AreQuantumStatesExponentiallyLongVectors?ScottAaronsonI'mgratefultoOd - PDF document

AreQuantumStatesExponentiallyLongVectors?ScottAaronsonI'mgratefultoOdedGoldreichforinvitingmetothe2005OberwolfachMeetingonComplexityTheory.Inthisextendedabstract,whichisbasedonatalkthatIgavethere,IdemonstratethatgratitudebyexplainingwhyGoldreich'sviewsaboutquantumcomputingarewrong.Whyshouldanyonecare?Becauseinmyopinion,Goldreich,alongwithLeonidLevin[10]andother\extreme"quantumcomputingskeptics,deservescreditforfocusingattentiononthekeyissues,theonesthatoughttomotivatequantumcomputingresearchintherstplace.Personally,Ihaveneverlainawakeatnightyearningforthefactorsofa1024-bitRSAinteger,letalonetheclassgroupofanumbereld.Therealreasontostudyquantumcomputingisnottolearnotherpeople'ssecrets,buttounraveltheultimateSecretofSecrets:isouruniverseapolynomialoranexponentialplace?LastyearGoldreich[7]camedownrmlyonthe\polynomial"side,inashortessayexpressinghisbeliefthatquantumcomputingisimpossiblenotonlyinpracticebutalsoinprinciple:AsfarasIamconcern[ed],theQCmodelconsistsofexponentially-longvectors(possiblecongurations)andsome\uniform"(or\simple")operations(computationsteps)onsuchvectors...Thekeypointisthattheassociatedcomplexitymeasurepostulatesthateachsuchoperationcanbeeectedatunitcost(orunittime).Mymainconcerniswiththispostulate.Myownintuitionisthatthecostofsuchanoperationorofmaintainingsuchvectorsshouldbelinearlyrelatedtotheamountof\non-degeneracy"ofthesevectors,wherethe\non-degeneracy"mayvaryfromaconstanttolinearinthelengthofthevector(dependingonthevector).Needlesstosay,Iamnotsuggestingaconcretedenitionof\non-degeneracy,"Iammerelyconjecturingthatsuchexistsandthatitcapture[s]theinherentcostofthecomputation.Myresponseconsistsoftwotheorem-encrustedprongs:1rst,thatyou'dhavetroubleexplainingevencurrentexperiments,ifyoudidn'tthinkthatquantumstatesreallywereexponentiallylongvectors;andsecond,thatformostcomplexity-theoreticpurposes,theexponentialityofquantumstatesisnotthatmuch\worse"thantheexponentialityofclassicalprobabilitydistributions,whichnobodycomplainsabout.Therstprongisbasedonmypaper\MultilinearFormulasandSkepticismofQuantumComputing"[1];thesecondisbasedonmypaper\LimitationsofQuantumAdviceandOne-WayCommunication"[2].Prong1:QuantumStatesAreExponentialForme,themainweaknessintheargumentsofquantumcomputingskepticshasalwaysbeentheirfailuretosuggestananswertothefollowingquestion:whatcriterionseparatesthequantumstateswe'resurewecanprepare,fromthestatesthatariseinShor'sfactoringalgorithm?Icallsuchacriteriona\Sure/Shorseparator."Tobeclear,I'mnotaskingforaredlinepartitioningHilbertspaceintotworegions,\accessible"and\inaccessible."Butaskepticcouldatleastproposeacomplexitymeasureforquantumstates,andthendeclarethatastateofnqubitsis\ecientlyaccessible"onlyifitscomplexityisupper-boundedbyasmallpolynomialinn.Inhisessay[7],GoldreichagreesthatsuchaSure/Shorseparatorwouldbedesirable,butaversthatit'snothisjobtoproposeone: CurrentlyattheUniversityofWaterlooandsupportedbyARDA.Email:aaronson@ias.edu.ThisabstractismostlybasedonworkdonewhileIwasastudentatUCBerkeley,supportedbyanNSFGraduateFellowship.1SanjeevAroraaskedwhyIdon'thavethreeprongs,therebyforminga -shapedpitchfork.1 MymaindisagreementwithScottisconceptual:Hesaysthatitisuptothe\skeptics"tomakea[concrete]suggestion(ofsucha\complexity")andviewstheir[arguments]asweakwithoutsuchasuggestion.Incontrast,Ithinkitisenoughforthe\skeptics"topointoutthatthereisnobasistothe(over-simpliedandcounter-intuitivetomytaste)speculationbywhichaQCcanmanipulateormaintainsuchhugeobjects\freeofcost"(i.e.,atunitcost).Motivatedbythe\hands-o"approachofGoldreichandotherskeptics,in[1]Itriedtocarryouttheskeptics'researchprogramforthem,byproposingandanalyzingpossibleSure/Shorseparators.Thegoalwastoillustratewhatascienticargumentagainstquantumcomputingmightlooklike.Forstarters,suchanargumentwouldbecarefultoasserttheimpossibilityonlyoffutureexperiments,notexperimentsthathavealreadybeendone.Asanexample,itwouldnotdismissexponentially-smallamplitudesasphysicallymeaningless,sinceonecaneasilyproducesuchamplitudesbypolarizingnphotonseachat45.Norwoulditappealtothe\absurd"numberofparticlesthataquantumcomputerwouldneedtomaintainincoherentsuperposition|sinceamongotherexamples,theZeilingergroup'sC60double-slitexperiments[4]havealreadydemonstrated\Schrodingercatstates,"oftheformj0i\nn+j1i\nn p 2,fornlargeenoughtobeinterestingforquantumcomputation.Ofcourse,therealproblemisthat,onceweacceptj iandj'iintooursetofpossiblestates,consistencyalmostforcesustoacceptj i+j'iandj i\nj'iaswell.Soisthereanydefensibleplacetodrawaline?Thisconundrumiswhatledmetoinvestigate\treestates":theclassofn-qubitpurestatesthatareexpressiblebypolynomial-sizetreesoflinearcombinationsandtensorproducts.Asanexample,thestatej0i+j1i p 2\n


\nj0i+j1i p 2isatreestate.Forthatmatter,soisanystatethatcanbewrittensuccinctlyintheDiracketnotation,usingonlythesymbolsj0i;j1i;+;\n;(;)togetherwithconstants(noP'sareallowed).InevaluatingtreestatesasapossibleSure/Shorseparator,weneedtoaddresstwoquestions:rst,shouldallquantumstatesthatariseinpresent-dayexperimentsbeseenastreestates?Andsecond,wouldaquantumcomputerpermitthecreationofnon-treestates?Myresultsimplyapositiveanswertothesecondquestion:notonlycouldaquantumcomputerecientlygeneratenon-treestates,butsuchstatesarisenaturallyinseveralquantumalgorithms.2Inparticular,letCbearandomlinearcodeoverGF2.Thenwithoverwhelmingprobability,auniformsuperpositionoverthecodewordsofCcannotberepresentedbyanytreeofsizen"logn,forsomexed"�0.3Indeed,n"lognsymbolswouldbeneededeventoapproximatesuchastatewellinL2-distance,andevenifwereplacedtherandomlinearcodebyacertainexplicitcode(obtainedbyconcatenatingtheReed-SolomonandHadamardcodes).Ialsoshowedann"lognlowerboundforthestatesarisinginShor'salgorithm,assumingan\obviouslytrue"butapparentlydeepnumber-theoreticconjecture:basically,thatthemultiplesofalargeprimenumber,whenwritteninbinary,constituteadecenterasurecode.Alloftheseresultsrelyonaspectacularrecentadvanceinclassicalcomputerscience:therstsuperpolynomiallowerboundson\multilinearformulasize,"whichwereprovenbyRanRaz[12]aboutamonthbeforeIneededthemformyquantumapplication.Incidentally,inallofthecasesdiscussedabove,Iconjecturethattheactualtreesizesareexponentialinn;currently,though,Raz'smethodcanonlyprovelowerboundsoftheformn"logn.4Perhapsmorerelevanttophysics,Ialsoconjecturethat2-Dand3-D\clusterstates"(informally,2-Dand3-Dlatticesofqubitswithpairwisenearest-neighborinteractions)haveexponentialtreesizes.5Iftrue,thisconjecturesuggeststhatstateswithenormoustreesizesmighthavealreadybeenobservedincondensed-matterexperiments|forexample,thoseofGhoshetal.[6]onlong-rangeentanglementinmagneticsalts. 2Ontheotherhand,Idonotknowwhetheraquantumcomputerrestrictedtotreestatesalwayshasanecientclassicalsimulation.AllIcanshowisthatsuchacomputerwouldbesimulablein3\3,thethirdlevelofthepolynomial-timehierarchy.3InRaz'sproof,"wasabout106,butwhat'saconstantbetweenfriends(ormoreprecisely,betweentheoreticalcomputerscientists)?4Ididmanagetoproveanexponentiallowerbound,providedwerestrictourselvestolinearcombinationsj i+j'ithatare\manifestlyorthogonal"|whichmeansthatforallcomputationalbasisstatesjxi,eitherh jxi=0orh'jxi=0.5Bycontrast,Icanshowthat1-DclusterstateshavetreesizeO n4 .2 Inmypersonalfantasyland,oncetheevidencecharacterizingthegroundstatesofthesecondensed-mattersystemsbecameoverwhelming,theskepticswouldcomebackwithanewSure/Shorseparator.Thentheexperimentalistswouldtrytorefutethatseparator,andsoon.Asaresult,whatstartedoutasaphilosophicaldebatewouldgraduallyevolveintoascienticone|onwhichprogressnotonlycanbemade,butis.Prong2:It'sNotThatBadTodescribeastateofnparticles,weneedtowritedownanexponentiallylongvectorofexponentiallysmallnumbers,whichthemselvesvarycontinuously.Moreover,theinstantwemeasureaparticle,we\collapse"thevectorthatdescribesitsstate|andnotonlythat,butpossiblythestateofanotherparticleontheoppositesideoftheuniverse.Quick,whattheoryhaveIjustdescribed?Theanswerisclassicalprobabilitytheory.Themoralisthat,beforewethrowupourhandsoverthe\extravagance"ofthequantumworldview,weoughttoask:isitsomuchmoreextravagantthantheclassicalprobabilisticworldview?Afterall,bothinvolvelineartransformationsofexponentiallylongvectorsthatarenotdirectlyobservable.Bothallowfaulttolerance,instarkcontrastwithanalogcomputing.Neitherletsusreliablypackn+1bitsintoann-bitstate.Andneither(apparently!)wouldprovideenoughpowertosolveNP-completeproblemsinpolynomialtime.Butnoneofthisaddressesthecentralcomplexity-theoreticquestion:ifsomeonegivesyouapolynomial-sizequantumstate,howmuchmoreusefulisthatthanbeinggivenasamplefromaclassicalprobabilitydistribution?Intheirtextbook[11],NielsenandChuangweregettingatthisquestionwhentheymadethefollowingintriguingspeculation:[W]eknowthatmanysystemsinNature\prefer"tositinhighlyentangledstatesofmanysystems;mightitbepossibletoexploitthispreferencetoobtainextracomputationalpower?Itmightbethathavingaccesstocertainstatesallowsparticularcomputationstobedonemuchmoreeasilythanifweareconstrainedtostartinthecomputationalbasis.Toacomplexitytheoristlikeme,them'sghtin'words|oratleast,complexity-class-denin'words.Inparticular,let'sconsidertheclassBQP=qpoly,whichconsistsofallproblemssolvableinpolynomialtimeonaquantumcomputer,ifthequantumcomputerhasaccesstoapolynomial-size\quantumadvicestate"j nithatdependsonlyontheinputlengthn.(Fortheuninitiated,BQPstandsfor\Bounded-ErrorQuantumPolynomial-Time,"and=qpolymeans\withpolynomial-sizequantumadvice.")Notethatj nimightbearbitrarilyhardtoprepare;forexample,itmighthavetheform2n=2Pxjxijf(x)iforanarbitrarilyhardfunctionf.Wecanimaginethatj niisgiventousbyabenevolentwizard;theonlydownsideisthatthewizarddoesn'tknowwhichinputx2f0;1gnwe'regoingtoget,andthereforeneedstogiveusasingleadvicestatethatworksforallx.Theobviousquestionisthis:isquantumadvicemorepowerfulthanclassicaladvice?Inotherwords,doesBQP=qpoly=BQP=poly,whereBQP=polyistheclassofproblemssolvableinquantumpolynomialtimewiththeaidofpolynomial-sizeclassicaladvice?Asusualincomplexitytheory,theansweristhatwedon'tknow.Thisraisesadisturbingpossibility:couldquantumadvicebesimilarinpowertoexponential-sizeclassicaladvice,whichwouldletussolveanyproblemwhatsoever(sincewe'dsimplyhavetostoreeverypossibleanswerinagiantlookuptable)?Inparticular,couldBQP=qpolycontaintheNP-completeproblems,orthehaltingproblem,orevenallproblems?Ifyouknowme,youknowI'dsooneracceptthatpigscan\ry.Buthowtosupportthatconviction?In[2],Ididsowiththehelpofyetanothercomplexityclass:PostBQP,orBQPwithpostselection.Thisistheclassofproblemssolvableinquantumpolynomialtime,ifatanystageyoucouldmeasureaqubitandthenpostselectonthemeasurementoutcomebeingj1i(inotherwords,ifyoucouldkillyourselfiftheoutcomewasj0i,andthenconditiononremainingalive).MymainresultwasthatBQP=qpolyiscontainedinPostBQP=poly.Looselyspeaking,anythingyoucandowithpolynomial-sizequantumadvice,youcan3 alsodowithpolynomial-sizeclassicaladvice,providedyou'rewillingtouseexponentiallymorecomputationtime(orsettleforanexponentiallysmallprobabilityofsuccess).6Ontheotherhand,sinceNPPP,thisresultstillsaysnothingaboutwhetheraquantumcomputerwithquantumadvicecouldsolveNP-completeproblemsinpolynomialtime.Toaddressthatquestion,in[2]Ialsocreateda\relativizedworld"whereNP6BQP=qpoly.Thismeans,roughly,thatthereisno\brute-force"methodtosolveNP-completeproblemsinquantumpolynomialtime,evenwiththehelpofquantumadvice:anyproofthatNPBQP=qpolywouldhavetousetechniquesradicallyunlikeanyweknowtoday.Inmyview,theseresultssupporttheintuitionthatquantumstatesare\morelike"probabilitydis-tributionsovern-bitstringsthanlikeexponentially-longstringstowhichonehasrandomaccess.Ifexponentially-longstringswererocketfuel,andprobabilitydistributionsweregrapejuice,thenquantumstateswouldbewine|thealcoholic\kick"inthisanalogybeingtheminussigns.Icanimaginesomeoneobjecting:\Whataloadofnonsense!Whetherquantumstatesaremorelikegrapejuiceorrocketfuelisnotamathematicalquestion,aboutwhichtheoremscouldsayanything!"TowhichI'drespond:ifresultssuchasBQP=qpolyPostBQP=poly,whichsharplylimitthepowerofquantumadvice,donotcountasevidenceagainstGoldreich'sviewofquantumstates,thenwhatwouldcountasevidence?Andifnothingwouldcount,thenhowscienticallymeaningfulisthatviewintherstplace?AmeatierobjectioncentersaroundarecentresultofRaz[13],thataquantuminteractiveproofsystemwheretheveriergetsquantumadvicecansolveanyproblemwhatsoever|orincomplexitylanguage,thatQIP=qpolyequalsALL.(HereALListheclassofallproblems,whichmeans,literally,theclassofallproblems.)Imadearelatedobservationin[2],whereIpointedoutthatPostBQP=qpolyequalsALL.7However,akeypointaboutbothresultsisthattheyhavenothingtodowithquantumcomputing,andindeed,wouldworkjustaswellwithclassicalrandomizedadvice.Inotherwords,theclassesIP=rpolyandPP=rpolyarealsoequaltoALL.SinceIlikemakingconjectures,I'llconjecturemoregenerallythatquantumadvicedoesnotwreakmuchhavocinthecomplexityzoothatisn'talreadywreakedbyrandomizedadvice.Soforexample,I'llconjecturethatjustastheclassMA=rpolyisstrictlycontainedinALL,8sotooitsquantumanalogueQMA=qpolyisstrictlycontainedinALL.Imightbeprovenwrong,butthat'sthewholepoint!ConclusionForalmostacentury,quantummechanicswaslikeaKabbalisticsecretthatGodrevealedtoBohr,Bohrrevealedtothephysicists,andthephysicistsrevealed(clearly)tonoone.Solongasthelasersandtransistorsworked,therestofusshruggedatallthetalkofcomplementarityandwave-particleduality,takingforgrantedthatwe'dneverunderstand,orneedtounderstand,whatsuchthingsactuallymeant.Buttoday|largelybecauseofquantumcomputing|theSchrodinger'scatisoutofthebag,andallofusarebeingforcedtoconfronttheexponentialBeastthatlurksinsideourcurrentpictureoftheworld.Andasyou'dexpect,noteveryoneishappyaboutthat,justasthephysiciststhemselvesweren'tallhappywhentheyrsthadtoconfrontitinthe1920's.Yetthisuneasehastocontendwithtwotraditionsoftechnicalresults:therstshowingthatmanyoftheobviousalternativestoquantummechanicsarenonstarters;andthesecondshowingthatquantummechanicsisn'tquiteasstrangeasonewouldnavelythink.Bothtraditionsaredecadesold:therstincludesBell'stheorem[5]andtheKochen-Speckertheorem[9],whilethesecondincludesHolevo'stheorem[8]andtheresultsofdecoherencetheory.Buttheoreticalcomputerscientistscometoquantummechanicswiththeirownsetofassumptions(somewouldsayprejudices),sointhisabstractI'vetriedtoindicatehowthey,too,mighteventuallybeshovedintothevastquantumocean,whichisn'tthatcoldonceonegetsusedtoit. 6In[3]IcharacterizedPostBQPexactlyintermsoftheclassicalcomplexityclassPP(ProbabilisticPolynomial-Time),whichconsistsofalldecisionproblemssolvableinpolynomialtimebyarandomizedTuringmachine,whichacceptswithprobabilitygreaterthan1=2ifandonlyiftheansweris\yes."Thus,amoreconventionalwaytostatemyresultisBQP=qpolyPP=poly.7Here'saone-sentenceproof:giventheadvicestate2n=2 xjxijf(x)i,toevaluatetheBooleanfunctionfonanygiveninputxwesimplyneedtomeasureinthestandardbasis,thenpostselectonseeingthejxiofinterest.8IndeedMA=rpoly=MA=poly;thatis,wecanreplacetherandomizedadvicebydeterministicadvice.4 References[1]S.Aaronson.Multilinearformulasandskepticismofquantumcomputing.InProc.ACMSTOC,pages118{127,2004.quant-ph/0311039.[2]S.Aaronson.Limitationsofquantumadviceandone-waycommunication.TheoryofComputing,1:1{28,2005.quant-ph/0402095.[3]S.Aaronson.Quantumcomputing,postselection,andprobabilisticpolynomial-time.Proc.Roy.Soc.London,2005.Toappear.quant-ph/0412187.[4]M.Arndt,O.Nairz,J.Vos-Andreae,C.Keller,G.vanderZouw,andA.Zeilinger.Wave-particledualityofC60molecules.Nature,401:680{682,1999.[5]J.S.Bell.SpeakableandUnspeakableinQuantumMechanics.Cambridge,1987.[6]S.Ghosh,T.F.Rosenbaum,G.Aeppli,andS.N.Coppersmith.Entangledquantumstateofmagneticdipoles.Nature,425:48{51,2003.cond-mat/0402456.[7]O.Goldreich.Onquantumcomputing.www.wisdom.weizmann.ac.il/~oded/on-qc.html,2004.[8]A.S.Holevo.Someestimatesoftheinformationtransmittedbyquantumcommunicationchannels.ProblemsofInformationTransmission,9:177{183,1973.Englishtranslation.[9]S.KochenandS.Specker.Theproblemofhiddenvariablesinquantummechanics.J.ofMath.andMechanics,17:59{87,1967.[10]L.A.Levin.Polynomialtimeandextravagantmodels,inThetaleofone-wayfunctions.ProblemsofInformationTransmission,39(1):92{103,2003.cs.CR/0012023.[11]M.NielsenandI.Chuang.QuantumComputationandQuantumInformation.CambridgeUniversityPress,2000.[12]R.Raz.Multi-linearformulasforpermanentanddeterminantareofsuper-polynomialsize.InProc.ACMSTOC,pages633{641,2004.ECCCTR03-067.[13]R.Raz.QuantuminformationandthePCPtheorem.InProc.IEEEFOCS,2005.Toappear.quant-ph/0504075.5