetensorproduct&7hereσsignifiesth&7 hermeasure5[hermeasurementfirst5[ hermitian&)%3?hermitianmatrixymakestheanalysisthatfollowsvastlymorecumbersome&7herself hertz5S% heterogene&)heterostructure !heurist&3K3hewouldhavegot1whetherornotaliceappliedahadamardtoh&7hi hick&)#hide high !higher highest highli  hill&)him hint hisitsowninvers&78hisknowledgeisaboutsomefeaturealreadypossessedbythequbit5[hisresponsewasthatanyappl&7hit hizi5Khoffmann 9hofstett:holcomb5Ihold hole hom home   homebound homepage >homework homogene&)-sinceth-Csincequantumcomputerscanonlydothispartoftheiroperationthroughrevers-6sinceonecanimaginealiceandbobrepeatedlyplayingthisgame-:sincemultiplicationbythescalar1andactionbytheunitoperator1-sinceitcanhappen-9sinceifbitjis0onlythefirsttermactswhichleavesbitiunchang-sinceh-sinceappliedto-Esinceanyunitarymatrixremainsunitaryifitismultipliedbyanoverallmultipl- sinceanyone-since1-sinasinglebiggerboxoftheform-sin-Hsimilarsitiuationshavebeenknownsinceafamouspaperbyjohnbellappearedin1964- significantli-siggia- shundrovskaya,*shouldbeaunitvectorinthecomplexvectorspace,shewouldhavegot1whetherornotbob, shemustalso,shear,sethna, serioustroubl,serguei,sergei,serg, semiclass,self, seesection2, secondari,seaw,&sdisparagingtermspukhaftefernwirkungen,sdecisionwhether,scott,!sciencenotationwithoutgettinginto,scienceandphysicsnot, sciencealias,schwarz,schwartz,scatter,scale,scalar,sby,sbit,say01100,sato,satisfiesanessentialconsist,sasconstitutingth,sandx,Asandwichinganarbitraryoperatorabetweentwosuchexpansionsoftheident,sand1,sand,sampl,salina, sagoodpoint,rough,rot,rossi,rose,rolf,role,rna,ritchi,riemann,5richervarietyoftransformationsonecanapplytoquantumbit,richard,rich, revoltingli,reu, restructur,respect,resolv,resolut,%requiringthatthedeterminantofumustbe1,'requirestherealandimaginarypartsofutobe,!representsthesquareoftheflipoper,representsthe2,representsthe1, representedbi,representbinaryinteg,remindyouofsomeofthem,remarksthatfollowina,Gremainsthesameinbothworldsandiscapableofcompletelydeterminingtheoutcome,"relevanttothetheoryofquantumcomput,Frelationslikethesepermitustoextendtocombinationsofstatesandoperatorsth, relatingtheamplitudesintheexpans,regime,regardlessofwheth,refresh,referringbacktoit,referencetowhichbititactson,6recentlyimentionedtoadistinguishedtheoreticalphysicist,3recallthatthetraceofamatrixisthesumofitsdiagonalele,0recallthattheelementsofahermitianmatrixasatisfya,9recallthattheadjointofamatrixisthetransposedcomplexconjug,ray, ratherthanhav, ratherthan,umcomputersdoanimportantpartoftheirmagicthroughreversibleopera0ulrich0ugrad0uau09uationinwhichbothapplyhadamardtransformationstotheirqubit0#uareclearlyassociatedwiththesamerot0uand08ualmeasurementareindependentofactionsappliedtootherqubit0tye0*twothingskeepthesituationfrombeinghopeless0 twosubset0twostate0 tworkatal0twhilethesituationisnothopeless0turbul0tunnel0tun0tumphysicsisthatinsomewai0tuesdai0tube0+trytocallattentiontosomeofthisstrangebehavi0?tryingineachcasetochoosetheformthatmakesthecontenteasiesttoread0:trivialreversibleoperationwecanapplytoasinglebitistheflip0 trevised10treatclassicalbit0transposedmatrix0 transport05transformationtotheirownqubitbeforemakingtheirmeasure0transformationisappliedtothen0tofindthemhere0tofind0"tofigureouthowyoucoulduseaquantum0towithinanoverallphasefact0(towhichthequantumformalismcanthenbeappli0$tounderstandhowtobuildaquantumcomput0Ctotheprobabilitiesofmeasuring0or1isoftenstatedintermsofinnerproduct0 totheform0 tothecase0tosomequiteextraordinarybehavi0toseethisweintroduceanotherof0tosai0Btoregardingtheordinarycomplexnumbersasanextensionoftherealrealnumb0 torefertotheactualphysicalsystem0toproduceoutputsfrom0+toprepareforthisenrichmentiintroducewhatmai0tomultiplyitsstateby00tomasi0!tomakethisevidentithelpstoexpandh0 tomaketherule0tom02toldthatthesesymbolsareallexpressingthestateof3bit0$tohelpyoubecomemoreathomewiththisnot0(togiveyousomepracticinginusingthequantum0#togettheangletakethetraceofbothsidecthespecialonezbeingsought%Gthespecificcharacterofthetestdependsonthekindsofphysicalsystemsonwhich/7thespellingisabomin%@thesquaredmagnitudesoftheamplitudesgivetheprobabilitiesofoutcome/7thestate% thestate1/7thestatedeterminesonlyth/7thestateofclass/7Ethestatesofnclassicalbitsarethusanextremelylimitedsubsetofthestatesof/7thestatesofthequbit8[+thestatisticaldistributionofcombinedoutcome/7thestatisticsofth/7!thestrangepropertiesofafamilyofst/7thestructureof%.thestructureofeachencodedqubitremainsoftheform%thestructureofsh% thesubgroupgeneratedbyacontainsa%9thesubgroupgeneratedbyanelementisthesetofallpowersoftheel%thesubjectofsectioniv% thesumoverxin% thesuperposit%Hthesurprisingkindsofphenomenathatthequantumbehaviorofmattercangiveriseto/7 thesymbol/7Bthesymmetryisreflectedinthefactthatthealternativechoiceoffieldsh% thetargetbit/7 thetargetqub%%thetelephonecompanyinitsmanymanifesta%thetensorproductnotationin/7"thetermisatranslationofschr¨oding/7 thetheori/7thethirdisequivalentto%thethre%%thethreeoperatorsz%thethreepaulimatrice/7-thethreepaulimatricesmustthereforebeoftheform/7thetotalangularmomentum%Athetransformationoftheinitialstatebroughtaboutbyaquantumcomputeri% thetrickisthi%thetwo%thetwobi/7thetwoorthonormalst%+thetwopossibilities10and01occurringrandomli8[/thetwoqubitsareseparatelypossessedbyaliceandbob8[thetwostatesareorthogon%theunitaryextensionofu% theunitaryone/7&theunitarytransformationsvandwsatisfyv%)theunitarytransformationthattakesthetwost%#theunitarytransformationuisgivenbyu%'theunitoperator1andthethreepaulimatrice/7theunitoperatorinthiscase/7>theuseofquantumcomputationincodebreakingisdescribedinsectioniv% theuwasappar%thevalue1evenifalicehadnot/7 thevalueoff%thevectorperpendicularto%Itheverymuchrichervarietyofthingsthatacomputercandoifitisdesignedtotakeful/7thewholestringof0/7they Ktheyalwaysgetoppositeresult8[ theyareact/74theyareconstrainedbytherequirementthatthetwostateson%theyarejustthesu% theyarenot8[*theyareprovidedwithalargenumberofqubitpair8[theycan$theydonotexist/7theyear/7theyhaveinvers%theyhavemanyotherusefulpurpose/7theymustbeident% theyweredo8[theywillallbezeroifandonlyif%Ktheywouldhaveyieldedactingonstatesthatwereeigenstatesoftheindividualqubitσ%2theyyieldthecorrespondingsuperpositionsoftheresult%thez%thezero/7 thefifteenpossiblecorruptionsof%thefigureshowshowthis3%6thefinalmeasurementtodistinguishbetweenthetwoorthogon% thefinalst%thefinalstateof%thefirstequalityisequivalentto%Jthefirstfewtimesilearneditiwasunabletoreconstructthetrickadaylaterwithout%thefirstngive%thefirstofthesefour%thefirstofthest8[thefirstsetofn%theflipoperatorisitsowninvers/7thibault8Kthickthin!Gthing;Bthingaboutaquantumcomputeristhatonecanpreparethesecondqubitinthest%think  4thinkaboutitforalittleyouwillseethatthisisquitegener/7+thinkofaasactingeithertotherightortotheleft/7thinkofthefirstgroupasunderth/7third jCthisalmostinvariablyinvolvesformulatingoversimplifiedabstractmodel/7thisambiguitycanberemov/7thisargumentwasfirstmade8[Fthisbasicruleforhowinformationcanbeextractedfromaquantumstatewasfirst/7thisbringsustoasecondcatch%thiscan%$thiscanalsobeseendirectlyfromtheform%3thiscanalsobewrittenintermsoftheprojectionoperatorp/7$thiscanbeconstructedfromtheoperatorv%Dthischangeofstateattendantuponameasurementisoftenreferredtoasareduct/76thiscontradictsthefactwhichwehaveextracteddirectlyfrom/7thiscouldbethoughtofas2%&thisdeterminestheactionofxandzonanyone/7Hthisdiscoveryinstantlytransformedquantumcomputationandallrelatedinvestig%*thisdoesnotimplythatpriortothetestthequbit/7Kthisexcessivelystrongrestrictiononthenatureoffcaniscircumventedbydoublingth%thisextendedbornrule/7?thisfollowsfromtheformsofthepowerseriesexpansionsoftheexponenti/7%thisformintroducesinaclassicalcontext/7&thisgeneralizesintheobviouswaytonqubit/7-thisisagoodexampleoftheprimarypointofdiracnot/7thisisaratherpeculiaroper/7$thisisastandardresultoflinearalgebra07thisisatranslationofeinstein07 thisisbecaus%@thisisbecausethequantumfouriertransformyieldswithappreciableprob%=thisisenoughtoprovideuswithvaluableinformationabouttheperiodr%(thisisequivalenttoactingonlyonthefirstn%4thisisincompatiblewiththestatisticsoftheremaining6po8[-thisisirreversiblebecausegivenonlythefinalst07thisisjusttheassertioninsu%thisismassivequantumparallel%thisisorthogonalto07 thisispuzzl8[Bthisisthesimplestexampleofsomethingaquantumcomputercandothataclass%,thisisthestatewhichmustberestoredtotheorigin%Gthislastexerciserequiresnofurtheruseoftheformalapparatusofquantummechan8[thisleavesthei07+thismakessenseforaclassicaloraquantumcomput%thismightappeartobeasimpletask%8thisnotationcanbeambiguousifthenumbernofqubitsisnotknown%5thispartoftheoperationisunimportantinaclassicalcomput07thispossibilitywasdiscoveredbyg%2thisrealizesthequantumfouriertransformasasetofnone%thisrelationneednothold071thissamedevicecanbegeneralizedtoassociateaunitari%thisstateisofthegener074thisstatesuffersachangeofsignwhen0and1areinterchang%=thistakesorthogonalstatesintoorthogonalstateswhetherornotthef%thistemptationmustberesist07'thistrivialidentitycanbesurprisinglyuse07thisturnsouttobeposs07&thisturnsouttobethegreatworkhorseoftwo07thiswillrequireponderingth8[thisworksasfollow%thiswouldbejust%thiswouldbenonspooki07=thisfictitiousworldrequiresmorethanjusttheirrelevanceofalice07thoma!thompson0) 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