Table Of Content

All Possible Coupling Schemes in XY Spin Chains for Perfect State Transfer Yaoxiong Wang and Feng Shuang Institute of Intelligent Machines, Chinese Academy of Sciences, Hefei, China 230031 Herschel Rabitz Department of Chemistry, Princeton University, Princeton, NJ 08544 We investigate quantum state transfer in XY spin chains and propose a recursive procedure to construct the nonuniform couplings of these chains with arbitrary length to achieve perfect state transfer(PST). We show that this method is capable of finding all possible coupling schemes for PST. These schemes, without external control fields, only involve preengineered couplings but not dynamical control of them, so they can besimply realized experimentally. Theanalytical solutions 1 provideall information for coupling design. 1 0 PACSnumbers: 03.67.Hk,05.50.+q 2 n Quantum information and quantum computation can "verifiable" but not "constructive" way. Thus, we have a J process lots of tasks which are intractable with classi- not yet got all possible coupling schemes for PST. 6 cal technologies. Although many schemes such as quan- In this Letter, we start from the necessary and suf- tum dots[1], ion trap[2], NMR[3] have been discussed ficient conditions of PST. After preselecting the eigen- ] extensively, a macroscopic scalable quantum computer h values of a XY spin chain Hamiltonian, we propose two p still seems to need a channel, often known as quantum recursiveformulasofthecouplingsforbothevenandodd - wire, to transmit or exchange quantum states between N casesandprovethembymathematicalinduction. Fur- t n inner parts of the quantum computer. These architec- therdiscussionsdemonstratethatthismethodiscapable a tures require to implement a transmission process for of finding all possible coupling schemes for PST in XY u an unknown quantum state from one place to another q chain with arbitrary length. Experimentally, our PST whichisoftencalledquantumstatetransfer. Inaseminal [ schemes can be realized, for example, by superconduct- paper[4],Boseproposedaspinchainmodel,whoseevolu- ingcircuitsandquantumbus[14],nanoelectromechanical 1 tionwasgovernedbyareasonableHamiltonian,andcon- resonator arrays[15] or cold-atom optical lattice[16]. v sideredthefidelityofstatetransferinthismodel. Similar 6 Next, we first review some basic concepts of state 5 results were also derived by studying dynamical proper- transfer protocol using spin chain as the channel[4, 17]. 1 ties of entanglement transition in Heisenberg XY spin An unknown qubit, as encoded in site 1, is attached to 1 chain[5]. This model, in which two processors are con- . one end of a spin chain when the chain is initialized to 1 nectedthroughaspinchainasquantumwire,isusefulfor the allspin-downgroundstate (state initializationis not 0 quantumcomputationbasedonHeisenberginteraction[6] necessary[18],andourresultscanbegeneralizedtothese 1 or measurements[7]. 1 cases). Due to the coupling between site 1 and 2, free Although some important and significant results have : evolutionofthe system causesthe unknown state to dis- v been found, see for example[4, 5], all of the available i tribute among the chain. After a specific interval, we X results are just concerned with uniform interaction, i.e. want to recover this unknown state at the opposite end the couplingsbetweenanytwonearest-neighborsitesare r of the chain to achieve state transfer. a the same. For this case, however, it is shown that when N ≥4, where N is the number of the sites in XY chain, A reasonable Hamiltonian for this task is XY type PST is impossible[8]. This drawback of uniform inter- Hamiltonian action motivates people to find some modified models to achieve "long" distance PST. Some works considered 1N−1 1 N long-rangeinteractions[9],andsomeconcentratedonnu- H = 2 X Ji(σixσix+1+σiyσiy+1)−2XBi(σiz−1), (1) mericalsimulations[10]. Onefeasiblechoiceistopreengi- i=1 i=1 neerthecouplings[8],i.e. choosespecialnonuniformcou- plings to achieve PST, and some specific analytical cou- whereJiisthecouplingstrengthbetweensitesiandi+1, pling schemes were found[8, 11, 12]. The necessary and andBi istheexternalstaticpotential,orcontrolfield,at sufficient conditions for the couplings of PST, which can site i. σx,σy,σz are the three Pauli matrices. One im- provideacriteriontoverifythepreengineeredschemesas portant observation is that Hamiltonian(1) commutates well as to find new analytical ones, were derived from a withthetotalz-spinoperator N σz. Thus, N σz is Pi=1 i Pi=1 i more systematical treatment of this problem[13] by mir- a conservation, and the evolution of the system in these ror inversion[11] and quantum computation. However, statetransfercaseswilljustinvolvethesubspacespanned all these preengineered schemes are obtained through a by ground state and N one-site excited states. By the 2 Jordan-Wigner transform, which maps(1) to For even N cases, we assume the eigenvalues of H N are {±Λ ,...,±Λ } where n = N/2 , Λ ∈ N and Λ > 1 n i 1 Λ > ... > Λ > 0 (if none of {J } is zero, then the N−1 N 2 n i H = X Ji(a†iai+1+a†i+1ai)+XBia†iai, (2) eigenvalues oτf HN are nondegenerate[19]), and omit the scalefactor . {J }and{Λ } areconnectedthroughthe i=1 i=1 π i i characteristic polynomial of the Hamiltonian(4): XY model can be solved exactly. Hamiltonian(2) de- scribes an N-site hopping model subjects to nonuniform n external fields. Let |ii denotes the single excited state Det(H −λI)= (λ2−Λ2) (5) N Y i at site i, Hamiltonian(2) in a 2N-dimensional space will i=1 reduce to an N-dimensional subspace spanned by {|ii}. which, by expanding it with respect to λ2, is equivalent Explicitly, to a set of equations: B J 0 ··· 0  1 1  N−1 n J1 B2 J2 ··· 0 J2 = Λ2 (6a) H = 0 J2 B3 ··· 0  (3) X i X i N   i=1 i=1  ... ... ... ... JN−1  ...  0 0 0 JN−1 BN  J2 ···J2 = Λ2 ···Λ2 , (6b) X k1 kn X k1 kn in {|ii} basis. The fidelity of this transfer procedure can ki+1−ki≥2 ki+1>ki be expressed as hN|e−iHNτ |1i, where τ is the time in- . . terval of the free evolution. The equivalent conditions . for PST, i.e. (cid:12)hN|e−iHNτ|1i(cid:12) = 1, are: (a) the reflec- J2J2...J2 = n Λ2 (6c) tion symmetry(cid:12)Bi = BN+1−i(cid:12)and Ji = JN−i. (b) after Y 1 3 N−1 Y i τ i=1 sorting the eigenvalues of H in decreasing order, the N π and we want to derive {J } from {Λ }. This is often differencebetweenanytwoadjacenteigenvaluesisanodd i i called an inverse problem. Notice that we still use J number[13]. All schemes discovered before required (a) N−i rather than J when i ≤ N/2 despite they are equal aspartoftheirprotocolsanddesignedtheeigenvaluesof i τ just for elegance of the expressions. We first introduce HNπ to be {−k,−k+1,···k−1,k} for 2k ∈ N[8, 11], some notations for convenience. Denote jn[N] = Jn[N] {q(k2 +k)+(2p+1)k} for k = 0,...,N[11] or {−k+ and j[N] = (J[N])2 for 1 ≤ i ≤ n − 1 whose mean- 12−n,...,−k−23,−k−12,k+21,k+23,...,k−12+n}[12] ing wiill becomie clear soon. Here, the superscript N whichallsatisfy(b). Allthese coupling schemesarespe- denotes the dimension of the matrix H and we im- N cialsolutionsforPST, andourmainresultinthis Letter ply the eigenvalues of H are {±Λ ,...,±Λ } and D 1 D/2 is to show how to get all possible couplings for PST in its couplings are {J[D]}. The main idea is to obtain theabsenceofexternalfields,i.e. B =0. Becauseofthe i i [D] [D−2] {j } from {j } when we require the Hamiltonians perfecttransfercondition(a)andthepostulationBi =0, i i construct by them respectively share the same eigen- Hamiltonian(3) becomes values {±Λ ,...,±Λ }. Further, denote ΓN = 1 (D−2)/2 k  J01 J01 J02 ······ 00 00  jn[N−j−1n[N2]]jjn[n[NN−−−2]12·]····j·k[jN+k[N−]1−12] for 1 ≤ k < n, where j0[D] ≡ 0, and  0 J2 0 ··· 0 0  ΓN = j[N−2]. Denote ∆N = jk[N−2]jk[N+−22]jk[N+−42]··· and HN = ... ... ... ... J2 0  (4) ∆nN =nj−[1N−2] , ∆N = 1k where tjkh[N+e]2jpk[N+ro]4jdk[Nu+]c6·t·s· in the  0 0 0 J 0 J  n−1 n−1 n  2 1  numerators and denominators involve terms only if the  0 0 0 0 J1 0  indices of them are not larger than n−1 and n respectively. With these notations, we will show the following whose eigenvalues are symmetric about zero. Owing to equation permits us to get {J } from {Λ } directly: this symmetry, there are only [N/2] independent cou- i i plings and [N/2] independent eigenvalues in(4) (0 is always an eigenvalue when N is odd). Our purpose is to j[N] = ΓN −(−1)iΛ ∆N i=1,··· ,n. (7) construct the couplings {J } from a set of preselected i i n i i eigenvalues {Λ } satisfy (b). We will first consider even i Eq.(7) allows to construct j[N] from {j[N−2],...,j[N−2]} N cases and show how to derive {J } effectively. Then, i n−1 i−1 i we generalize these results to odd N cases, and finally ,{jn[N],··· ,ji[+N1]}andΛn. Thus,whenweknow{ji[N−2]}, show the completeness of this method, i.e. it can get all by adding one more parameter Λn, we can derive possible coupling schemes for PST. j[N],j[N] ,···j[N] one by one explicitly. Now, we need n n−1 1 3 to proveEq.(7)is consistentwith Eq.(6). Direct calcula- 1,...,n are all eigenvalues of (4). Fig.(1) shows idea of tion shows Λ satisfies a continued fraction: the proof. n For odd N cases, we assume Λ >Λ >...>Λ > j[N] n n−1 1 jn[N−]1 n+(−1)n−1Λ =1. (8) n0.−D1e,finne=j(n[NN]−=12)/(J2,n[Na]n)d2 danefidnjei[NΓ]N=an(Jdi[N∆]N)2afosrth1e≤saim≤e . n k k .. as even N cases. The corresponding recursive formula jΛ1[Nn]−Λn+Λn for odd N is Eq.(8) is equivalentto Det(H −Λ I)=0. Actually, by j[N] = Λ2∆N −ΓN (10a) N n k n k k expanding Det(Hi −ΛI) in terms of order i−1 deter- j[N] = ∆N −ΓN (10b) minants, we will find the original continued fraction for k−1 k−1 k−1 Det(HN −ΛnI)=0 is where k = n,n−2,n−4,··· till we get j[N]. The dif- 1 (J[N])2 ference between n and k in Eq.(10a) is an even number (J.N[N−]1)2 N +(−1)N−1Λn+(−1)NΛn =0. (9) awlhteicrhnaitmelpyl.ieJsusΛt2nlikaeppeveaenrsNincathsees,riwgehtcahnanddireocftlEyqc.h(1ec0k) . . Λ is an eigenvalue of (4) by the continued fraction rep- (J1[Λ(NJn2[]N)2])−2Λn+Λn−Λn fraencsetonrta2tiaopnpweahresnin{Jtih}eadreefienxitpiroenssoefdjbn[Ny]Eaqls.o(1c0o)m, aensdfrtohme DuetothesymmetrybetweenJ andJ ,wecanmove thecontinuedfractionstructure. Althoughthemainidea i N−i upper half of the continued fraction to the right hand of is the same,there arestill somedifferences betweeneven the equal sign. After taking a square root on both sides, andoddcases. First,{j[N]}arenolongerunchangedun- i we obtain Eq.(8), which means Λn is actually an eigen- der the permutation of Λn and −Λn−1 when N is odd. value of (4). The square root operation is exactly the Instead, the permutations of {Λn,Λn−2,Λn−4,...} and origin of why we denote jn[N] = Jn[N] but ji[N] = (Ji[N])2 {Λn−1,Λn−3,Λn−5,...} form two groups keep {ji[N]} in- for i 6= n before. Next, we will prove Eq.(7) is correct variant respectively (if we consider all the eigenvalues for arbitrary N by mathematical induction. We assume {±Λi}, then both even and odd cases have two groups the permutations of {Λ ,−Λ ,Λ ,...,(−1)nΛ } form formedby interlacedeigenvaluesrespectively whichkeep 1 2 3 n−1 a groupkeeps {j[N−2]} unchangedwhichis actually true {j[N]} unchanged, see Fig.(1)). Second, it’s interesting i i for {j[4]}. The following step is to prove {j[N]} are also to see j3[N] and j2[N−2] have the same winivtahritaihnetuasnsduemrpthtieonpearmbouvtea,tidoinreocftlΛyninadnudc−eisΛ{n±−1Λwi}hicfohr, Λjn1[N−]j12[N−Λ]n−1+Λn−1 jΛ1[Nn−−22]−Λn−2 i = 1,...,n are the eigenvalues of H when {J[N]} are structure when a similar replacement as in even N cases N i beenmade. With the helpofthis property,we canprove [N] constructed from Eq.(7). Obviously, j is unchanged n Λ isalsoaneigenvaluesofH . Combiningitwiththe under the permutation of Λn and −Λn−1. For jn[N−]1, fanc−t1that Λn is an eigenvalue oNf HN and the symmetry we can expand it using Λ , Λ and {j[N−4]} in which property between Λ and Λ , by means of mathemat- n n−1 i n n−2 {ji[N−4]} are irrelevantto Λn and Λn−1. Notice that this ical induction, we assert the eigenvalues of HN whose expression has similar form with j[N−2] when j[N−2] is off-diagonal elements are constructed from Eq.(10) are n−5 n−5 expandedbyΛ ,Λ and{j[N−6]},andifwereplace n−1 n−2 i Λ , Λ and {j[N−4]} in j[N] by −Λ , −Λ and n n−1 i n−1 n−1 n−2 {j[N−6]} respectively, we will find they are indeed the -Λ -Λ-Λ -Λ ...-Λ -Λ 0 Λ Λ ... Λ Λ Λ Λ i 1 2 3 4 n-1 n n n-1 4 3 2 1 same one. Owing to the assumption that the permutation of Λ and −Λ keeps j[N−2] unchanged, we (a) n−1 n−2 n−5 concludethatj[N] isalsounchangedunderthepermuta- tion of Λn andn−−Λ1n−1. This method, demonstrating the -Λn -Λn-1-Λn-.2..0 Λ1 Λ2 Λ3 Λ4 Λ5 ... Λn-2 Λn-1 Λn invariance by replacement, is applicable for other ji[N], (b) and we can further prove all {j[N]} are invariant under i the permutation of Λ and −Λ . Figure 1: (a) and (b) are recursive procedures for even and Combining this pronof and then−fa1ct that {Λ ,−Λ } ac- oddN casesrespectively. Permutationsof{Λi}withthesame 1 2 colour form one group. Although 0 is always an eigenvalue tually form a group for {j1[4],j2[4]}, we can prove the in (b), the permutation group does not contain it. Arrows permutationsof{Λ1,−Λ2,...,(−1)n+1Λn}formagroup onthestraightlinesindicatetherecursivedirections,e.g. the keeps {j[N]} unchanged. Furthermore, if Λ is an eigen- rightestarrowin(b)impliesifΛnisaneigenvalueof (4)when value ofi(4), then, according to Eq.(8), {±nΛ } for i = {Ji} are constructed from Eq.(10) then so does Λn−2. i 4 {0,±Λ } for i=1,...,n indeed. chains can not afford PST. We also prove these formu- i The completeness of this method comes from the fact las are complete. Although this method is numerically that (3) is uniquely determined by its eigenvalues when effective, there are still some interesting issues. We set {J } are all positive[20]. This also implies all real cou- thediagonalelementstobezeros,i.e. thereisnoexternal i pling schemes for (4) are uniquely determined by its controlfieldinspinchainorthelaserresonanceswithany eigenvalues. Since the completeness is available only if twoadjacentlevelsinanN-levelsystem. Thisisnotnec- all {J } are real, we need to prove the positivity of {J2} essary for PST or perfect population transfer. Non-zero i i for Eq.(7) and Eq.(10). This is more apparent when diagonalelementsbreakthesymmetryofthespectrumof we factor out the common factors of each equation in theHamiltonian,andtheeigenvaluesnolongerappearin Eq.(7) and Eq.(10). After factorization, we will find pairs. Nevertheless, the continued fraction is also avail- each expression contains two factors, one is positive and able when we replace Λ by B −Λ. We expect similar i the other is monotone with respect to Λ . Considering formula for cases involve control fields which, of course, n Λ > Λ > 0 and Λ > Λ in even and odd cases will contain N recursive equations but not [N/2] for an n−1 n n n−1 respectively, we assert {j[N]} are all positive and {J } N-sitespinchain. Anotherquestioniswhether thereare n i are all real which satisfy the completeness condition. In other simple coupling schemes for special selected eigen- a word, Eq.(7) and Eq.(10) are complete for all possible values. We have tested some simple sets of eigenvalues, coupling schemes. but the couplings still seem complicated. Upto now,we havesolvedbothevenandodd N cases This work was supported in part by Foundation of in the absence of external control fields {B }. This con- i President of Hefei Institutes of Physical Science CAS, structive method allows us to calculate the couplings one of us (F.S.) was also partly supported by National from a set of preselected eigenvalues. We have chosen Natural Science Foundation of China (No. 61074052). several sets of 50 numbers whose interval between any two adjacent ones in each set is a random odd number in the domain [1,100]. In general, we got the couplings within 10 seconds. This numerical calculationshows our method is effective. Although the resultantcouplings of- [1] D.LossandD.DiVincenzo,Phys.Rev.A57,120(1998). tenhaveenormousnumeratorsanddenominatorscaused [2] J. Cirac and P. Zoller, Phys.Rev.Lett.74, 4091 (1995). by the continued fraction structure of the constructive [3] D. Cory et al., Proc. Natl. Acad. Sci. 94, 1634 (1997). [4] S. Bose, Phys. Rev.Lett.91, 207901 (2003). method, we can choose some specific eigenvalues and [5] V. Subrahmanyam,Phys.Rev.A 69,034304 (2004). thengetcompactcouplingschemes. Forexample,choos- [6] D. DiVincenzo et al., Nature408, 339 (2000). ing the eigenvalues as {±(T + 12 +i(2S+1))}, where T [7] R.RaussendorfandH.Briegel,Phys.Rev.Lett.86,5188 and S are two non-negative integers, for i = 1,2...,N (2001); R. Raussendorf et al., Phys. Rev. A 68, 022312 2 when N is even, we will find J2 are i(N−i)(1+2S)2 and (2003). i 4 [8] M.Christandletal.,Phys.Rev.Lett.92,187902(2004); ((1+2T)+(1+i)(1+2S))((41+2T)+(N+1−i)(1+2S)) for even and M. Christandl et al., Phys. Rev.A 71, 032312 (2005). odd i respectively. [9] A. Kay, Phys. Rev. A 73, 032306 (2006); G. Gualdi et The model used here is also similar to that we en- al., Phys. Rev.A 78, 022325 (2008). counter in population transfer in an N-level system in [10] P. Karbach and J. Stolze, Phys. Rev. A 72, 030301 which N discrete energy levels are equivalent to N sin- (2005);A.Wójciketal.,Phys.Rev.A72,034303(2005). [11] C. Albanese et al., Phys.Rev.Lett. 93, 230502 (2004). gleexcitedstates{|ii}. Assumingtheonlyinteractionto [12] T. Shiet al., Phys. Rev.A 71, 032309 (2005). be that of electric-dipole transitions and each frequency [13] M.-H.YungandS.Bose,Phys.Rev.A71,032310(2005). of the laser to be close to resonance with two adjacent [14] J. Q. You and F. Nori, Phys. Today 58, 42 (2005); J. states, after rotating wave approximation (a general re- Majer, et al., Nature449, 443 (2007). viewofthistopic,see[21]),Hamiltonianofthisproblemis [15] P. Rabl, et al., Nat. Phys.6, 602 (2010). identical to Eq.(4) when we treat the dipole interactions [16] I. Bloch, Nature453, 1016 (2008). [17] S. Bose, Contemporary Physics 48, 13 (2007); A. Kay, as the couplings in XY chain. Our results for PST can Int.J. Quan.Info. 8, 641 (2010). alsobeusedtodesigntheamplitudeofeachfrequencyof [18] C.DiFrancoetal.,Phys.Rev.Lett.101,230502(2008). the control laser to achieve perfect population transfer. [19] J. Wilkinson,The Algebraic eigenvalue problem (Claren- InthisLetter,wehaveconsideredtheproblemoftrans- don Press, 1965). ferring an unknown state from one end of a spin chain [20] H.Hochstadt,LinearAlgebraandItsApplications8,435 to the other end, and proposed two recursive formulas (1974). for designing the couplings since uniform coupled XY [21] B. Shore, ActaPhysica Slovaca 58, 243 (2008).