Quantum computer.docx
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Quantum computer.docx
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Quantumcomputer
Quantumcomputer
TheBlochsphereisarepresentationofaqubit,thefundamentalbuildingblockofquantumcomputers.
Aquantumcomputer(alsoknownasaquantumsupercomputer)isacomputationdevicethatmakesdirectuseofquantum-mechanicalphenomena,suchassuperpositionandentanglement,toperformoperationsondata.[1]Quantumcomputersaredifferentfromdigitalcomputersbasedontransistors.Whereasdigitalcomputersrequiredatatobeencodedintobinarydigits(bits),eachofwhichisalwaysinoneoftwodefinitestates(0or1),quantumcomputationusesqubits(quantumbits),whichcanbeinsuperpositionsofstates.AtheoreticalmodelisthequantumTuringmachine,alsoknownastheuniversalquantumcomputer.Quantumcomputerssharetheoreticalsimilaritieswithnon-deterministicandprobabilisticcomputers;oneexampleistheabilitytobeinmorethanonestatesimultaneously.ThefieldofquantumcomputingwasfirstintroducedbyYuriManinin1980[2]andRichardFeynmanin1982.[3][4]Aquantumcomputerwithspinsasquantumbitswasalsoformulatedforuseasaquantumspace–timein1969.[5]
Asof2014quantumcomputingisstillinitsinfancybutexperimentshavebeencarriedoutinwhichquantumcomputationaloperationswereexecutedonaverysmallnumberofqubits.[6]Bothpracticalandtheoreticalresearchcontinues,andmanynationalgovernmentsandmilitaryfundingagenciessupportquantumcomputingresearchtodevelopquantumcomputersforbothcivilianandnationalsecuritypurposes,suchascryptanalysis.[7]
Large-scalequantumcomputerswillbeabletosolvecertainproblemsmuchmorequicklythananyclassicalcomputerusingthebestcurrentlyknownalgorithms,likeintegerfactorizationusingShor'salgorithmorthesimulationofquantummany-bodysystems.Thereexistquantumalgorithms,suchasSimon'salgorithm,whichrunfasterthananypossibleprobabilisticclassicalalgorithm.[8]Givensufficientcomputationalresources,however,aclassicalcomputercouldbemadetosimulateanyquantumalgorithm;quantumcomputationdoesnotviolatetheChurch–Turingthesis.[9]
Basis[edit]
Aclassicalcomputerhasamemorymadeupofbits,whereeachbitrepresentseitheraoneorazero.Aquantumcomputermaintainsasequenceofqubits.Asinglequbitcanrepresentaone,azero,oranyquantumsuperpositionofthesetwoqubitstates;moreover,apairofqubitscanbeinanyquantumsuperpositionof4states,andthreequbitsinanysuperpositionof8.Ingeneral,aquantumcomputerwith
qubitscanbeinanarbitrarysuperpositionofupto
differentstatessimultaneously(thiscomparestoanormalcomputerthatcanonlybeinoneofthese
statesatanyonetime).Aquantumcomputeroperatesbysettingthequbitsinacontrolledinitialstatethatrepresentstheproblemathandandbymanipulatingthosequbitswithafixedsequenceofquantumlogicgates.Thesequenceofgatestobeappliediscalledaquantumalgorithm.Thecalculationendswithameasurement,collapsingthesystemofqubitsintooneofthe
purestates,whereeachqubitispurelyzeroorone.Theoutcomecanthereforebeatmost
classicalbitsofinformation.Quantumalgorithmsareoftennon-deterministic,inthattheyprovidethecorrectsolutiononlywithacertainknownprobability.
Anexampleofanimplementationofqubitsforaquantumcomputercouldstartwiththeuseofparticleswithtwospinstates:
"down"and"up"(typicallywritten
and
or
and
).ButinfactanysystempossessinganobservablequantityA,whichisconservedundertimeevolutionsuchthatAhasatleasttwodiscreteandsufficientlyspacedconsecutiveeigenvalues,isasuitablecandidateforimplementingaqubit.Thisistruebecauseanysuchsystemcanbemappedontoaneffectivespin-1/2system.
Bitsvs.qubits[edit]
Aquantumcomputerwithagivennumberofqubitsisfundamentallydifferentfromaclassicalcomputercomposedofthesamenumberofclassicalbits.Forexample,torepresentthestateofann-qubitsystemonaclassicalcomputerwouldrequirethestorageof2ncomplexcoefficients.Althoughthisfactmayseemtoindicatethatqubitscanholdexponentiallymoreinformationthantheirclassicalcounterparts,caremustbetakennottooverlookthefactthatthequbitsareonlyinaprobabilisticsuperpositionofalloftheirstates.Thismeansthatwhenthefinalstateofthequbitsismeasured,theywillonlybefoundinoneofthepossibleconfigurationstheywereinbeforemeasurement.Moreover,itisincorrecttothinkofthequbitsasonlybeinginoneparticularstatebeforemeasurementsincethefactthattheywereinasuperpositionofstatesbeforethemeasurementwasmadedirectlyaffectsthepossibleoutcomesofthecomputation.
Qubitsaremadeupofcontrolledparticlesandthemeansofcontrol(e.g.devicesthattrapparticlesandswitchthemfromonestatetoanother).[10]
Forexample:
Considerfirstaclassicalcomputerthatoperatesonathree-bitregister.Thestateofthecomputeratanytimeisaprobabilitydistributionoverthe
differentthree-bitstrings000,001,010,011,100,101,110,111.Ifitisadeterministiccomputer,thenitisinexactlyoneofthesestateswithprobability1.However,ifitisaprobabilisticcomputer,thenthereisapossibilityofitbeinginanyoneofanumberofdifferentstates.WecandescribethisprobabilisticstatebyeightnonnegativenumbersA,B,C,D,E,F,G,H(whereA=probabilitycomputerisinstate000,B=probabilitycomputerisinstate001,etc.).Thereisarestrictionthattheseprobabilitiessumto1.
Thestateofathree-qubitquantumcomputerissimilarlydescribedbyaneight-dimensionalvector(a,b,c,d,e,f,g,h),calledaket.Here,however,thecoefficientscanhavecomplexvalues,anditisthesumofthesquaresofthecoefficients'magnitudes,
thatmustequal1.Thesesquaremagnitudesrepresenttheprobabilityamplitudesofgivenstates.However,becauseacomplexnumberencodesnotjustamagnitudebutalsoadirectioninthecomplexplane,thephasedifferencebetweenanytwocoefficients(states)representsameaningfulparameter.Thisisafundamentaldifferencebetweenquantumcomputingandprobabilisticclassicalcomputing.[11]
Ifyoumeasurethethreequbits,youwillobserveathree-bitstring.Theprobabilityofmeasuringagivenstringisthesquaredmagnitudeofthatstring'scoefficient(i.e.,theprobabilityofmeasuring000=
theprobabilityofmeasuring001=
etc..).Thus,measuringaquantumstatedescribedbycomplexcoefficients(a,b,...,h)givestheclassicalprobabilitydistribution
andwesaythatthequantumstate"collapses"toaclassicalstateasaresultofmakingthemeasurement.
Notethataneight-dimensionalvectorcanbespecifiedinmanydifferentwaysdependingonwhatbasisischosenforthespace.Thebasisofbitstrings(e.g.,000,001,...,111)isknownasthecomputationalbasis.Otherpossiblebasesareunit-length,orthogonalvectorsandtheeigenvectorsofthePauli-xoperator.Ketnotationisoftenusedtomakethechoiceofbasisexplicit.Forexample,thestate(a,b,c,d,e,f,g,h)inthecomputationalbasiscanbewrittenas:
where,e.g.,
Thecomputationalbasisforasinglequbit(twodimensions)is
and
.
UsingtheeigenvectorsofthePauli-xoperator,asinglequbitis
and
.
Operation[edit]
Listofunsolvedproblemsinphysics
Isauniversalquantumcomputersufficienttoefficientlysimulateanarbitraryphysicalsystem?
Whileaclassicalthree-bitstateandaquantumthree-qubitstatearebotheight-dimensionalvectors,theyaremanipulatedquitedifferentlyforclassicalorquantumcomputation.Forcomputingineithercase,thesystemmustbeinitialized,forexampleintotheall-zerosstring,
correspondingtothevector(1,0,0,0,0,0,0,0).Inclassicalrandomizedcomputation,thesystemevolvesaccordingtotheapplicationofstochasticmatrices,whichpreservethattheprobabilitiesadduptoone(i.e.,preservetheL1norm).Inquantumcomputation,ontheotherhand,allowedoperationsareunitarymatrices,whichareeffectivelyrotations(theypreservethatthesumofthesquaresadduptoone,theEuclideanorL2norm).(Exactlywhatunitariescanbeapplieddependonthephysicsofthequantumdevice.)Consequently,sincerotationscanbeundonebyrotatingbackward,quantumcomputationsarereversible.(Technically,quantumoperationscanbeprobabilisticcombinationsofunitaries,soquantumcomputationreallydoesgeneralizeclassicalcomputation.Seequantumcircuitforamorepreciseformulation.)
Finally,uponterminationofthealgorithm,theresultneedstobereadoff.Inthecaseofaclassicalcomputer,wesamplefromtheprobabilitydistributiononthethree-bitregistertoobtainonedefinitethree-bitstring,say000.Quantummechanically,wemeasurethethree-qubitstate,whichisequivalenttocollapsingthequantumstatedowntoaclassicaldistribution(withthecoefficientsintheclassicalstatebeingthesquaredmagnitudesofthecoefficientsforthequantumstate,asdescribedabove),followedbysamplingfromthatdistribution.Notethatthisdestroystheoriginalquantumstate.Manyalgorithmswillonlygivethecorrectanswerwithacertainprobability.However,byrepeatedlyinitializing,runningandmeasuringthequantumcomputer,theprobabilityofgettingthecorrectanswercanbeincreased.
Formoredetailsonthesequencesofoperationsusedforvariousquantumalgorithms,seeuniversalquantumcomputer,Shor'salgorithm,Grover'salgorithm,Deutsch-Jozsaalgorithm,amplitudeamplification,quantumFouriertransform,quantumgate,quantumadiabaticalgorithmandquantumerrorcorrection.
Potential[edit]
Integerfactorizationisbelievedtobecomputational
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