Quantum Computation Beyond the Standard Circuit Model

Construction of explicit quantum circuits follows the notion of the "standard circuit model" introduced in the solid and profound analysis of elementary gates providing quantum computation. Nevertheless the model is not always optimal (e.g. concerning the

QuantumComputation

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aBeyondthe”StandardCircuitModel” K.Ch.Chatzisavvasa ,C.Daskaloyannisb ,C.P.Panosa§aDepartmentofTheoreticalPhysics,bDepartmentofMathematics,AristotleUniversityofThessaloniki,54124Thessaloniki,GreeceAbstractConstructionofexplicitquantumcircuitsfollowsthenotionofthe”standardcircuitmodel”introducedinthesolidandprofoundanaly-sisofelementarygatesprovidingquantumcomputation.Neverthelessthemodelisnotalwaysoptimal(e.g.concerningthenumberofcom-putationalsteps)anditneglectsphysicalsystemswhichcannotfollowthe”standardcircuitmodel”analysis.Weproposeacomputationalschemewhichovercomesthenotionofthetranspositionfromclassicalcircuitsprovidingacomputationschemewiththeleastpossiblenum-berofHamiltoniansinordertominimizethephysicalresourcesneededtoperformquantumcomputationandtosucceedaminimizationofthecomputationalprocedure(minimizingthenumberofcomputa-tionalstepsneededtoperformanarbitraryunitarytransformation).

Itisageneralschemeofconstruction,independentofthespeci csys-temusedfortheimplementationofthequantumcomputer.TheopenproblemofcontrollabilityinLiegroupsisdirectlyrelatedandrisestoprominenceinane orttoperformuniversalquantumcomputation.Keywords.QuantumGates,QuantumComputation,QuantumControlTheory.

Construction of explicit quantum circuits follows the notion of the "standard circuit model" introduced in the solid and profound analysis of elementary gates providing quantum computation. Nevertheless the model is not always optimal (e.g. concerning the

1The”StandardCircuitModel”

The”standardcircuitmodel”isanestablishedproposaltoimplementquantumgatesinquantumcomputation[1].Inthismodelessentialisthenotionoftheuniversalgate[2].Thus,anygivenquantumgate(anygivenunitarytransformationofthequantumsystemthatimplementsthequantumcomputer)canbeanalyzedusingasetofbasicgates,knownasuniversalgates.Theselectionofthesetofuniversalgatesisnotunique[3].One-qubitgatescanbeanalyzedusingonlyHadamardandphasegates.Two-qubitgatescanbeanalyzedusingHadamard,phaseandtheCNOTgateandthisisgeneralizedinthecaseofN-qubitgates,whileitwasnotedthatinthegeneralcaseanin nitenumberofstepsareneededtoperformagateexplicitly[4].

Inthe”standardcircuitmodel”,physicalsystemsareneglectediftheycannotcopyeasilythemodel(ifsomeonecannotperformeasilyoneoftheselecteduniversalgates).Also,neitherthenumberofcomputationalstepsnorthetotaltimetoperformcomputationareoptimal[5].

2QuantumControlTheory

Inquantumcontroltheory,thegeneralizationofthecontroltheoryinquantumsystems,asystemissaidtobecontrollableifanarbitraryLiegroupelementW∈SU(2N)canbedecomposedin nitetimeas

W=e ianJ(n)...e ia2J(2)e ia1J(1)(1)

whereJ(k)∈{J1,J2,...,Jm}aregeneratorsofthecorrespondingsu(2N)Liealgebraandai∈R.Inthecaseofquantumcomputation,Wisequivalentwithanarbitraryunitarytransformation(uptoaglobalphase)soitisequivalentwithanarbitraryN-qubitgate.J(k)correspondstotheHamiltoniansdescribingthesystemunderconsiderationwhileaiareequivalentwithtimeparametersti.ThecontrollabilityonLiegroupsfromamathe-maticalpointofviewwasstudiedin[6,7,8,9].ThisdirectrelationbetweentheproblemofcontrollabilityinLiegroupsandtheproblemofuniversalquantumcomputationallowsustoapproachquantumcomputationwithanalternativewaybeyondthe”standardcir-cuitmodel”.InthisapproachiftheselectedHamiltoniansJ1,J2,...,Jmformacompletesetofoperators,theneveryW∈SU(2N)canbeexactlyrealizedusinga nitenumberofsteps,althoughthisnumberofstepsisnot xed,whereinthecaseofthe”standardcir-cuitmodel”thesameelementSU(2N)couldbeapproximatelyrealizedusinganin nitelynumberofsteps.Theorderofgeneration(thenumberofcomputationalstepsrequiredtoperformanarbitraryN-qubitgate)isavailableforarbitraryHamiltoniansonlyinthecaseoftheSU(2)group(one-qubitgates)viatheLowental’scriterion[10].Inthiscase,onlytwoHamiltonians{J1,J2}aresu cienttoformacompleteset.IftheHamiltoniansareorthogonal,i.e.Trace(J1J2)=0,thenthreeatmoststepsarerequired,torealizinganyW∈SU(2)).WhenTrace(J1J2)=0,thenumberofstepsaregivenbytheLowental’scriterion,butthealgorithmtoobtainthesolutionisnotknown.

InthecaseofhigherordergroupsthereisananalysisbasedontheCartandecomposi-tionofthesu(2N)algebras[8].Thisanalysisprovidesalsoananalyticalwayofcalculating

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Construction of explicit quantum circuits follows the notion of the "standard circuit model" introduced in the solid and profound analysis of elementary gates providing quantum computation. Nevertheless the model is not always optimal (e.g. concerning the

thecorrespondingtimeparameters(Eulerangles)inthecaseoftheSU(4)(2-qubitgates).OnthesamespiritistheproposalforexactcomputationbyWhaleyandcollaborators

[11].OpenproblemsinLiegroupscontrollabilityare:

a)SU(2)group(one-qubitgates).Analgorithmwhich,givenanarbitrarycoupleofgenerators–Hamiltonians,willbeabletoprovideanalyticallythetimeparameterstoperformuniversalcomputation,ifthenumberofrequiredstepsaremorethanthree.

b)Higherordergroups.Acriterionforminimumnumberofstepstogeneratean

arbitraryelementofthegroup(whichcorrespondstoanarbitraryN-qubitgate,respectively)inthecasewherethegenerators–Hamiltoniansarenotorthogonal.Algorithmstoevaluatethecorrespondingtimeparameters.

3QuantumGatesUsingtheIntrinsicAbili-

tiesofaPhysicalSystem

Insteadofforcingaphysicalsystemtoactasapredeterminedsetofuniversalgateswefocusontheabilityofthephysicalsystemtoactasaquantumcomputerusingonlyitsnaturalavailableinteractions(encodeduniversality[12]).

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