BLONDEL,JEANDEL,KOIRAN,ANDPORTIERComplexversusrealentries.Throughoutth - PDF document

BLONDEL,JEANDEL,KOIRAN,ANDPORTIERComplexversusrealentries.Throughoutthepaperwewillassumethattheinitialstate,theunitarymatrices,andtheprojectionmatrixhaverealratherthancomplexentries(i.e.,thesematricesareactuallyorthogonal).Thisisnotasigni“cantrestrictionsinceanyquantumautomaton(withpossiblycomplexentries)canbesimulatedbyanotherquantumautomatonwithrealentriesbydoublingthenumberofstates.Moreprecisely,letbethesetofstatesof.Wereplaceeachelementbytwostates.Letbethe-linearmapwhichsendsacon“guration.Wereplacetheinitialcon“guration).LetbeoneofthematricesofTherowsandcolumnsofareindexedbyelementsof.Letbetheentryatrowandcolumn.Recallthatacomplexnumbercanbeidenti“edtothe22matrixItisthereforenaturaltoreplacethisentrybythe22matrixThetworowsandtwocolumnsofthismatrixareindexed,respectively,by.Byabuseofnotationwealsodenotebythemapwhichsends.Itiseasybutinstructivetocheckthatforanyandforanycomplexmatricesthefollowingrelationshold:,and).Nowrecallthatunitarymatrices,orthogonalmatrices,complexmatricesoforthogonalprojection,andrealmatricesoforthogonalprojectionare,respectively,characterizedbythefollowingrelations:,and.Itfollowsthatsendsunitarymatricestoorthogonalmatrices,andcomplexmatricesoforthogonalprojectiontorealmatricesoforthogonalprojection.Thequantumautomatonde“nedbytheorthogonalmatrices),theprojectionmatrix),andtheinitialcon“gurationforanyword.HenceVal)=Val)foranyword2.Undecidabilityfornonstrictinequality.Weproveinthissectionthattheproblemofdeterminingifaquantumautomatahasawordofvaluelargerthanorequaltosomethresholdisundecidable.TheproofisbyreductionfromPostscorrespondenceproblem(PCP),awell-knownundecidableproblem.AninstanceofPCPisgivenbya“nitealphabetandpairsofwords(.Asolutiontothecorrespondenceisanynonemptywordoverthealphabetsuchthat,where.Thiscorrespondenceproblemisknowntobeundecidable:thereisnoalgorithmthatdecidesifagiveninstancehasasolution[Pos46].Itiseasytoseethattheproblemremainsundecidablewhenthealphabetcontainsonlytwoletters.Theproblemisalsoknowntobeundecidablefor=7pairs[MS05]butisdecidablefor=2pairs;thedecidabilityofthecases27isnotyetknown.Wearenowreadytostateour“rstresult.Theorem2.1.ThereisnoalgorithmthatdecidesforagivenautomatonthereexistsanonemptywordforwhichVal,orifthereexistsoneforVal.Thesetwoproblemsremainundecidableeveniftheautomatonisgivenbyorthogonalmatricesindimension BLONDEL,JEANDEL,KOIRAN,ANDPORTIERwehavethatthesolutionsoftheoriginalPCPproblemarethewordsforwhich=0,whichisequivalenttoValThevaluestakenbyVal)arenonnegativeandsotheproblemofdeterminingifthereexistsanonemptywordsuchthatVal0isundecidable.Noticealso=1andsowithequalityonlyforThus,theproblemofdeterminingifthereexistsanonemptywordsuchthatVal1isundecidabletoo. Theorem2.1dealsonlywithnonemptywords.Weremovethisrestrictioninthenextresult,andwereducethenumberofmatricesfrom7to2.Corollary2.2.ThereisnoalgorithmthatdecidesforagivenautomatonifthereexistsawordforwhichVal,orifthereexistsoneforwhichVal.Theseproblemsremainundecidableeveniftheautomatonisgivenbyorthogonalmatricesindimension,orbyorthogonalmatricesindimensionProof.AsintheproofofTheorem2.1,theundecidabilityresultsfortheconditionVal1followfromthosefortheconditionVal0.Hencewesupplytheproofsforthelatterconditiononly.WeproceedbyreductionfromtheproblemVal0for7matricesindimension6,whichisundecidablefornonemptywordsasshowninTheorem2.1.NotethatthelanguageofthenonemptysuchthatVal0istheunionofthesevenlanguagesde“nedbytheconditionsVal0forpossiblyemptywords.Hencetheemptinessofoneoftheselanguages(say,the“rstone)mustbeundecidable.Thus,theproblemofdeterminingifthereexistsawordsuchthatVal0isundecidable.Foreachautomaton=((,s,P)wecannowconstructthequantum=((,y,P),where.ThenVal0ifandonlyifValThefollowingproblemisthereforeundecidable:givenaquantumautomatonde“nedby7orthogonalmatricesindimension6,istherea(possiblyempty)wordsuchthatValFinally,weshowhowtoreducethenumberofmatricesto2.WeuseaconstructionfromBlondelandTsitsiklis[BT97]andBlondelandCaterini[BC03].Giventheaboveandtheprojectionmatrix,wede“ne...YWhentakingproductsofthesetwomatricesthematrixactsasaselectingmatrixŽontheblocksof.Letusde“ne...P Itisnotdiculttoshowthatthe6otherproblemsmustbeundecidableaswell. BLONDEL,JEANDEL,KOIRAN,ANDPORTIERisclearlysemidecidable.Inordertoshowthatitisdecidable,itremainstoexhibitaprocedurethathaltswhenValforallLetaquantumautomatabegivenbya“nitesetoforthogonaltransition,aninitialcon“gurationofunitnorm,andaprojectionmatrix.ThevalueofthewordisgivenbyVal.Letbethesemigroupgeneratedbythematrices,andletbethefunctionde“nedbysXP.WehavethatValandtheproblemisnowthatofdeterminingifforall.Thefunctionisa(continuous)polynomialmapandsothisconditionisequivalenttoforall ,where istheclosureof.Theset hastheinterestingpropertythatitisalgebraic(seebelowforaproof),andsothereexistpolynomial,suchthat isexactlythesetofcommonzeros.Ifthepolynomialsareknown,theproblemofdeterminingforall canbewrittenasaquanti“ereliminationproblem···)=0)=Thisisa“rst-orderformulaovertherealsandcanbedecidedeectivelybyTarski…Seidenbergeliminationmethods(see[Ren92a,Ren92b,Ren92c,BPR96]forasurveyofknownalgorithms).Ifweknewhowtoeectivelycomputethepolynomialsfromthematrices,adecisionalgorithmwouldthereforefollowimmediately.Inthefollowingwesolveasimplerproblem:weeectivelycomputeasequenceofpolynomialswhosezerosdescribethesameset after“nitelymanyterms(butwemayneverknowhowmany).Itturnsoutthatthisissucientforourpurposes.Wewillusesomebasicalgebraicgeometry.Inparticular,wewillusetheNoether(ordescendingchainŽ)property:inany“eld,thesetofcommonzerosofasetof-variatepolynomialsisequaltothesetofcommonzerosofasubsetofthesepolynomials(seeanytextbookonalgebraicgeometry,forinstance,[CLO92,Prop.1,sect.4.6]).Theorem3.1.Letbeorthogonalmatricesofdimensionandlet theclosureofthesemigroup.Theset isalgebraic,andiftherationalentries,wecaneectivelycomputeasequenceofpolynomialssuchthat forallthereexistssomesuchthat Proof.We“rstprovethat isalgebraic.Itisknown(see,e.g.,[OV90])thateverycompactgroupofrealmatricesisalgebraic.Infact,theproofofalgebraicityin[OV90]revealsthatanycompactgroupofrealmatricesofsizeisthezerosetsetX]G={fR[X]:f(I)=0andisthezerosetofthepolynomialsinvariableswhichvanishattheidentityandareinvariantundertheactionof.Wewillusethispropertylaterintheproof.Toshowthat isalgebraic,itsucestoshowthat iscompactandisagroup.Theset isobviouslycompact(boundedandclosedinanormedvectorspaceof“nitedimension).Letusshowthatitisagroup.Itisinfactknownthateverycompactsubsemigroupofatopologicalgroupisasubgroup.Hereisaself-containedproofinoursetting:Foreverymatrix,thesequenceadmitsasubsequencethatisa BLONDEL,JEANDEL,KOIRAN,ANDPORTIERalgebraicnumbersover.Thispurelyalgebraicinformationissucienttocomputethesequenceofpolynomials()inTheorem3.1.Wealsoneedtodecideforeveryinitialsegmentwhether(3.1)holds.Afterquanti“erelimination,thisamountstocomputingthesignofa“nitenumberofpolynomialfunctionsoftheelementsofInordertodothisweneedonlyassumethatwehaveaccesstoanoraclewhichforanyelementandany0outputsarationalnumbersuchthat(suchanoraclecanbeeectivelyimplementediftheentriesarecomputablerealnumbers).Weusethealgebraicinformationtodeterminewhetherapolynomialtakesthevaluezero,andifnotweuseapproximationstodetermineitssign.IntheproofofTheorem3.2wehavebypassedtheproblemofexplicitlycomputinga“nitesetofpolynomialsde“ning .Itisinfactpossibletoshowthatthisproblemisalgorithmicallysolvable[DJK03].Thisimpliesinparticularthatthefollowingtwoproblemsaredecidable:(i)Decidewhetheragiventhresholdisisolated.(ii)DecidewhetheragivenQFAhasanisolatedthreshold.AthresholdissaidtobeisolatedifVal�.Itisknownthatthesetwoproblemsareundecidableforprobabilisticautomata[Ber75,BMT77,BC03].Thealgorithmof[DJK03]forcomputing alsohasapplicationstoquantumcircuits:thisalgorithmcanbeusedtodecidewhetheragivensetofquantumgatesiscomplete(completemeansthatanyorthogonaltransformationcanbeapproximatedtoanydesiredaccuracybyaquantumcircuitmadeupofgatesfromtheset).Mucheorthasbeendevotedtotheconstructionofspeci“ccompletesetsofgates[DBE95,95],butnogeneralalgorithmfordecidingwhetheragivensetiscompletewasknown.Finally,wenotethattheproofofTheorem3.2doesnotyieldanyboundonthecomplexityofproblems(i)and(ii).Wehopetoinvestigatethisquestioninfuturework.Acknowledgment.P.K.wouldliketothankEtienneGhysforpointingoutreference[OV90].Wearealsogratefultotheanonymousrefereeforhisverycarefulreadingofthemanuscript.uscript.M.AmanoandK.IwamaUndecidabilityonquantum“niteautomata,inProceedingsofthe31stACMSymposiumonTheoryofComputing,ACM,NewYork,1999,pp.368…375.368…375.+95]A.Barenco,C.H.Bennett,R.Cleve,D.P.DiVincenzo,N.H.Margolus,P.W.Shor,T.Sleator,J.A.Smolin,andH.WeinfurterElementarygatesforquan-tumcomputation,Phys.Rev.A,52(1995),pp.3457…3467.3457…3467.S.Basu,R.Pollack,andM.-F.RoyOnthecombinatorialandalgebraiccomplexityofquanti“erelimination,J.ACM,43(1996),pp.1002…1045.1002…1045.V.D.BlondelandV.CanteriniUndecidableproblemsforprobabilisticautomataof“xeddimension,TheoryComput.Syst.,36(2003),pp.231…245.231…245.A.BertoniThesolutionofproblemsrelativetoprobabilisticautomataintheframeoftheformallanguagestheory,inVierteJahrestagungderGesellschaftf¨urInformatik,LectureNotesinComput.Sci.26,Springer,Berlin,1975,pp.107…112.107…112.A.Bertoni,G.Mauri,andM.TorelliSomerecursivelyunsolvableproblemsrelatingtoisolatedcutpointsinprobabilisticautomata,inProceedingsofthe4thInternational