Efficient Quantum Algorithm for Computing n-time Correlation Functions

1 MB·

by  J. S. Pedernales

EfficientQuantumAlgorithmforComputingn-timeCorrelationFunctions
J. S. Pedernales,1 R. Di Candia,1 I. L. Egusquiza,2 J. Casanova,1 and E. Solano1,3
1DepartmentofPhysicalChemistry,UniversityoftheBasqueCountryUPV/EHU,Apartado644,48080Bilbao,Spain
2Department of Theoretical Physics and History of Science,
University of the Basque Country UPV/EHU, Apartado 644, 48080 Bilbao, Spain
3IKERBASQUE,BasqueFoundationforScience, AlamedaUrquijo36, 48011Bilbao, Spain
Weproposeamethodforcomputingn-timecorrelationfunctionsofarbitraryspinorial,fermionic,andbosonic
operators, consisting of an efficient quantum algorithm that encodes these correlations in an initially added
ancillaryqubitforprobeandcontroltasks. Forspinorialandfermionicsystems,thereconstructionofarbitrary
n-timecorrelationfunctionsrequiresthemeasurementoftwoancillaobservables,whileforbosonicvariables
time derivatives of the same observables are needed. Finally, we provide examples applicable to different
quantumplatformsintheframeofthelinearresponsetheory.
4 PACSnumbers:03.67.Ac,75.10.Jm,42.50.Dv,71.27.+a
1
0
2 Quantum mechanics is a recipe for computing probability and cloning methods are available [29]. On the other hand,
distributionsofmeasurementoutcomesingivenexperiments, inquantumcomputersciencetheSWAPtest[30]representsa
l
u
typically, at the microscopic scale [1]. In development since standardwaytoaccessn-timecorrelationfunctionsifaquan-
J
the early years of the twentieth century, quantum mechanics tum register is available that is, at least, able to store two
4
has allowed us to describe the most fundamental properties copiesofastate,andtoperformageneralized-controlledswap
1
oflightandmatter,suchasquantumsuperpositionandentan- gate[31]. However,thiscouldbedemandingifthesystemof

] glement[2],orthebehaviorofelementaryparticlesemerging interestislarge,forexample,foranN-qubitsystemtheSWAP
h
fromscatteringprocesses[3]. Morerecently,withtheadvent test requires the quantum control of a system of more than
p
of modern quantum technologies [4–7], quantum mechanics 2N qubits. AnotherpossibilitycorrespondstotheHadamard
-
t hasbecometheroadmapforthedesignofcomputationalpro- test [32] that requires controlled-time evolutions. The latter
n
a tocols and simulations of physical systems beyond the capa- is demanding if the dynamics of interest involve many-body
u bilitiesofclassicaldevices[2,8].Inthisrenewedview,proof- ortime-dependentHamiltonians. Incontrasttothis, herewe
q of-principleexperimentshaveimplementedquantumsimula- present a protocol that exploits the natural evolution of the
[
tions with the promise of an exponential speed-up in the in- systemandthatrequirestheadditionofonlyonequbit.

2 formationprocessing[10–16]. Let us thus consider a two-time correlation function
v Accordingtoquantumtheory, allinformationaboutasys- A(t)B(0) whereA(t)=U†(t)A(0)U(t),U(t)beingagiven
0 (cid:104) (cid:105)
3 tem, its stationary states and its evolution, is encoded in the unitary operator, while A(0) and B(0) are both Hermitian.
4 Hamiltonian. Nonetheless, for most cases, the extraction of Remark that, generically, A(t)B(0) will not be Hermitian.
2 thisinformationmaynotbestraightforward[17,18]. There- However, one can always construct two self-adjoint opera-
1. fore, alternative strategies are needed to identify and obtain torsC(t)= 21{A(t),B(0)}andD(t)=(1/2i)[A(t),B(0)]such
0 measurable quantities that characterize the relevant physical that A(t)B(0) = C(t) +i D(t) .Accordingtothequantum
(cid:104) (cid:105) (cid:104) (cid:105) (cid:104) (cid:105)
4 information [19]. A case of particular importance is given mechanical postulates, there exist two measurement appara-
1 by response functions and susceptibilities, which in the lin- tusassociatedwithobservablesC(t)andD(t).Inthisway,we
:
v ear response theory are computed in terms of two-time cor- mayformally compute A(t)B(0) fromthe measured C(t)
(cid:104) (cid:105) (cid:104) (cid:105)
Xi relation functions [20–22]. For example, the knowledge of and(cid:104)D(t)(cid:105). However,thedeterminationof(cid:104)C(t)(cid:105)and(cid:104)D(t)(cid:105)
two-time correlation functions of the form ΨA(t)B(0)Ψ , dependsnontriviallyonthecorrelationtimesandonthecom-
ar stemmingfromperturbationtheory,provides(cid:104)us|withami|cro(cid:105)- plexityofthespecifictime-evolutionoperatorU(t). Further-
scopicderivationofusefulquantitiessuchasconductivityand more, we point out that the computation of n-time correla-
magnetization [23]. The reconstruction of time-correlation tions, as ΨΨ(cid:48) = ΨU†(t)AU(t)BΨ , is not a trivial task
(cid:104) | (cid:105) (cid:104) | | (cid:105)
functions,however,neednotbetrivialatall,andcouldprofit evenifonehasaccesstofullstatetomography,duetotheam-
from quantum algorithm and simulation protocols for their biguity of the global phase of state Ψ(cid:48) =U†(t)AU(t)BΨ .
| (cid:105) | (cid:105)
determination. The computation of time correlation func- Therefore, we are confronted with a cumbersome problem:
tions for propagating signals is at the heart of quantum op- the design of measurement apparatus depending on the sys-
tical methods [24], including the case of propagating quan- temevolutionfordeterminingn-timecorrelationsofasystem
tum microwaves [25–27]. However, these methods are not whoseevolutionmaynotbeaccessible. Toourknowledge,a
necessarily easy to export to the case of spinorial, fermionic generalformalismtoattackthisproblemisstillmissing,while
and bosonic degrees of freedom of massive particles. In this alternativealgorithmicstrategies[33]maybeconsidered.
sense,recentmethodshavebeenproposedforthecaseoftwo- In this Letter, we propose an efficient quantum algorithm
time correlation functions associated to specific dynamics in forcomputinggeneraln-timecorrelationfunctionsofanarbi-
opticallattices[28],aswellasinsetupswherepost-selection traryquantumsystem,requiringonlyaninitiallyaddedprobe
2
and control qubit. Moreover, our method is applicable to a prepared in state 1 (e + g ) with g its ground state, as
√2 | (cid:105) | (cid:105) | (cid:105)
general class of interacting spinorial, bosonic, and fermionic in step 1 of Fig. 1, so that the whole ancilla-system quan-
systems. Finally,weprovideexamplesofourprotocolinthe tum state is 1 (e + g ) φ , where φ is the state of
√2 | (cid:105) | (cid:105) ⊗| (cid:105) | (cid:105)
frameofthelinearresponsetheory,wheren-timecorrelation
the system. Second, we apply the controlled quantum gate
functionsareneeded. U0=exp[ (i/h¯)g g H τ ],where,aswewillseebelow,
The protocol works under the following two assumptions. Hc is a H−amilton|ia(cid:105)n(cid:104) r|e⊗late0d0to the operator O , and τ is
0 0 0
First, we are provided with a controllable quantum sys- the gate time. As we point out in the Supplemental Mate-
tem undergoing a given quantum evolution described by the rial [36], this entangling gate can be implemented efficiently
Schrödingerequation with Mølmer-Sørensen gates for operators O that consist of
0
atensorproductofPaulimatrices[34]. Thisoperationentan-
ih¯∂t φ =H φ . (1) glestheancillawiththesystemgeneratingthestate 1 (e
| (cid:105) | (cid:105) √2 | (cid:105)⊗
Second, we require the ability to perform entangling opera- |φ(cid:105)+|g(cid:105)⊗U˜c0|φ(cid:105)), with U˜c0 =e−(i/h¯)H0τ0, step 2 in Fig. 1.
Next, we switch on the dynamics of the system governed
tions, for example Mølmer-Sørensen [34] or equivalent con-
by Eq. (1). For the sake of simplicity let us assume t =0.
trolled gates [3], between some part of the system and the 0
The effect on the ancilla-system wave function is to produce
ancillary qubit. More specifically, and as it is discussed in the state 1 (cid:0)e U(t ;0)φ + g U(t ;0)U˜0 φ (cid:1), step 3
the Supplemental Material [36], we require a number of en- √2 | (cid:105)⊗ 1 | (cid:105) | (cid:105)⊗ 1 c| (cid:105)
inFig.1. Notethat,remarkably,thislaststepdoesnotrequire
tangling gates that grows linearly with the order n of the n-
aninteractionbetweenthesystemandtheancillary-qubitde-
timecorrelationfunctionandthatremainsfixedwithincreas-
grees of freedom nor any knowledge of the Hamiltonian H.
ing system size. This protocol will provide us with the effi-
These techniques, as will be evident below, will find a nat-
cientmeasurementofgeneralizedn-timecorrelationfunctions
ural playground in the context of quantum simulations, pre-
of the form φ O (t )O (t ),...,O (t )O (t )φ ,
n 1 n 1 n 2 n 2 1 1 0 0
(cid:104) | − − − − | (cid:105) servingitsanalogueordigitalcharacter. Ifweiteratentimes
whereO (t ),...,O (t )arecertainoperatorsevaluatedat
n 1 n 1 0 0
different t−imes−, e.g., O (t )=U†(t ;t )O U(t ;t ),U(t ;t ) step 2 and step 3 with a suitable choice of gates and evolu-
k k k 0 k k 0 k 0
tiontimes, weobtainthestateΦ= 1 (e U(t ;0)φ +
being the unitary operator evolving the system from t0 to √2 | (cid:105)⊗ n−1 | (cid:105)
t . For the case of dynamics governed by time-independent g U˜n 1U(t ;t ),...,U(t ;t )U˜1U(t ;0)U˜0 φ ). Now,
Hkamiltonians, U(tk;t0)=U(tk t0)=e−(i/h¯)H(tk−t0). How- |we(cid:105)⊗targect−thequn−an1titny−T2r(e g 2Φ1Φc)bym1 easucri|ng(cid:105)the σx
− | (cid:105)(cid:104) || (cid:105)(cid:104) | (cid:104) (cid:105)
ever, our method applies also to the case where H =H(t), and σ corresponding to the ancillary degrees of freedom.
y
(cid:104) (cid:105)
and can be sketched as follows. First, the ancillary qubit is Simplealgebraleadsusto
1 1
Tr(|e(cid:105)(cid:104)g||Φ(cid:105)(cid:104)Φ|)= 2((cid:104)Φ|σx|Φ(cid:105)+i(cid:104)Φ|σy|Φ(cid:105))2(cid:104)φ|U†(tn−1;0)U˜cn−1U(tn−1;tn−2),...,U(t2;t1)U˜c1U(t1;0)U˜c0|φ(cid:105). (2)
It is easy to see that, by using the composition property Mølmer-Sørensen gates [4, 34, 37, 38]. In this way, we can
U(t ;t )=U(t ;0)U†(t ;0), Eq. (2) corresponds to a writethesecondlineofEq.(2)as
k k 1 k k 1
− −
generalconstructionrelatingn-timecorrelationsofsystemop-
erators U˜ck with two one-time ancilla measurements. In or- (−i)n(cid:104)φ|On−1(tn−1)On−2(tn−2)...O0(0)|φ(cid:105), (4)
der to explore its depth, we shall examine several classes of whichamountstothemeasuredn-timecorrelationfunctionof
systemsandsuggestconcreterealizationsoftheproposedal- HermitianandunitaryoperatorsO . Wecanalsoapplythese
k
gorithm. The crucial point is establishing a connection that ideastothecaseofnon-Hermitianoperators, independentof
associatestheU˜ck unitarieswithOk operators. theirunitarycharacter,byconsideringlinearsuperpositionsof
Startingwiththediscretevariablecase, e.g., spinsystems, theHermitianobjectsappearinginEq.(4).
andprofitingfromthefactthatPaulimatricesarebothHermi- We show now how to apply this result to the case of
tianandunitary,itfollowsthat fermionic systems. In principle, the previous proposed steps
would apply straightforwardly if we had access to the corre-
U˜cm(cid:12)(cid:12)Ωτm=π/2=exp[−(i/h¯)Hmτm](cid:12)(cid:12)Ωτm=π/2=−iOm, (3) sponding fermionic operations. In the case of quantum sim-
ulations, a similar result is obtained via the Jordan-Wigner
where Hm = h¯ΩOm, Ω is a coupling constant, and mappingoffermionicoperatorstotensorialproductsofPauli
Om is a tensor product of Pauli matrices of the form matrices,b†p→Πrp=−11σ+pσzr [40]. Here,b†p andbq arecreation
OI.mI=nσcoimn⊗seσqujmen,.c.e.,,σtkhmewciothntirmo,lljemd,.q..u,kamnt∈um0,gx,ayte,zs,ianndstσep0=2 atinodnarenlnaithioilnasti,onb†fe,rbmio=nicδop.erFaotorrstraopbpeeydinigonasn,tiacoqmuamntuutma-
correspondtoUcm(cid:12)(cid:12)Ωτm=π/2=exp(−i|g(cid:105)(cid:104)g|⊗ΩOmτm),which algorithmforth{epeffiqc}ientimp,qplementationoffermionicmod-
canbeimplementedefficiently,uptolocalrotations,withfour els has recently been proposed [4, 41, 42]. Then, we code
Embedding Quantum Simulators for Quantum Computation of Entanglement
R.DiCandia,1 B.Mejia,2 H.Castillo,2 J.S.Pedernales,1 J.Casanova,1 andE.Solano1,3
1DepartmentofPhysicalChemistry,UniversityoftheBasqueCountryUPV/EHU,Apartado644,48080Bilbao,Spain
2DepartamentodeCiencias,PontificiaUniversidadCat´olicadelPeru´,Apartado1761,Lima,Peru´
3IKERBASQUE,BasqueFoundationforScience,AlamedaUrquijo36,48011Bilbao,Spain
(Dated:October12,2013)
Weintroducetheconceptofembeddingquantumsimulators,aparadigmallowingthee�cient
quantumcomputationofaclassofbipartiteandmultipartiteentanglementmonotones.Itconsistsin
thesuitableencodingofasimulatedquantumdynamicsintheenlargedHilbertspaceofanembed-
3
dingquantumsimulator. Inthismanner,entanglementmonotonesareconvenientlymappedonto
physicalobservables,overcomingthenecessityoffulltomographyandreducingdrasticallytheexper-
imentalrequirements. Furthermore,thismethodisdirectlyapplicabletopurestatesand,assisted
byclassicalalgorithms,tothemixed-statecase. Finally,weexpectthattheproposedembedding Themethodpresentedhereworksaswellwhenthesystem
frameworkpavesthewayforageneraltheoryofenhancedone-to-onequantumsimulators.
Step1 Step2 Step3 ispreparedinamixed-stateρ0,e.g. astateinthermalequilib-
PACSnumbers:03.67.-a,03.67.Ac,03.67.Mn e rium[20,21]. Accordingly, forthecaseofspincorrelations,
| i |fi |fi U(t1;0)|fi wehave
Entanglement is considered one ogf tfhe most rempark-
ablefeaturesofquantummechanics|[1i,|2],istemming4from One-to-onequantumsimulator Tr(e gρ˜)=( i)nTr(O (t )O (t ),...,O (0)ρ ),
bcoipuanrtteirtpearotr.mFiurlsttilpyarretviteealceodrrbeylaEtiionnssteiwni,tPhooudtolscklays,saicnadlg |fi (U˜c(0|fi U(t) U(t1;0)U˜c0|f��i | (cid:105)(cid:104) | − n−1 n−1 n−2 n−2 0 0(7)
Rosenasapossibledrawbackofquantumtheory[3],en|- i
tanglementwassubsequentlyidentifiedasalofucnadlarmoteanttiaoln ((Uc0 U(t1;0) with
resourceforquantumcommunication[4,5]andquantum
cmoemnptuatsinagppuurreplyostehse[o6r,e7ti]c.aBlefeyaotnudrec,otnhseidSdeteerpvine2gloSepntmetpae3nngtloe-f Step2 Step3 (S(tepq2uaSnEttmuempbe3dsidminuglator Step2Enmtoannogtleomneesnt ρ˜ =(cid:2)...U(t2;t1)Uc1U(t1;0)Uc0(cid:3)ρ˜0(cid:2)Uc0†U(t1;0)†Uc1†U(t2;t1)†...(cid:3)
quantum technologies has allowed us to create, manip- ˜(t)
ulate, anddetectentangledstatesindi↵erentquantum ... U �� (8)
platforms. Amongthem,wecanmentiontrappedions,
whereeight-qubitWandfourteen-qubitGHZstateshave andρ˜0= 21(|e(cid:105)+|g(cid:105))((cid:104)e|+(cid:104)g|)⊗ρ0. Ifbosonicvariablesare
been created [8, 9], circuit QED (cQED) where seven involved,theanaloguetoEq.(6)reads
spguleeprmceorencnodtnuhdcautiscntgbinecgeinreclureeimtasleinzwethsdehrineavcpeornbotpeienanguaoetunisnt-agvUnaqgcr1ulieaaUdbnlt(e[u1t2e0m;n]t,1tma)suni--- UFetc2hmIeGbUd.eyd(1tnd.3ai;n(mtcg2oi)cqloaurlaenovntouUllimuncnte�is)oi2nmOUuonflea(-tttthnoo�er-.o1qn;Tuteanh�nqetu2uc)aomnntUvuseicmnmy�ous1rliambteuolrltass.troTerphvreeesrreseunastl ∂Ωτj,...,∂ΩτkTr(|e(cid:105)(cid:104)g|ρ˜)(cid:12)(cid:12)Ω(τα,...,τβ)=π/2,Ω(τj,...,τk)=0=
cgrleomwaevnetsb[e1t1w],eeanndeibghutlkp-ohpottiocnbsahsaesdbFseeIetGunp.as1chw.ih(ecveroeedloe[rn12toa].nnl-ine)(cloroQwemduinp)agoannnttedhuneitmmse�aaargceilingesanportlyritci(otibhnmlmtupheue)tfeapomtarirobtcnesodoomdffitnephnguetqatsuniimnagnlguetlmuanmte-entdstiimmwmaueovlnaecotovtooerrcn,-teaoslr-. (−i)nTr(On−1(tn−1)On−2(tn−2),...,O0(0)ρ0). (9)
Quantifyingentanglementisconsirdeelraetdioanpfaurtniccutiloarnlys. Theancillastate 1 (e + g )generatesthe e Wewillexemplifytheintroducedformalismwiththecase
di�cult task, both from theoretical and experimental √2 | (cid:105) | (cid:105) | (cid:105)
viewpoints. Infact,itischallengingantoddegfinepaenthtasn,gslet-ep 1t,umforsimthuelaatonrciwllhae-rsey,sftoermexcaomuppleli,nag.twAo-fletveerltshyastt,em ofquantumcomputingofspin-spincorrelationsoftheform
mentmeasuresforanarbitrarynumbceornoftr|poal(cid:105)rlteieds[g1a3t,e1s4]U. cmianntdheusnimitualrayteedvdoylnuatimoincssiUsd(itrmec;ttlmyr1ep)reaspenptleieddbytoan-
Moreover,theexistingentanglementoumrosnyostotenmes,[s1t5e]pdso2anodth3e,rptrwood-luecveeltshyestfiemnailnsttahteesΦim.uFlait−noarl.lyI,nththeismLeeatt-er, σk(t)σl(0) , (10)
nHoetrmcoitriraenspoopnedratdoirrec[1t6ly].tAoctchoerdeinxgpsleyuc,traethtmieoencnovtmaoplufuetthaoteifoanancillwtaoerrysi,nsatprlloiondwuoicnpegettrhhaeetoceor�nsccσieepxnttaocnfodmemσpbuyetdlaedtiainodngsoquufsaantwotuidmne-tsciilmmasueslao-f (cid:104) i j (cid:105)
ofmanyentanglementmeasures, seecoRrerfe.la[1t7io]nfofrulnowcteirons.entanglementmonotones[15]. Thismethodcanbeap- where k,l = x,y,z, and i,j = 1,...,N, N being the number
bound estimations, requires previously the reconstruc- pliedatanytimeoftheevolutionofasimulatedbipartite of spin particles involved. In the context of spin lattices,
tionofthefullquantumstate,whichcouldbeacumber- ormultipartitesystem,withthepriorknowledgeofthe where several quantum models can be simulated in differ-
siqitwsmhouimeltaeahnnerdttgNiaupmelmr.,leoyaInbftnuslowedinmomefnteohcaigoeofsrfinatnbshtpuliheehdmeyefsboriHtz,ereeriflcNobohoreffn⇠irointtqhbsu1stes0peaesarnaqvsccbsuaeeoebb,c(cid:104)wNccgliaoeitbroanssmhoo†t.prwneweNer(dTeserta,-eehl)qeHldtaribxuaes(eitpbldqakσobioid(steirtn+y0pnorbset)ygen⊗erb(cid:105)scsxetpt†aiicp=aeσnuoa(emlncslttrzyeepo--,)(cid:104)−bΦa1c|,((HTs|(lc. a0stσ.oharae0.)emen+,uitpdhσFenii⊗claso�itzt1grsaoedce.t)σnnmih1teiaqza)bnabpsu=.nsec−taeodyInHσ1ncedwtoi,eu±athf¯i.dahtrmH.tneii=.hsidtd,tnsetσciσewmtapa21hn+zsipn1rute(eoh)la,⊗σtaectaosntaoxnσiclrσotaoc±trhn−ireleqzlpgsresilin,et−p⊗iσeadetsos1ihaunnyσH,reid)nm.cizoei,q.llnbpvt.a−tehg,ooeshr1eσlrsftuiea,nfozttf1sa.iiffopto.oceeir.aetnau−s,nrcl,tσemrot(hisafintztad/1niteathy¯opgh|t,r)ΦnlaeenmeHuarmsin|t(cid:105)mt tcisl,.eacii0knuctieest-. eoroenpfl,ttaiftcqoiaourlnaenslxataultitmmkicepepls(el1a,[01ttf)h0oe,armr4me9sa,aga5nsc0ert]utr,icacipaasnpludesedclceeiimproctneiusbniitt[li1Qitn1yE–[tD1h23e0,[–5c121o62–m,]5.p44uI7]nt,,acp4tioa8orr]n--,
structthequantumstatescalesas2o2Nf�th1e.kind(cid:104)apppearqinmgon(cid:105)iontoEneqs.,(c4an).bTehe�iscireenstluyltenecxodteedndinstonaphtuysriacalllyob- ticular, with our protocol, we have access to the frequency-
Fromageneralpointofview,astandardquantumsim- servables,overcomingthereforetheneedoffullstatere-
ulationismeanttobeimplementedintoamonue-lttoi-toinmeequcaon-rrelactoinosntrsucotifonf.erTmheiosinmiuclastyinsgteqmuasn.tumdynamics,which dependent susceptibility χσω,σ that quantifies the linear re-
Thecaseofbosonicn-timecorrelatorsrequiresavariantin sponseofaspinsystemwhenitisdrivenbyamonochromatic
the proposed method, due to the nonunitary character of the field. ThissituationisdescribedbytheSchrödingerequation
associated bosonic operators. In this sense, to reproduce a ih¯∂t|ψ(cid:105)=(H+ fωσljeiωt)|ψ(cid:105), where, for simplicity, we as-
linearizationsimilartothatofEq.(3),wecanwrite sumeH=H(t). Withaperturbativeapproach,andfollowing
(cid:54)
the Kubo relations [20, 21], one can calculate the first-order
∂ΩτmU˜cm(cid:12)(cid:12)Ωτm=0=∂Ωτmexp[−(i/h¯)Hmτm](cid:12)(cid:12)Ωτm=0=−iOm, effectofamagneticperturbationactingonthe jthspininthe
(5) polarizationoftheithspinas
withHm=h¯ΩOm. Consequently,itfollowsthat
σk(t) = σk(t) +χω f eiωt. (11)
(cid:12) (cid:104) i (cid:105) (cid:104) i (cid:105)0 σ,σ ω
∂Ωτj,...,∂ΩτkTr(|e(cid:105)(cid:104)g||Φ(cid:105)(cid:104)Φ|)(cid:12)Ω(τα,...,τβ)=π/2,Ω(τj,...,τk)=0= Here, σk(t) correspondstothevalueoftheobservableσk
( i)n φ O (t )O (t ),...,O (0)φ , (6) (cid:104) i (cid:105)0 i
− (cid:104) | n−1 n−1 n−2 n−2 0 | (cid:105) in the absence of perturbation, and the frequency-dependent
where the label (α,...,β) corresponds to spin operators and susceptibilityχσω,σ is
(j,...,k) to spin-boson operators. The right-hand side is
(cid:90) t
atencdoirnreglaotiuornporefvHioeursmriteisaunltso.perFaotorrse,xatmhupsles,ubOstanwtioaulllydeinx-- χσω,σ = 0 dsφσ,σ(t−s)eiω(s−t) (12)
m
cal†u)d.eTshpeinw-baoysoonf gceonueprlaintignsgatsheOamss=ociσaitmed⊗eσvojmlu,.t.i.o,nσkomp(eara+- bwehwerreitφteσn,σin(tt−erms)sisofctawlloe-dtitmheecreosrpreolnastieofnufnucnticotino,nwshich can
torU˜m=exp( iΩO τ ) has been shown in [4, 41, 43], see
also tche Suppl−ementmalmMaterial [36]. Note that, in general, φσ,σ(t−s)=(i/h¯)(cid:104)[σik(t−s),σlj(0)](cid:105)
dealingwithdiscretederivativesofexperimentaldataisanin- =(i/h¯)Tr(cid:0)[σk(t s),σl(0)]ρ(cid:1), (13)
volvedtask[44,45]. However,recentexperimentsintrapped i − j
ions [14, 15, 46] have already succeeded in the extraction of with ρ = U(t)ρ U†(t), ρ being the initial state of the
0 0
preciseinformationfromdataassociatedtofirst-andsecond- systemandU(t)theperturbation-freetime-evolutionoperator
orderderivatives. [20]. Note that for thermal states or energy eigenstates,
4
we have ρ = ρ . According to our proposed method, and the computation of the susceptibilities, χω and χω ,
0 σ,σ a+a†,σ
assuming for the sake of simplicity ρ = Φ Φ, the mea- only requires the knowledge of the time correlation func-
| (cid:105)(cid:104) |
surement of the commutator in Eq. (13), corresponding to tions [σk(t s),σl(0)] and [(a+a†) ,σl(0)] , which
the imaginary part of σk(t s)σl(0) , would require the can be(cid:104) efifici−ently cjalcul(cid:105)ated w(cid:104)ith the pr(to−tos)coljdesc(cid:105)ribed in
(cid:104) i − j (cid:105)
followingsequenceofinteractions: Φ U1U(t s)U0 Φ , Fig 1. In this manner, we provide an efficient quantum al-
| (cid:105)→ c − c| (cid:105)
Uwch1e=ree−Ui|gc0(cid:105)(cid:104)=g|⊗eσ−ikΩi|gτ(cid:105),(cid:104)g|⊗foσrljΩΩτ,τ =U(πt−/2s.)=Ae−ft(ei/rh¯)sHu(ct−hs),a gaantde gslayotsretidethmtmosttthooeecqxhutaearrnantcautlmerpizecerotmutrhpbeuattarieotisnopsno.nosOfeutrroafnmsdeititifhofoenrdepnmrtoabqyaubabinelittuiremes-
sequence, the expected value in Eq. (13) corresponds to
α (t)2= f U(t)i 2= iP (t)i , between initial and fi-
c−o1m/p2u(cid:104)Φta|tσioyn|Φo(cid:105)f.hiIgnhtehr-eosrdamerecowrareyc,tKiounbsoofretlhaetiopnesrtuarlbloewd dthye- |nalf,istat|es, ||i(cid:104)(cid:105) |and ||f(cid:105)(cid:105)|, wit(cid:104)h|Pff(t)|(cid:105)=U(t)†|f(cid:105)(cid:104)f|U(t), and
to transition or decay rates ∂ α (t)2 in atomic ensembles.
namicsintermsofhigher-ordertimecorrelationfunctions. In t| f,i |
These questions are of general interest for evolutions per-
particular, second-order corrections to the linear response of
turbedbyexternaldrivingfieldsorbyinteractionswithother
Eq.(11)canbecalculatedthroughthecomputationofthree-
quantumparticles.
time correlation functions of the form σk(t )σl(t )σl(0) .
(cid:104) i 2 j 1 j (cid:105) In conclusion, we have presented a quantum algorithm
Using the method introduced in this paper, to measure
to efficiently compute arbitrary n-time correlation functions.
such a three-time correlation function one should perform
the evolution Φ U1U(t t )U0U(t )U0 Φ , where The protocol requires the initial addition of a single probe
Uc0=e−i|g(cid:105)(cid:104)g|⊗σ|ljΩτ(cid:105),→U(t)c=e−2(i/−h¯)H1t ancdUc11=ce−|i|g(cid:105)(cid:105)(cid:104)g|⊗σikΩτ atinodnsc.oFnutrrothleqrmuboirte,anwdeihsavvealaidppfloierdatrhbiistrmareythuondittaoryinteevroalcut--
for Ωτ = π/2. The searched time correlation then corre-
ingfermionic,spinorial,andbosonicsystems,showinghowto
spondstothequantity1/2(i Φσ Φ Φσ Φ ).
(cid:104) | x| (cid:105)−(cid:104) | y| (cid:105) computesecond-ordereffectsbeyondthelinearresponsethe-
Our method is not restricted to corrections of observables
ory. Moreover,ifusedinaquantumsimulation,theprotocol
thatinvolvethespinorialdegreeoffreedom. Indeed, wecan
preserves the analogue or digital character of the associated
showhowthemethodapplieswhenoneisinterestedintheef-
dynamics. We believe that the proposed concepts pave the
fectoftheperturbationontothemotionaldegreesoffreedom
wayformakingaccessibleawideclassofn-timecorrelators
of the involved particles. According to the linear response
inawidevarietyofphysicalsystems.
theory, correctionstoobservablesinvolvingthemotionalde-
TheauthorsacknowledgesupportfromSpanishMINECO
greeoffreedomenterintheresponsefunction,φ (t s),
a+a†,σ − FIS2012-36673-C03-02, UPV/EHU UFI 11/55, UPV/EHU
as time correlations of the type (a +a†) σl , where
(cid:104) i i (t−s) j(cid:105) PhDgrant,BasqueGovernmentIT559-10andIT472-10,and
(ai+a†i)(t s)=e(i/h¯)H(t−s)(ai+a†i)e−(i/h¯)H(t−s). The re- CCQED,PROMISCE,andSCALEQITEuropeanprojects.
sponse fun−ction can be written as in Eq. (13) but replacing
the operator σk(t s) with (a +a†) . The corrected ex-
pectationvalueiis−now i i (t−s)
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SUPPLEMENTALMATERIALFOR
“EFFICIENTQUANTUMALGORITHMFORCOMPUTINGN-TIMECORRELATIONFUNCTIONS”
Inthissuplementalmaterialweprovideadditionaldiscussionsabouttheefficiencyofourmethodtogetherwithcomparisons
ofourmethodtoexistingonesandspecificcalculationsconcerningtheimplementationofgates.
I.EFFICIENCYOFTHEMETHOD
Ouralgorithmisconceivedtoberuninasetupcomposedofasystemundergoingtheevolutionofinterestandanancillary
qubit. Thus,thesizeofthesetupwherethealgorithmistoberunisalwaysthatofthesystemplusonequbit,regardlessofthe
order of the time correlation. For instance, if we are considering an N-qubit system then our method is performed in a setup
composedofN+1qubits.
With respect to time-efficiency, our algorithm requires the performance of n controlled gatesUi and n 1 time-evolution
c −
operatorsU(t ;t ),nbeingtheorderofthetime-correlationfunction.
j+1 j
6
If we assume that q gates are needed for the implementation of the system evolution and m gates are required per control
operation, our algorithm employs (m+q) n q gates. As m and q do not depend on the order of the time correlation we
∗ −
can state that our algorithm needs a number of gates that scales as a first-order polynomial with respect to the order n of the
time-correlationfunction.
Thescalingofqwithrespecttothesystemsizedependsonthespecificsimulationunderstudy. However,formostrelevant
cases it can be shown that this scaling is polynomial. For instance, in the case of an analogue quantum simulation of unitary
dynamics, what it is usually called an always-on simulation, we have q=1. For a model requiring digital techniques q will
scale polinomially if the number of terms in the Hamiltonian grows polinomially with the number of constituents, which is a
physically-reasonableassumption[2–4]. Inanycase,wewanttopointoutthatthewayinwhichqscalesisaconditioninherent
toanyquantumsimulationprocess,and,hence,itisnotanadditionaloverheadintroducedbyourproposal.
With respect to the number m, and as explained in the next section, this number does not depend on the system size, thus,
fromthepointofviewofefficiencyitamountstoaconstantfactor.
In order to provide a complete runtime analysis of our protocol we study now the number of iterations needed to achieve a
certainprecisionδ inthemeasurementofthetimecorrelations. AccordingtoBerntein’sinequality[1]wehavethat
(cid:34)(cid:12)(cid:12)1 L (cid:12)(cid:12) (cid:35) (cid:18) Lδ2(cid:19)
Pr (cid:12) ∑X X (cid:12)>δ 2exp − , (17)
(cid:12)(cid:12)Li=1 i−(cid:104) (cid:105)(cid:12)(cid:12) ≤ 4σ02
where X are independent random variables, and σ2 is a bound on their variance. Interpreting X as a single observation of
i 0 i
therealorimaginarypartofthetime-correlationfunction,wefindthenumberofmeasurementsneededtohaveaprecisionδ.
Indeed,wehavethat(cid:12)(cid:12)L1∑Li=1Xi−(cid:104)X(cid:105)(cid:12)(cid:12)≤δ withprobabilityP≥1−e−c,providedthatL≥ 4(1δ+2c),wherewehavesetσ02≤1,
aswealwaysmeasurePauliobservables. Thisimpliesthatthenumberofgatesthatweneedtoimplementtoachieveaprecision
δ fortherealortheimaginarypartofthetime-correlationfunctionis 4(1+c)[(m+q)n q]. Again,cisaconstantfactorwhich
δ2 −
doesnotdependnorontheorderofthetimecorrelationneitheronthesizeofthesystem.
II.N-BODYINTERACTIONSWITHMØLMER-SØRENSENGATES
Exponentials of tensor products of Pauli operators, exp[iφσ σ ... σ ], can be systematically constructed, up to local
1 2 k
⊗ ⊗ ⊗
rotations,withaMølmer-Sørensengateappliedoverthekqubits,onelocalgateononeofthequbits,andtheinverseMølmer-
Sørensengateonthewholeregister. Thiscanbeschematizedasfollows,
U =U ( π/2,0)U (φ)U (π/2,0)=exp[iφσz σx ... σz], (18)
MS − σz MS 1⊗ 2⊗ ⊗ k
wk=her4enU+M1S,(aθn,dφφ)==expφ[−foirθk(c=os4φnSx+1,swinitφhSpyo)2s/it4iv],eSinx,tyeg=er∑nki.=F1oσrixe,yveanndk,UUσz((φφ))=iserxeppl(aiφce(cid:48)σd1zb)yfUor o(dφd)=k,ewxhpe(irφe σφ(cid:48)y)=,wφheforer
(cid:48) − − σz σy (cid:48) 1
φ =φ fork=4n,andφ =φ fork=4n 2,withpositiveintegern. Subsequentlocalrotationswillgenerateanycombination
(cid:48) (cid:48)
−
ofPaulimatricesintheexponential.
The replacement in the previous scheme of the central gate U (φ) by an interaction containing a coupling with bosonic
σz
degreesoffreedom,forexampleU (φ)=exp[iφ σz(a+a†)],willdirectlyprovideuswith
σz,(a+a†) (cid:48) 1
U =U ( π/2,0)U (φ)U (π/2,0)=exp[iφσz σx ... σz(a+a†)]. (19)
MS − σz,(a+a†) MS 1⊗ 2⊗ ⊗ k
InordertoprovideacompleterecipeforsystemswhereMølmer-Sørenseninteractionsarenotdirectlyavailable,wewantto
comment that the kind of entangling quantum gates required by our algorithm, see the right hand side of Eq. (19) above, are
alwaysdecomposableinapolynomialsequenceofcontrolled-Zgates[3]. Forexample,inthecaseofathree-qubitsystemwe
have
CZ1,3CZ1,2e−iφσ1yCZ1,2CZ1,3=exp(−iφσ1y⊗σ2z⊗σ3z) (20)
Here,CZi,j isacontrolled-Zgatebetweenthei,jqubitsande−iφσ1y islocalrotationappliedonthefirstqubit. Thisresultcanbe
easilyextendedton-qubitsystemswiththeapplicationof2(n 1)controlledoperations[3].
−
Therefore, it is demonstrated the polynomial character of our algorithm, and hence its efficiency, even if Mølmer-Sørensen
gatesarenotavailableinoursetup.
7
III.OURPROTOCOLVSHADAMARDANDSWAPTESTS
TwotypicalapproachesforthemeasurementofcorrelationsinthecontextofquantumcomputingaretheHadamardandthe
SWAP tests. The Hadamard test is performed in a setup consisting of the system of interest and a qubit, and thus in terms of
spaceisasefficientasouralgorithm. TheperformanceofaHadamardgatefollowedbyacontrolledunitaryevolution,another
Hadamard gate and the measurement of two ancilla operators will lead to the real and imaginary parts of a correlation of the
type U ,whereU correspondstothecontrolledunitary. Whiletheevolutionsofthesystemofinterestarenotcontrolledinour
(cid:104) (cid:105)
protocol,theHadamardtestneedstoperformcontrolunitaryoperationswhichmaynotbetrivialformanybodyHamiltonians
or Hamiltonians depending in time. In this sense our method supposes a significant step forward in simplicity and a notable
reduction in the requirements of our setup. It is noteworthy to mention that our algorithm could access time correlations of
systemsthatundergonon-unitarydynamics.
InthecaseoftheSWAPtest,correlationsbetweentwostatesaremeasuredfollowingasimilarscheme,inthiscaseaHadamard
gateisperformedontheancillaqubit,afterthatacontrolSWAPgateisimplementedbetweenthetwostatesofinterestandfinally
asecondHadamardgateisperformedontheancillaryqubit. Againlocalmeasurementsontheancillaryqubitwillprovidereal
andimaginarypartsofthecorrelationbetweenthetwostates. Whileourprotocolinvolvesonlyoneancillaryqubit,N+1,the
SWAPtestneedstwocopiesofthesystemandtheancillaryqubitwhichmakesatotalof2N+1qubits,thismakesourprotocol
significantlymorespacesaving.
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