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1 NP-completeproblemstractable Copyingquantumcomputermakes MikaHirvensalo February1998 ISBN TurkuCentreforComputerScience ISSN TUCSTechnicalReportNo161
2 superpositions,weshowthatnp-completeproblemscanbesolvedprobabilisticallyinpolynomialtime.wealsoproposetwomethodsthatcould potentiallyallowtoavoidtheuseofaquantumcopymachine. SupportedbytheAcademyofFinlandundergrant Undertheassumptionthataquantumcomputercanexactlycopyquantum Abstract TobepresentedatMCU'98,March1998,Metz,France. TheoryGroup:MathematicalMethodsinComputerScience TUCSResearchGroup
3 simulatequantumphysicalphenomenabyanordinarycomputerwithoutan ItwasconjecturedbyR.Feynman[5]in1982thatitmaybeimpossibleto 1 Introduction exponentialslowdownintheeciencyofthesimulation.inhiswork,feynmanalsosuggestedthattheslowdowncouldbeavoidedbyallowingthe PeterW.Shordiscoveredhisfamousalgorithmsforfactoringintegersand computerrunaccordingtorulesofquantummechanics,thusintroducing quiteamarginalphenomenoninthetheoryofcomputinguntil1994,when ndingdiscretelogarithmsinpolynomialtimebyaquantumcomputer[10]. theideaofquantumcomputer.however,quantumcomputationremained beensurprisinglylittleprogressintheeldofquantumcomputation.one lievedtobeclassicallyintractable.however,aftershor'sresults,therehas Bothtasksplayanessentialroleinpublic-keycryptographyandarebegestthatNP-completeproblemscannotbesolvedbyaquantumcomputer inpolynomialtime.ontheotherhand,berthiaumeandbrassard[3]arguedthatquantumcomputationmaybeexponentiallymorepowerfulthan theclassicalcomputation.thisapparentdiscrepancybecomesunderstandablewhenwenoticethatbothoftheseresultswererelative,i.e.theywere obtainedusingoraclecomputations.thestrongestnon-relativeresultso faristhequadraticspeed-upgivenbygrover[6],whichtellsusthatnp- remarkableresultwasgivenbybennett&al.in[2],wheretheauthorssug- 2wherep(n)isapolynomialdependingontheproblem. completeproblemscanbesolvedbyaquantumcomputerintimep2np(n), Inthissectionwewillshortlyoutlinethebasicfeaturesofquantumcomputation.Nopreviousknowledgeaboutquantumphysicsisneeded,nor Quantumcomputationsbriey computationisgiveninapurelymathematicalway. anyconnectionstophysicsaregivenhere,buttheformalismofquantum plexhilbertspace,sayj0iandj1i(socalledket-notationjiisduetop. tumbitsaredescribedbyorthonormalbasisvectorsintwo-dimensionalcom- quantumbits,qubitscanappearinsuperpositionsofclassicalstates.quan- Themaindierencebetweenquantumandclassicalcomputationisthat bitsandisdepictedbyann-foldtensorproductoftwo-dimensionalhilbert sitionsofbasisstates.aquantumregisteroflengthnconsistsofnquantum referredasbasisstatesandtheirunit-lengthlinearcombinationsassuperpo- Dirac).ThisHilbertspaceiscalledthestatespaceandthesetwovectorsare spaces.weusenotations plexhilbertspacewithorthonormalbasisstatesthatcanbeexpressedas Thestatespaceofaquantumregisterthusbecomesa2n-dimensionalcom- jx1ijx2i:::jxni=jx1ijx2i:::jxni=jx1x2:::xni: (1) 1
4 Theunit-lengthlinearcombinationsofthesestatesarecalledsuperpositions (replacingthebinarystringswiththenumberstheyrepresent) aswell. j0i;j1i;j2i;:::;j2n?1i: (2) thonormalset).theunitarityimpliesthatquantumcomputationisre- versible,butthisisnotarestriction:foranyturingmachinethereisa reversibleturingmachinethatdoesthesamecomputationthattheoriginal onedid(see[1]or[8]). isaunitarymappinguthatcomputesafunctionf:n!f0;1g(thismeans thatujjij0i=jjijf(j)i).then Quantumparallelismisaconsequenceofthelinearity:Assumethatthere viaunitarymappings(recallthatunitarityinnitedimensionmeansthat therowsandthecolumnsofthematrixofthemappingbothformanor- Thenatureofquantumcomputationislinear:Quantumregistersevolve = U1 p2n?j0ijf(0)i+j1ijf(1)i+:::+j2n?1ijf(2n?1)i; 1p2n?j0ij0i+j1ij0i+:::+j2n?1ij0i costofonesinglecomputation.butthisisnotthewholetruth:observation Whichmeansthatwecancomputeexponentiallymanyvaluesoffwitha (3) (3)tocollapseintostatejjijf(j)ithatwasobserved,destroyingtherestof (3)irreversibly. eachwithprobability1 of(3)willgiveusjustoneofstatesjjijf(j)i,wherej2f0;1;:::;2n?1g, Ingeneral,observationofk:thqubitwhentheregisteroflengthnisin 2n.Moreover,theobservationprocedureshallcause willyield1or0withprobabilities state P(1)=Xj2Jjjj2;whereJ=fjjthekthqubitofxjis1g; 0jx0i+1jx1i+:::+2n?1jx2n?1i; (5) (4) Moreover,thepost-observationstatewillbe P(0)=Xj2J0jjj2;whereJ0=fjjthekthqubitofxjis0g:(6) dependingonwhichoneofvalues1or0wasseen. qpj2jjjj2xj2jjjxjior 1 qpj2j0jjj2xj2j0jjxji; 1 (7) destroytheadvantageobtainedinsuperposition(3),namelythatall2n Thereductionofthestateduringtheobservationprocedureseemsto 2
5 featuresoffthatmayinterestus:wemayask,forexample,isfaconstant valuesoffwerecomputedsimultaneously.sowecannotdirectlyreveal function?consider,forinstance,superposition(3)withf(j)=0forallj: p2n2n?1 1Xj=0jjij0i andwithafunctionf(k)=1butf(j)=0forj6=k: (8) Ifweobservethelastqubitofsuperposition(9),theprobabilitiesofseeing1 p2n?xj6=kjjij0i+jkij1i: 1 isonly1 iterativemethodfor\amplifying"this1.ontheotherhand,thedistance alwayspreservethedistance,sowemaybelievethatitisextremelydicult of(8)and(9)inthestatespaceisonly1 2n,so1istoo\faint"tobeseen.Inthenextsectionweintroducean mappings(theprocedureintroducedin[6]doestheamplicationinp2n tondoutanyrapidamplicationprocedurebasedonlyontheunitary 2n?1andtheunitarymappings iterations).therefore,wewilltrysomehowtoutilizethestatereductions 3intheobservations. Inthissectionweconsiderproblem\generalizedsatisability",gsat:Let =f0;1gbethebinaryalphabetandfacompletelydenedblackbox Amplicationprocedure functionf:!.theconcept\blackboxfunction"heremeansthat problemgsatistodecidewhether,fora\given"blackboxfunctionthereis function.weshallagaindenotetherestrictionoffintonbyf.the structure"offisnotknown-wedonotassumethatfisacomputable thevaluef(x)isobtainedinasinglecomputationstepandthatthe\inner awordx2nsuchthatf(x)=1. theproblemaboveisnp-complete,sincesatisabilityofapropositional expressionwouldclearlybecomeasubcaseofgsat. Ifweassumethatfisagivenfunctioncomputableinpolynomialtime, 1f(k)=1butf(j)=0forallk6=j2nand Letusrstconsiderhowtotellthedierencebetweenthecases canbe\called"instantaneously,whenqubitsrepresentthevaluej.wealso Ifweassumeftobeablackboxfunction,wesupposethatthevaluef(j) 2f(j)=0forallj2n. 3
6 assumethatfwillrespectlinearnatureofquantumphysics,thatis,asingle calloffonsuperposition p2n2n?1 willproduce 1Xj=0jjij0i (10) p2n2n?1 If,ontheotherhand,weassumethatfiscomputableinpolynomialtime, 1Xj=0jjijf(j)i: (11) performsthecomputationujjij0i=jjijf(j)i: manyunitarymatricesthatoperateonsubspacesofboundeddimensionand thenthereisamatrixu(see[7]or[13])thatisaproductofpolynomially Moreover,Ucanbeecientlyfound.Letalso (12) bethehadamardtransformationand H=1p21?1 1 (13) D= 0 B@ 1 1p2?1p beamatrixthatactsoncomputationalbasisfj00i;j01i;j10i;j11ig,soj00i 1 CA (14) ismappedinto1p2j00i+1p2j01iandj11iisaxedpoint. wepreparethequantumsuperposition Wewillintroducetheprocedurewhilestudyingcase1.Inthebeginning p2n2n?1 usinghadamardtransformations(see[11]).functionfisthenappliedon 1Xj=0jjij00i (15) then-qubitstringjjitoobtain p2n2n?1 1Xj=0jjijf(j)f(j)i=1 p2nxj6=kjjij00i+jkij11i 4 (16)
![thenthereisamatrixu(see[7]or[13])thatisaproductofpolynomially Moreover,Ucanbeecientlyfound.Letalso (12) bethehadamardtransformationand H=1p21?1 1 (13) D= 0 B@ 1 1p2?](/docs-images/58/4026497/images/page_6.jpg "1p200 1 0 0 10 01 beamatrixthatactsoncomputationalbasisfj00i;j01i;j10i;j11ig,soj00i 1 CA (14) ismappedinto1p2j00i+1p2j01iandj11iisaxedpoint.")
7 ApplyingDonthelasttwoqubitswillyield p2nxj6=k?1p2jjij00i+1 1 p2jjij01i+jkij11i seeingoneandzeroare2n+1 anditiseasytoseethat,ifweobservethelastqubit,theprobabilitiesof 22nand2n?1 (17) one,thenthepost-observationstateis p2n+1xj6=kjjij01i+p2jkij11i; 1 22nrespectively.Iftheobservationyields aseasilycalculated.theprocedurecontinuesbyaddingtwoextraqubits (18) thusobtainingstate(wecanaswellthinkthattheseextraqubitsexistedin thebeginning)1 Thenwecomputethefunctionfagain,thistimesavingtheresultinthe p2n+1xj6=kjjij01ij00i+p2jkij11ij00i: (19) newqubits.theresultingstateis ApplyingDonlasttwoqubitswillnowyield p2n+1xj6=kjjij01ij00i+p2jkij11ij11i: 1 (20) p2n+1xj6=k(1p2jjij01ij00i+1 1 p2jjij01ij01i)+p2jkij11ij11i: Ifwenowobservethelastqubit,theprobabilitiesofseeingoneandzeroare 2(2n+1)and2n?1 2n+3 2(2n+1)respectivelyandtheresultingstate,ifonewereobserved, (21) willbe Wecontinueagainaddingtwoextraqubits,applyingDandmeasuringthe p2n+3xj6=kjjij01ij01i+2jkij11ij11i: 1 (22) inductionthatthestateafterl?1thiterationis lastqubit.supposingthatthemeasurementalwaysyields1,weseeusing p2n+2l?1?1xj6=kjjij01il?1+p2l?1jkij11il?1; 1 5 (23)
8 probabilitiesofseeing1and0arethen whereja0a1il?1meansl?1copiesofja0a1i.againweaddtwoextra qubits,computefandapplydonthelastqubits.itiseasytoseethatthe Pl(1)= 2(2n+2l?1?1)andPl(0)= 2n+2l?1 2(2n+2l?1?1): 2n?1 ofseeing1atthen+1thmeasurementisgivenby Ifwesucceedtomeasure1atnconsecutiveiterations,thentheprobability (24) Ontheotherhand,letuslookatwhathappensincase2.Thestate Pn+1(1)=2n+2n+1?1 2(2n+2n?1)=3?2?n correspondingto(17)willbe 4?21?n34: (25) p2n2n?1 1Xj=0?1p2jjij00i+1p2jjij01i = p2n+12n?1 Heretheprobabilitiesofseeing1and0inthelastqubitareboth12,andthe 1 Xj=0?jjij00i+jjij01i (26) post-measurementstate,if1wereobserved,willbe p2n2n?1 1Xj=0jjij01i probabilityofseeing1inthelastqubit-itwillbe12eachtime. Weseeimmediatelythattheproceduredescribedabovedoesnotchangethe (27) dierthecases1and2bythepropertythatafternthiterationthesecond casegives1withprobability12buttherstonecasewithprobabilityatleast 34.Inthenextsectionitwillbeshownthatthesecasescanbedistinguished So,ifwecould\force"1tobeobservedintherstniterations,wecould reliablybyconstantnumberoftrials.moreover,wecanimmediatelysee thatiftherearemorethanonevaluekgivingf(k)=1,thentheprobability observedeverytime.infact,theprobabilityofobserving1atnconsecutive ofseeing1afterniterationsisatleast34. measurementsisgivenby(seeequations(24)) However,thisprocedurewillnotwork,sincewecannotforce1tobe threewayshowwecouldtrytoavoidthisproblem(therstonebeingthe whichisexponentiallysmall(withrespectton).therefore,wepropose P1(1)P2(1):::Pn(1)=22n?1 22n =1 2n?1?1 22n; (28) extensionofmodel,thelasttwoonesremainasfutureresearch): 6
9 Ifwehadaquantumcopymachine,whichcouldproduceacopyof reader),sincetheprobabilityofseeingmzerosconsecutivelyisatmost?12m.unfortunately,exactquantumcopymachinesdonotexist[12], notlengthentheexpectedrunningtimeessentially(detailsareleftto (0wasobserved)restore(23),copyit,andobserveagain.Thiswould eachsuperposition(23)wecouldafteranunsuccessfulmeasurement Maybeacleverquantumerror-correctionschemecoulddotherestoratumcopymachinetionafteranunsuccessfulmeasurementwithoutanassumptionofquan- butmaybewecouldsomehowutilizeapproximatecopying[4]. Infact,wecanforce1tobeobservedeachtime:Weshouldrst preparesuperposition thatwouldcorrespondto(15).thenwewouldruntheprotocoldescribedintheprevioussection,usingthequbitaddedinthefront (29) 1nj1i?1 p2n2n?1 Xj=0jjij00in+r1?1n2j0ij0nij01in abilityofseeing1oneachmeasurementwillbeatleast1?1n2.now, protocol,butthe1'sonsuitableplacestherewillensurethattheprob- sothelatter,heavily-weightedpartof(29)willnotchangeduringthe asacontrolbit:thecomputingisdoneonlyifthecontrolbitis1, thisapproachishowto\annihilate"theheavily-weightedpartof(29) theprobabilityofseeing1innconsecutivemeasurementsisatleast afterniterations. (1?1n2)n1?1nbyBernoulli'sinequality.Thegreatdicultyin Theproblemoftellingthedierencebetweencases1and2afternsuc- Fairandfakecoins 4cessfuliterationsisequivalenttothefollowingone:Wehaveapileofcoins, wechoosearbitrarilyacoin,howmanytossesweneedtotell,withprobability1?",whetherthecoinwasfairorfakeabilitypandtail(0)withprobabilityq=1?p.weexaminetherandom variablexdescribingmtosses,say Letusassumethatthecointhatwaschosengiveshead(1)withprob- wheresomecoinsarefairbutsomegiveheadwithprobabilityatleast34.if whereeachxiis1withprobabilitypand0withprobabilityq.theprobabilitythattheexperimentgiveskheadsisgivenby X=x1+x2+:::+xm; (30) pk=p(x=k)=mkpkqm?k: 7 (31)
10 Moreover,theexpectedvalueandthevarianceofXaregivenby and Var(X)=E(X2)?E(X)2 E(X)=p00+p11+p22+:::+pmm=pm (32) ByChebyshev'sInequality[9]wehave =p002+p112+:::+pmm2?(pm)2=pqm: (33) whichyields,fork=m8(thischoisebecausem2+m8=5m8=3m4?m8), P(jX?E(X)jk)Var(X) k2; (34) Aftermtosseswelookatthenumberofheads.Ifthisnumberiscloserto m2thanto3m4,wedecidethatthecoinwasfair,otherwiseweconcludethat P(jX?pmjm8)64pq m16m: (35) thecoinwasfake.by(35),ourconclusionisfalsewithprobabilityatmost References m.takingm>16"ourdecisioniscorrectwithreliabilityatleast1?". [1]C.H.Bennett:LogicalReversibilityofComputation,IBMjournalofRes. [2]C.H.Bennett,EthanBernstein,G.BrassardandUmeshV.Vazirani:StrengthsandWeaknessesofQuantumComputation,LosAlamos Preprint,Dec1,1994. Development17,p ,Nov1973. [3]AndreBerthiaumeandGillesBrassard:OracleQuantumCompu- [4]V.BuzekandM.Hillery:Universaloptimalcloningofqubitsandquantumregisters,tobepresentedattheFirstNASAConferenceofQuantum PhysComp'92,October1992,IEEEPress,pp tation,proceedingsoftheworkshoponphysicsandcomputation- [5]RichardP.Feynman:SimulatingPhysicswithComputers,International ComputingandQuantumCommunications,17-20,February1998,Palm SpringsU.S.A. JournalofTheoreticalPhysics,Vol21,N6/7,
11 [6]LovK.Grover:Afastquantummechanicalalgorithmfordatabase [7]MikaHirvensalo:OnQuantumComputation,Licentiate'sThesis,UniversityofTurku,1997. search,proceedings,stoc1996,philadelphiapausa,pp [9]SheldonM.Ross:Introductiontoprobabilitymodels,4thedition,AcademicPress,Inc.,USA,1989. ProceedingsofDLT'97,toappear. [8]MikaHirvensalo:TheReversibilityinQuantumComputationTheory, [10]Peter.W.Shor:AlgorithmsforQuantumComputation:DiscreteLog [11]Daniel.R.Simon:OnThePowerofQuantumComputation,Proceedingsofthe35thAnnualSymposiumonFoundationsofComputerSciencedationsofComputerScience,pp.22-22,1994. andfactoring,proceedingsofthe35thannualsymposiumonthefoun- [12]W.K.WoottersandW.H.Zurek,Nature299,802(1982) pp ,1994. [13]AndrewChi-ChihYaoQuantumCircuitComplexity,FOCS(pp )
![Shor:AlgorithmsforQuantumComputation:DiscreteLog [11]Daniel.R.Simon:OnThePowerofQuantumComputation,Proceedingsofthe35thAnnualSymposiumonFoundationsofComputerSciencedationsofComputerScience,pp.](/docs-images/58/4026497/images/page_11.jpg "22-22,1994. andfactoring,proceedingsofthe35thannualsymposiumonthefoun- [12]W.K.WoottersandW.H.Zurek,Nature299,802(1982) pp.116-123,1994. [13]AndrewChi-ChihYaoQuantumCircuitComplexity,FOCS(pp.")
12 TurkuCentreforComputerScience FIN-20520Turku Finland Lemminkaisenkatu14 UniversityofTurku DepartmentofMathematicalSciences AboAkademiUniversity DepartmentofComputerScience InstituteforAdvancedManagementSystemsResearch TurkuSchoolofEconomicsandBusinessAdministration InstituteofInformationSystemsScience
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NO. HU-DOP_TN-212-25_001 Product type 212 (TN) 3,5x25 mm EN 14566:2008+A1:2009 NO. HU-DOP_TN-212-35_001 Product type 212 (TN) 3,5x35 mm EN 14566:2008+A1:2009 NO. HU-DOP_TN-212-45_001 Product type 212 (TN)
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