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QuantumgatesbyreverseengineeringofaHamiltonian Alan C. Santos1,∗ 1Instituto de F´ısica, Universidade Federal Fluminense, Av. Gal. MiltonTavaresdeSouzas/n, Gragoata´, 24210-346Niteroí, RiodeJaneiro, Brazil (Dated:January17,2017) ReverseengineeringofHamiltonian(REH)fromanevolutionoperatorisanusefultechniqueforprotocols ofquantumcontrolwithpotentialapplicationstoquantuminformationprocessing. Inthispaperweintroduce aparticularprotocoltoperformREHandweshowasthisschemecanbeusedtoperformuniversalquantum computing by using minimal quantum resources (such as entanglement, interactions between more than two qubitsorauxiliaryquits). Remarkable,whilepreviousprotocolrequiresthree-qubitsinteractionsandauxiliary quitstoimplementsuchgates,ourprotocolrequiresjusttwo-qubitinteractionsandnoadditionalresource. We applyourresultstoshowasimplementthequantumFouriertransform(QFT)motivedbyitsapplicabilityto Shor’salgorithm. Byusingthisapproach,wecanobtainalargeclassofHamiltoniansthatallowusimplement 7 singleandtwo-quitgatesnecessaryinsubroutinesofquantumcircuitsofquantumalgorithmsas,forexample, 1 0 theShor’salgorithm. 2 n I. INTRODUCTION required[14–17]. a In this paper we introduce a alternative way to perform J Currently, protocol of quantum control with time- universal QC based on REH from evolution operators. Dif- 4 1 dependent Hamiltonians, as adiabatic passage [1], Lewis- ferently from methods developed recently for REH, our ap- Riesenfeld invariants [2, 3], transitionless quantum driving proach is independent of the initial state of the system and, ] (TQD) [4–6] and the current propose of REH from unitary therefore,ancillaqubitsisnotnecessaryinourscheme. Inthe h evolutions operators [7], has play an important role in quan- sectionIIwediscussthegeneralaspectsofourapproachand p - tuminformationprocessing(seeRef.[8]forareviewofmany weshowasobtainasetofHamiltonianwhichallowusimple- t applicationsofthisthreefirsttechniques). Inaddition,inthe mentthequantumgatesofaquantumcircuit.Becauseourap- n a lastyears,manyexperimentalandtheoreticalstudieshasbeen proachisindependentofinitialstate,whilemanyprotocolsre- u performedinordertoanalyzetherobustnessofsuchprotocols quiresauxiliaryqubitstoperformuniversalQC,ourapproach q againstdecoherenceeffects[9–12]. need not help of auxiliary elements to implement single and [ Remarkable, in quantum computation (QC) and informa- controlled quantum gates. In this sense, the scheme present 2 tionthesetechniquesprovidesHamiltoniansabletosolvesat- hereisanenhancedwaytoimplementquantumgates. Inthe v isfiability problems [13], or to introduce new hybrid models section III we illustrate the results obtained here through the 8 ofQC.Infact,throughquantumcontrolprotocolswecanob- Shor’salgorithm[19]whichmakeusesofQFTassubroutines 4 tain controlable time-dependent Hamiltonians to implement ofsuchalgorithm[20,21]. 8 single and controlled quantum gates. For example, quantum 1 gates of a circuit can be implemented via adiabatic Hamilto- 0 nians [14, 15] or via its superadiabatic counterpart [16–18]. II. QUANTUMGATESBYREVERSEENGINEERINGOF . 1 However, these schemes requires auxiliary qubits in order to AHAMILTONIAN 0 canimplementuniversalQC.Wesayuniversalinsensethat, 7 1 given an unknown input state, we should be able to imple- Let us start by considering the Schro¨dinger equation (we : ment any single- and two-qubit quantum gate on this qubit. set(cid:126)=1throughoutthemanuscript) v Ingeneral,theHamiltoniansthatimplementsanadiabaticdy- i X namics,aswellassuperadiabaticHamiltonianandthosepro- H(t)|ψ(t)(cid:105)=i|ψ˙(t)(cid:105), (1) videdbyREH,mustevolvesthesystemfrominitialstatethat r a iseigenstateoftheadiabaticHamailtonian. Thus,sometimes, suchthatthedynamicsisunitaryand, therefore, therearean suchHamiltoniansmustbehighlydegenerate[16,17]andso operator U(t) where we can write |ψ(t)(cid:105) = U(t)|ψ(0)(cid:105). From we can encode any unknown state in the initial state of the equation above (valid for any |ψ(0)(cid:105)), the Hamiltonian reads system. Consequently, ancillaqubitsarenecessaries. Onthe as otherhand,evenwhenwehaveancillaqubitsavailable,when wewantimplementtwo-qubitquantumgatestheproblemof H(t)=iU˙(t)U†(t), (2) manybodiesinteractionsmustbehighlight.Infact,ingeneral weneedinteractionsbetweenthequbitsofourtargetsystem whichisawayso-knowntoobtaintheHamiltonianassociate as well as between the target system and the ancilla qubits, with evolution operator U(t). In general, this equation is the such that sometimes interactions of three (or more) qubits is starting point in protocols of reverse engineering in closed quantum systems [7, 22, 23], as well as transitionless quantum driving [4–6, 8]. The operator U(t) can be defined of manydifferentway,buttherearepropertiesthatsuchoperator ∗[email protected]ff.br must satisfy, namely, the unitary condition (U(t)U†(t) = 1) 2 andU(0) = 1. Letusnowusetheseconditionstodefinethe Notice that the evolution operator is not ψ -independent, inp operatorU(t)as in sense that we do not need to know any information about (cid:88) thestate|ψinp(cid:105)tousethemethod. Therefore,theHamiltonian U(t)= eiϕn(t)|n(t)(cid:105)(cid:104)n(t)|, (3) H(t) associate with operator U(t) is also ψ -independent. n inp Thus, single-qubit universal operations can be performed by where |n(t)(cid:105) constitutes an orthonormal bases for the Hilbert using this approach, where no additional quantum resource space associated with the system and ϕn(t) are real free pa- (suchasentanglementorauxiliaryqubits,forexample)isre- rameters.ItiseasyshowthatU(t),definedabove,satisfiesthe quired. unitarity condition U(t)U†(t) = 1 for any set of parameters ϕ (t), but the imposition U(0) = 1 restricts the initial values n of ϕn(t) and it must comply with the condition ϕn(0) = 2nπ B. Two-qubitquantumgates forn∈Z. Differently from others protocols [1–7], the definition of To show that our protocol can be used to implement uni- the operator U(t) can not be view as an operator that drives versal QC, we must show as implement two-qubit quantum the system from initial state |n(0)(cid:105) to |n(T)(cid:105). This subtle dif- gatesthroughit. Bywritingatwo-qubitinputstateinitsmost ferencecharacterizesthepresentprotocolasanalternativeto generalformas|ψ (cid:105)=a|00(cid:105)+b|01(cid:105)+c|10(cid:105)+d|11(cid:105)andby inp,2 eachothersschemes.Moreover,asweshallsee,thisapproach definingtheoperator canbeusefulinsomeprotocolsofquantuminformationpro- (cid:88) creessosiunrgce,,sfuocrhexaasmimplpel.ementing universal QC using minimal U2(t)= k=1,2|nk,+(t)(cid:105)(cid:104)nk,+(t)|+eiϕk(t)|nk,−(t)(cid:105)(cid:104)nk,−(t)|, (10) whereweset(withk¯ =k−1) A. Single-qubitgates |nk,+(t)(cid:105)=cosθk(t)|k¯0(cid:105)+eiφk(t)sinθk(t)|k¯1(cid:105), (11) |nk,−(t)(cid:105)=−sinθk(t)|k¯0(cid:105)+eiφk(t)cosθk(t)|k¯1(cid:105), (12) Inthissectionwewillshow, withoutadditionalresources, whichsinglequantumgatescanbeimplementedbyusingthe with the initial conditions ϕ (0) = ϕ (0) = 2nπ (due the re- 1 2 scheme presented here. To this end, we start by considering quirement U (0) = 1). Now we have six free parameters 2 thatasingle-qubitgatecanbeviewasalineartransformation and we need set them in order to obtain the expected result on an arbitrary quantum state |ψ (cid:105) = a|0(cid:105)+b|1(cid:105), and so let at the end of the dynamics. The operator U (t) is a gen- inp 2 usconsiderthetransformation|ψ(t)(cid:105) = U (t)|ψ (cid:105),wherethe eraltransformoftwo-qubitstateand,byadjustingadequately 1 inp operatorU (t)isgivenby our free parameters, U (t) can be an entangled gate or only 1 2 a composition of two independent single-qubits gates (i.e., U1(t)=|n+(t)(cid:105)(cid:104)n+(t)|+eiϕ(t)|n−(t)(cid:105)(cid:104)n−(t)|, (4) U2(t) = A1(t)⊗ A2(t)). In order to analyze some results by using its most general form, we keep our discussion without where considersomeparticularcaseforU (t),butparticularizations 2 |n+(t)(cid:105)=cosθ(t)|0(cid:105)+eiφ(t)sinθ(t)|1(cid:105), (5) forU2(t)willbedoneinthesectionIII. |n (t)(cid:105)=−sinθ(t)|0(cid:105)+eiφ(t)cosθ(t)|1(cid:105), (6) FromEqs. (10-12)wecanwritetheevolvedstateas − |ψ (cid:105)=U (t)|ψ (cid:105) with θ(t), ϕ(t) and φ(t) being real free parameters. It is easy inp,2 2 inp,2 show that the conditions U (t)U†(t) = 1 and U (0) = 1 are =α(t)|00(cid:105)+β(t)|10(cid:105)+γ(t)|01(cid:105)+δ(t)|11(cid:105) (13) 1 1 1 satisfiedifwechooseϕ(t)suchthatϕ(0) = 2nπ, forintegers withthefollowingcoefficients n. The parameters associated with the quantum gate to be iTmopslheomwenttheadtawreeecnacnordeeadllyintiompplaermamenettesrisnθg(let)-,qϕu(bti)tagnadteφs(bt)y. α(t)= aσ1,+(t)−σ1,−(t)[acosθ1(t)+beiφ1(t)sinθ1(t)], (14) 2 ugsivinegntbhyeoperatorU1(t),letusconsidertheevolvedstate|ψ(t)(cid:105) β(t)= bσ1,+(t)+σ1,−(t)[bcosθ1(t)−ae−iφ1(t)sinθ1(t)],(15) 2 |ψ(t)(cid:105)=U1(t)|ψinp(cid:105)=α(t)|0(cid:105)+β(t)|1(cid:105), (7) γ(t)= cσ2,+(t)−σ2,−(t)[ccosθ2(t)+deiφ2(t)sinθ2(t)], (16) wherethecoefficientsα(t)andβ(t)aregiven,respectivelyby 2 α(t)= aσ+(t) − σ−(t)[acosθ(t)+be−iφ(t)sinθ(t)] (8) δ(t)= dσ2,+(t)+σ2,−(t)[dcos2θ2(t)−ce−iφ2(t)sinθ2(t)],(17) 2 2 β(t)= bσ+(t) + σ−(t)[bcosθ(t)−aeiφ(t)sinθ(t)] (9) wσkh,±er(e0)ag=ai2nδw+∓e.hAavseindetfihneecdasσek,o±f(ts)in=gl(ee-iϕqku(tb)i±t g1a)t,es,utchhetohpa-t 2 2 eratorU (t),aswellastheHamiltonianassociatetoU (t),is 2 2 withσ (t)=(eiϕ(t)±1). Byusingtheinitialconditionϕ(0)= ψ -independent. ± inp 2nπwecanseethatα(0) = aandβ(0) = b,becauseσ (0) = Inthisdiscussionwehavenotlabeledatargetandcontrol ± 2δ+±. Therefore, from Eqs. (7-9), an arbitrary single-qubit qubit and this choice can be done through the definition of rotationcanbeperformed. the free parameters. Thus, the operator U (t) encompasses a 2 3 largeclassoftwo-qubitgates, likeentanglingquantumgates Finally, let us discuss about implementation of the CNOT (as CNOT or some controlled single-qubit unitary rotations) gate. Without loss of generality, we consider a bipartite sys- ornon-entanglinggates(suchastheSWAPgate). teminitiallyinthestate |ψ (cid:105)=a|0(cid:105) |0(cid:105) +b|0(cid:105) |1(cid:105) +c|1(cid:105) |0(cid:105) +d|1(cid:105) |1(cid:105) (20) inp,2 c t c t c t c t III. UNIVERSALSETOFQUANTUMGATESAND where the subscript “c” and “t” labels the control and target QUANTUMFOURIERTRANSFORM qubit,respectively. Thus,underthisencodingthestateofthe systematendoftheevolutionmustbewrittenas Inordertoshowaswecantoimplementasetofuniversal |ψ (cid:105)=a|0(cid:105) |0(cid:105) +b|0(cid:105) |1(cid:105) +γ(T)|1(cid:105) |0(cid:105) +δ(T)|1(cid:105) |1(cid:105) (21) out,2 c t c t c t c t quantumgatesbyusingtheresultsdevelopedhere,inthissec- tionweconsider, withoutlossesgenerality, somechoicesfor whereγ(t)andδ(t)aregivenbytheEqs.(16)and(17),respec- the free parameters previously discussed. As application of tively. Noticethatthecoefficientsaandbwerenotchanged, thismethod,wediscussaboutthesetofquantumgatesneces- therefore, from Eqs. (14) and (15) we must set ϕ1(t) = 0. sarytoimplementthequantumFouriertransform[20],which Withthischoice,theparametersθ1(t)andφ1(t)becomesaddi- hasanimmediateapplicationtotheShor’salgorithm[19]. tionalfreeparametersthatmaybeusedtosimplifytheHamil- tonian. Inaddition,toobtaintheCNOToperationrightly,we needchooseourparametersuchthatγ(T) = d andδ(T) = c. A. Quantumgatesfor(approximately)universalQC Therefore,fromEqs. (16)and(17),thisresultcanbeachieve fromthechoicesθ (T)=π/2,ϕ (T)=πandφ (T)=0. 2 2 2 In thissection we provides aset of choices of thefree parameters of our approach which allow us implement an uni- B. ApplicationtoquantumFouriertransform versal set of quantum gates composed by phase shift gates, HadamardgateandCNOTgatethatmaybeusedtoperform universalQCwitharbitraryefficiency[20,24]. Now,letusdiscusstheapplicabilityofourapproachtoim- Forphaseshiftgates,givenanyentry|ψ (cid:105)=a|0(cid:105)+b|1(cid:105)we plement the discrete quantum Fourier transform (QFT). This inp havethecorrespondingoutput|ψ (cid:105) = a|0(cid:105)+eiξb|1(cid:105),forany exampleisconsideredagoodexampleduetheconnectionof out value 0 < ξ < 2π. Thus, from Eqs. (8) and (9), such a gate theQFTwiththeShor’salgorithm[19,20]. Toimplementthe may be implemented if we choose ϕ(T) = ξ and θ(T) = 0. quantum circuit for QFT we need of Hadamard, controlled Notice that we have boundary conditions for the parameter phase shift gates in addition with the SWAP gate [20]. The ϕ(t)givenbyϕ(0) = 0andϕ(T) = ξ,soϕ(t)cannotassume Hadamard gate was discussed in the previous section, there- anyform. Ontheotherhand, theparameterθ(t)hasasingle fore we will discuss about controlled phase shift and SWAP condition that must be satisfied, where such a condition can gate. beobtainedsettingθ(t)=0. Moreover,wecanuseitasafree Let us start from Eq. (21) to discuss the controlled phase parameter in order to simplify the Hamiltonian that evolves shiftgate. AsinthecaseoftheCNOTgate,herewewillcon- thesystem. Infact,puttingθ(t)=0theHamiltonianforphase sider our bipartite system as given by the Eq. (20) and the shift gates is given by (in order to obtain simple Hamiltoni- final state given by Eq. (21), however, here we need satisfy ans, throughout the manuscript we consider that the system γ(T) = c and δ(T) = eiξd. Now, the choice of the free pa- evolvesuptoaglobalphase) rametersnecessarytoachievethisresultmightbeθ2(T) = 0, ϕ (T) = ξ, ϕ (T) = 0 and φ (T) = 0. Under this considera- 2 1 2 ϕ˙(t) tions,wecansetφ (t)=θ (t)=0andtheHamiltoniancanbe H (t)= σ , (18) 2 2 ph 2 z writtenas whereσz = diag[1, −1]isthePaulimatrixandthesymbol H(t)= ϕ˙2(t)(cid:2)1c⊗σzt+σzc⊗1t−σzc⊗σzt(cid:3) , (22) 4 “˙” denotes temporal derivatives. This choice is not unique andwecanhavemanyotherspossibilitiesifwesetθ(t) (cid:44) 0, where the parameter θ1(t) has not any relevance because we butifwepickθ(t)(cid:44)0thecorrespondingHamiltonianmaybe put ϕ1(t) = 0. The Hamiltonian above is the more simple notassimpleastheHamiltonianobtainedabove. Hamiltonian that we can obtain with this method, but is not The Hadamard gate is an exclusive gate of quantum com- uniqueandcanassumeothersformfromdifferentchoicesof puters due its ability of generating quantum superpositions theparametersφ2(t)andθ2(t). with elements of the computational basis. More specifically, Byconsideringthestate|ψinp,2(cid:105),theSWAPgateimplement given a quantum state |ψ (cid:105) = a|0(cid:105)+b|1(cid:105) we have its corre- theexchangeb→candc→b.However,suchachangeisnot inp √ √ spondingoutput|ψ (cid:105) = (a+b)/ 2|0(cid:105)+(a−b)/ 2|1(cid:105). We possiblefromEqs. (14-17)andweneedredefineourscheme. out canimplementsuchoperationbysettingθ(T)=π/4,φ(T)=0 For this particular case of the SWAP gate, we redefine the andϕ(T) = π. Inthiscase,wehaveonlyonefreeparameter, states|nk,±(t)(cid:105)ofthefollowingway namely,φ(t)andsoweconsiderφ(t) = 0inordertosimplify |n1,+(t)(cid:105)=cosθk(t)|00(cid:105)+eiφk(t)sinθk(t)|11(cid:105), (23) theHamiltoniangivenby |n1,−(t)(cid:105)=−sinθk(t)|00(cid:105)+eiφk(t)cosθk(t)|11(cid:105), (24) H (t)= ϕ˙√(t)(σ +σ ). (19) |n2,+(t)(cid:105)=cosθk(t)|01(cid:105)+eiφk(t)sinθk(t)|10(cid:105), (25) Had 2 2 z x |n2,−(t)(cid:105)=−sinθk(t)|01(cid:105)+eiφk(t)cosθk(t)|10(cid:105), (26) 4 wherethecoefficientsgivenintheEqs. (14-17)becomes the initial state of the system. We illustrate the results obtained here through the Shor’s algorithm that makes uses of α(t)= aσ1,+(t)−σ1,−(t)[acosθ1(t)+deiφ1(t)sinθ1(t)], (27) QFTassubroutinesofsuchalgorithm.Whilethefirstpurpose 2 toimplementQFT,fromadiabaticdynamicsusingcontrolled β(t)= bσ2,+(t)+σ2,−(t)[bcosθ2(t)−ce−iφ2(t)sinθ1(t)],(28) quantumevolution,requiresthree-qubitsinteractions[15,21], 2 ourprotocolrequiresjusttwo-qubitinteractions.Byusingthis γ(t)= cσ2,+(t)−σ2,−(t)[ccosθ2(t)+beiφ2(t)sinθ2(t)], (29) applepmroeancths,inwgelecaanndobtwtaoin-qauiltarggaetecsl,atshseoreffHoraemtihlitsonmiaenthsotdoicman- 2 beausefultechniquetoperformuniversalQCinmanyphys- δ(t)= dσ1,+(t)+σ2,−(t)[dcosθ1(t)−ae−iφ1(t)sinθ1(t)].(30) icalsystems. 2 Whentheinevitableinteractionofoursystemwiththeex- Therefore,nowthenewcoefficientsallowusmaketheex- ternal medium (reservoir) is considered, decoherence effects change b → c and c → b. By considering ϕ (t) = 0 maybeabigproblemandthisproblemsincreaseswithnum- 1 we guarantee that α(t) = a and δ(t) = d that is a require- berofparticlesinoursystem. Ontheotherhand,large-scale ment of the SWAP gate. Finally, if we put θ (T) = π/2, quantumcomputingrequiresthemanipulationofmanyqubits 2 ϕ (T) = π and φ(T) = 0, we drive the system from the that remaining with a good robustness against decoherence 2 |ψ (cid:105) = a|0(cid:105) |0(cid:105) +b|0(cid:105) |1(cid:105) +c|1(cid:105) |0(cid:105) +d|1(cid:105) |1(cid:105) tothefinal effects. In this scenario, the scheme presented here may be in,2 c t c t c t c t state|ψ (cid:105)=a|0(cid:105) |0(cid:105) +c|0(cid:105) |1(cid:105) +b|1(cid:105) |0(cid:105) +d|1(cid:105) |1(cid:105). more robust to decoherence than the protocols that requires in,2 c t c t c t c t auxiliaryqubits,alreadywedonotneedauxiliaryqubitsand, consequently, the error produced by such qubits is removed IV. CONCLUSION here. Itwouldbeinterestingtomakethisanalysisinorderto comparativethisschemewithothersknownprotocols. Other In summary, in this paper we have introduced a scheme interesting point is the energetic resource to implement this toperformuniversalQCviareverseengineeringofaHamil- protocol. Can we optimize it, for example, from additional tonian from the evolution operator. We discuss the general free parameters? If we can, how can we? We leave these as aspects of our approach and show as obtain a set of Hamil- openproblemsforfutureresearch. tonian that allow us implement an universal set of quantum ACKNOWLEDGMENTS gates. Our method is an economic scheme that can be view asanalternativetoothersmethodpresentintheliterature. In WeacknowledgefinancialsupportfromtheBrazilianagen- fact, while many protocols requires auxiliary qubits to per- ciesCNPqandtheBrazilianNationalInstituteofScienceand form universal QC, our approach does not need help of aux- Technology for Quantum Information (INCT-IQ). 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