Asymptotically perfect efficient quantum state transfer across uniform chains with two impurities
aa r X i v : . [ c ond - m a t . qu a n t - g a s ] O c t Asymptotically perfect efficient quantum state transfer across uniform chains withtwo impurities
Xining Chen, Robert Mereau, and David L. Feder ∗ Institute for Quantum Science and Technology, Department of Physics and Astronomy,University of Calgary, Calgary, Alberta, Canada T2N 1N4 (Dated: October 18, 2018)The ability to transfer quantum information from one location to another with high fidelity isof central importance to quantum information science. Unfortunately for the simplest system of auniform chain (a spin chain or a particle in a one-dimensional lattice), the state transfer time growsexponentially in the chain length N at fixed fidelity. In this work we show that the addition of animpurity near each endpoint, coupled to the uniform chain with strength w , is sufficient to ensureefficient and high-fidelity state transfer. An eigenstate localized in the vicinity of the impurity canbe tuned into resonance with chain extended states by tuning w ( N ) ∝ N / ; the resulting avoidedcrossing yields resonant eigenstates with large amplitudes on the chain endpoints and approximatelyequidistant eigenvalues. The state transfer time scales as t ∝ N / and its fidelity F approachesunity in the thermodynamic limit N → ∞ ; the error scales as 1 − F ∝ N − . Thus, with the additionof two impurities, asymptotically perfect state transfer with a uniform chain is possible even in theabsence of external control. I. INTRODUCTION
The ability to transfer information is crucial for digi-tal communications. Likewise in quantum computationand communication, the ability to efficiently and reliablytransfer quantum information is central to both currentand future quantum technologies [1]. In the standardcircuit model of quantum computation, the quantuminformation is encoded on localized spins with two ormore accessible distinguishable quantum states (qubitsor qudits), and an algorithm is effected by manipulat-ing individual spins, performing two-spin operations, andmaking measurements [2]. The physical objects encod-ing the spins, for example spins in semiconductors [3] ornitrogen-vacancy centers in diamond [4], may be widelyseparated, requiring the development of robust quantumstate transfer protocols for spin networks [5].Schemes for perfect quantum state transfer (PST) inspin networks were developed over a decade ago [6, 7].Quantum information encoded on a spin at one end of alinear chain of nearest-neighbor coupled spins was foundto propagate perfectly to the opposite end, as long as thenon-uniform spin coupling coefficients could be carefullychosen. Subsequent work showed how to find all pos-sible coupling constants consistent with PST for a spinchain of arbitrary length [8, 9]. The ability to adjust alarge number of coupling constants is expected to be ex-perimentally challenging, and unfortunately PST is notpossible in uniform spin chains (i.e. where all couplingconstants are identical) longer than three sites. This hasprompted the investigation of different network topolo-gies with uniform (but possibly signed) couplings thatcan support PST [10–20].An alternative strategy is to relax the assumption of ∗ Corresponding author: [email protected] perfect quantum state transfer, replacing it with ‘prettygood state transfer’ (PGST) [21, 22] or equivalently ‘al-most perfect state transfer’ (APST) [23]. In these cases,one sets the desired fidelity F of the output state, anddetermines the time required (if any) to achieve it. Whilethese equivalent concepts in the literature are frequentlyreferred to as APST, we prefer to employ the term ‘im-perfect quantum state transfer’ (IST) to clearly distin-guish the behavior from PST. In IST there exists a timeat which the initial state at some site transfers to a dif-ferent site with probability approaching unity to withinsome error.For uniform spin chains, IST is only possible in princi-ple for particular values of the number of spins N [22, 24].For fixed minimum fidelity, however, the transfer time t increases exponentially with N [24]. A linear scaling of t with N can be achieved by coupling the initial and finalspins only weakly to the uniform chains [25–34]. Be-cause the quantum information is strongly localized inthe vicinity of the chain ends at all times, this modelis also more robust against noise than the bare uniformchain. Unfortunately, in this model after optimizing thestrength of the weak coupling parameter (which is foundto decrease like N − / ) and initializing the channel, themaximum output amplitude decreases with N , attain-ing an asymptotic value of 0 . N → ∞ [28].Adding additional weak links improves the results, butthe asymptotic error in fidelity remains finite [30]. In-stead applying a magnetic field in the vicinity of the chainends yields fidelities that approach unity in the large- N limit, but at the cost of t growing exponentially with N [33, 35–38].In this work we consider a uniform spin chain of length N , with sites labeled from 1 through N , with the minormodification of an additional impurity spin coupling tothe spin at site 3 and another to site N −
2, both withcoupling constant w . The value of w is left as a variableto be optimized. It is found that there exists a value of w ∼ √ N that yields a resonance between the quantumstate localized near the impurity and extended states ofthe chain. The resulting avoided crossing yields stronglymixed eigenvectors with equally spaced eigenvalues andlarge overlaps with the chain endpoints. Under theseconditions, the fidelity for quantum state transfer fromsite 1 to site N approaches unity in the thermodynamic(large- N ) limit, with error 1 − F ∝ N − . The systemtherefore exhibits ‘asymptotically perfect quantum statetransfer,’ a behavior previously unobserved for a spin net-work. The time is also found to scale efficiently with thechain length, t IST ∝ N / . In concrete terms, the out-put fidelity surpasses F = 0 . N >
99 and exceeds F = 0 .
99 for N ≥ w is found to be a strongly non-monotonic function of N .To find a functional form for w ( N ), the model is inves-tigated analytically in Sec. IV; high-fidelity IST in time t IST ∝ N / is found to occur if w ( N ) ∝ √ N . Section Vis devoted to a full numerical simulation of the evolutionunder the model Hamiltonian, validating and clarifyingthe analytical predictions. A brief discussion of the re-sults is found in Sec. VI. II. IMPERFECT QUANTUM STATETRANSFER
The time evolution of a quantum state | φ ( t ) i under theaction of a governing Hamiltonian H is given by | φ ( t ) i = e − itH/ ~ | φ (0) i . (1)Suppose that | φ ( t ) i is only defined at discrete sites j =1 , , . . . , N . Defining states | i i corresponding to unit ba-sis vectors associated with site i , one obtains the proba-bility amplitudes h i | φ ( t ) i = φ i ( t ). Given N eigenvalues λ ( n ) and orthonormal eigenvectors | ψ ( n ) i of H , Eq. (1)becomes | φ ( t ) i = N X n =1 exp (cid:20) − itλ ( n ) ~ (cid:21) | ψ ( n ) ih ψ ( n ) | φ (0) i . (2)Suppose furthermore that the initial state is completelylocalized at a particular site j , so that h i | φ (0) i = δ ij .Eq. (2) can then be rewritten as | φ ( t ) i = e − iλ ( m ) t/ ~ N X n =1 e − it ( λ ( n ) − λ ( m ) ) / ~ | ψ ( n ) ih ψ ( n ) | j i . (3)Evidently if ( λ ( n ) − λ ( m ) ) t/ ~ = 2 πs ( s ∈ Z ) for any n and m , then | φ ( t ) i = | j i for all j up to an unimportantoverall phase; that is, the discrete Hamiltonian is siteperiodic. This immediately implies that the Hamiltonian is site periodic if all the eigenvalues of H satisfy the ratiocondition λ ( n ) − λ ( m ) λ ( p ) − λ ( q ) ∈ Q , (4)for all possible indices { n, m, p, q } (except λ ( p ) = λ ( q ) ).For large systems in practice, condition (4) can only besatisfied if the spectrum is linear, i.e. the gap betweensuccessive non-degenerate eigenvalues is constant.In perfect quantum state transfer (PST) between sites i and j , a state initially localized in state | i i ends upin state | j i , j = i (and vice versa) after some elapsedtime t ; these sites are therefore periodic in time 2 t . PSTtherefore implies site periodicity, but the converse is notgenerally true. For space-symmetric Hamiltonians thatcommute with the parity operator, however, site period-icity implies PST. Suppose that the smallest difference ineigenvalues is ∆ = λ ( n ) − λ ( m ) for some m and n ; then siteperiodicity occurs in time t P = 2 π ~ / ∆. Consider an ini-tial state in a superposition state of sites i and j equidis-tant from an axis of symmetry, | φ (0) i = √ ( | i i + | j i ).Because | φ (0) i has even parity, only even-parity eigenvec-tors will contribute to the sum in Eq. (3). Site periodicitymust be maintained, but the absence of odd eigenvaluesimplies that the gap has now doubled to ∆ ′ = 2∆ (assum-ing the spectrum is linear); now t P = 4 π ~ / ∆ ′ . At halfthis time t = t P / π ~ / ∆ ′ the sum over n in Eq. (3)still resolves to the identity. Thus, even-parity states suchas the superposition state √ ( | i i + | j i ) evolve to them-selves in half the site periodicity time t = π ~ / ∆, whichis only possible if after this time | i i ↔ | j i , i.e. each siteundergoes PST to its mirror-symmetric counterpart. Tosummarize: mirror-symmetry and a linear spectrum aretogether sufficient to ensure that the Hamiltonian sup-ports PST.Unfortunately, most Hamiltonians do not possess aperfectly linear spectrum, even if they are mirror sym-metric. One may instead probe for IST, where the ini-tial state at some site transfers to a different site withhigh probability. A na¨ıve approach is to approximate alleigenvalues in the spectrum by rationals with the samecommon denominator, so that in principle the ratio con-dition (4) is automatically satisfied. Then in principlethere should exist some time at which the Hamiltonianis almost site periodic, where initial states localized ata given site return with probability proportional to theaccuracy of the rational approximation. By extension, ifparity is a commuting operator then at half this time theprobability at the mirror-symmetric vertex (the fidelity)should approach unity to within a similar error.In fact the criteria for IST are slightly more involvedthan the na¨ıve approach discussed above [23]. Recallthat for a site-periodic Hamiltonian there exists a time λ ( n ) t/ ~ = 2 πM n , M n ∈ Z for all n so that the sum over n in Eq. (2) resolves to the identity. For almost site period-icity, and allowing for an arbitrary n -independent phase,this criterion would become (cid:12)(cid:12) λ ( n ) t P / ~ + ϕ − πM n (cid:12)(cid:12) < δ ,where δ ≪
1. For reflection-symmetric Hamiltonians the
FIG. 1: Geometry of the model. Chain sites are labeled 1through N , and the tunneling amplitude between adjacentsites is constant J (not labeled in the figure). The impuritiesconsist of additional leaves at sites 3 and N − N + 1 and N + 2, respectively. The amplitude to tunnelbetween chain and impurity sites is w . eigenvalues of even and odd parity eigenstates interleave.At the IST time t = t P /
2, odd- n eigenstates map | i i to | N − i + 1 i but with an additional π phase. The ISTcondition then reads − δ < λ ( n ) t ~ − α n − πM n < δ, n = 1 , , . . . , N, (5)where α n = πn − ϕ . The first task would be to find in-tegers M n , the phase ϕ , and time t to satisfy this set ofinequalities for each n at fixed δ . Once this is accom-plished, one would calculate the output fidelity. Giveninitial occupation of vertex 1, the desired output vertexis N . The output fidelity is therefore F ( t ) = |h N | e − iHt | i| . (6)IST is said to occur at time t if F ( t ) exceeds some mini-mum threshold, for example F min = 0 . III. MODEL AND BEHAVIOR
Consider an array of spin- particles, each confinedto its own lattice site. As shown in Fig. 1, N sites arearranged in a one-dimensional chain with an additionalspin connected to the third site from each end. Givena spin-spin coupling constant J along the chain (not ex-plicitly labeled in the figure) and coupling constant w between the additional spins and their counterparts onthe chain, the XY Hamiltonian reads H = H + H ′ , (7)where H = J N − X i =1 ( X i X i +1 + Y i Y i +1 ) (8)corresponds to the one-dimensional uniform chain Hamil-tonian and the Hamiltonian for the two additional impu-rity sites is H ′ = w (cid:0) X X N +1 + Y Y N +1 + X N − X N +2 + Y N − Y N +2 (cid:1) ; (9) FIG. 2: The minimum time, in units of ~ /J , for IST on theimpurity-modified chain is shown as a function of the chainlength for N between 6 and 200 sites. The inset shows thevalue of the impurity hopping parameter (impurity spin cou-pling constant) w associated with the minimum time, in unitsof the chain parameter J . here X = σ x and Y = σ y are two-dimensional Pauli ma-trices. The total spin projection is a good quantum num-ber and the Hamiltonian diagonalizes into blocks with afixed number of excitations. Because only one excitationis required in order to effect state transfer, it is conven-tional to work in the single-excitation subspace [7, 35]. Inthis case, the spin Hamiltonian (7) is equivalent to a sin-gle particle hopping via a tight-binding Hamiltonian onan array with the same geometry, but with spin-couplingconstants replaced by hopping amplitudes [39]: H = J N − X i =1 ( | i ih i + 1 | + | i + 1 ih i | ) ; (10) H ′ = w ( | ih N + 1 | + | N − ih N + 2 | + H.c.) . (11)Here, states | i i are unit vectors associated with site i ,and H.c. stands for the Hermitian conjugate.Determination of the conditions under which IST canoccur (if any) for the geometry shown in Fig. 1 hingeson the diagonalization of the spin Hamiltonian (7) oralternatively (and more simply) the equivalent hoppingHamiltonian, Eqs. (10) and (11). To probe numericallyfor IST, we explicitly obtain the time-dependence of theprobability on the output vertex using Eq. (2), and cal-culate the time-dependent fidelity F ( t ), Eq. (6).Figure 2 shows the minimum time for which IST ispossible as a function of chain length N , for 6 ≤ N ≤ w = w/J in the range 0 ≤ ˜ w ≤ . N . For each ˜ w , the value of tJ/ ~ wasincreased in 0 . F ( t ) was found to exceed F min = 0 .
9. Though IST is found to occur for manychoices of t , only the lowest value of t for each ˜ w is shownin Fig. 2; the value of ˜ w that minimizes t is shown in theinset. The data clearly show that the minimum time (andthe impurity coupling constants associated with these)are nonmonotonic in N . There are intervals where theminimum time appears to scale linearly with N , but theseare interrupted and interspersed with different trends.Likewise, the impurity coupling constants seem to scaleroughly as √ N , but the data are not clean.In principle, one could try to determine the conditionson ˜ w that yield the absolute lowest-slope curve. Thiswould yield a slightly modified spin chain where the ISTtime would scale linearly with length, albeit with restric-tions on the values of N for which IST is possible. Inthis work, however, we pursue a different tack; namely,determining values of w valid for any instance of N andfor which the IST time scales efficiently with N (i.e. asa power-law with N preferably with a low exponent).The analytical treatment discussed in the next sectionaddresses this strategy. IV. ANALYTICAL TREATMENT
While a complete analytical solution for arbitrary w (orfor arbitrary w/J choosing J as the characteristic energyscale) appears difficult to obtain, approximate solutionsmay be obtained by solving the problem in the vicinity ofthe additional site(s) and matching to the bulk solution.First consider the left block in Fig. 1, consisting only ofsites i = { , , , , , N + 1 } (the block with sites i → N − i + 1, i = 1 , , . . . , N , and N + 1 → N + 2 is whollyequivalent). Expressing the left block solution as | ψ L i = X i =1 a i | i i + a N +1 | N + 1 i , (12)and operating with the left-block hopping Hamiltonian H L = J X i =1 ( | i ih i + 1 | + | i + 1 ih i | )+ w ( | ih N + 1 | + | N + 1 ih | ) (13)such that H L | ψ L i = λ | ψ L i , one obtains { a , a , a , a , a N +1 } = a γ ( , ˜ λ, ˜ λ − , ˜ λ − γ ˜ λ , ˜ w ˜ λ − λ ) , (14)where γ = 1 + ˜ w − (3 + ˜ w )˜ λ + ˜ λ , and ˜ w = w/J and˜ λ = λ/J are the rescaled values of w and λ relative tothe characteristic energy scale J .The most important eigenvectors for IST are thosewith large amplitudes on the first and last sites of thechain, and by extension small amplitudes elsewhere.Small amplitude on the third site can be ensured if ˜ λ ∼ γ/ ˜ λ ≈ γ ≪
1; this is possible if one chooses ˜ w ≫
1. More concretely, suppose one sets a = αa as-suming α ≫
1; using the explicit coefficients in Eq. (14)one obtains ˜ w = s ˜ λ c ( α ˜ λ c − α ˜ λ c − α (˜ λ c − . (15)Setting ˜ λ c = 1 − ǫ and expanding to lowest order in ǫ ∼ w ≈ ± r α αǫ ≈ ± √ ǫ . (16)Note that necessarily ǫ > λ c . λ c ≈ − ǫ ≈ −
12 ˜ w . (17)The value of ˜ w can be positive or negative. Inserting thisexpression for ˜ λ into Eq. (14) and again assuming ˜ w ≫ { a , a , a , a , a N +1 } ∝ (cid:26) ˜ w , ˜ w , − , − , − ˜ w (cid:27) . (18)Note that the amplitudes on both the first and secondsites of the chain are much larger than those elsewhere,which implies that any possible IST could equally origi-nate at either of these sites. Thus, ˜ λ c is the eigenvalue fora state strongly localized in the vicinity of the impurity.Next consider the solutions | ψ i for the bulk one-dimensional chain H , Eq. (10): h i | ψ ( n )1D i = sin( k n i ); ˜ λ ( n )1D = 2 cos( k n ) . (19)The reflection symmetry imposes the constraint that h i | ψ ( n )1D i = ±h N − i + 1 | ψ ( n )1D i , i.e. that the solution isan eigenstate of the parity operator. Setting sin( k n i ) = − sin( k n i ) cos[ k n ( N + 1)] + cos( k n i ) sin[ k n ( N + 1)] for theeven-parity case gives the conditions cos[ k n ( N +1)] = − k n ( N +1)] = 0 which is satisfied by k n = π (2 n +1) N +1 , n = 1 , , . . . . Likewise, the odd-parity solution requires k n = πnN +1 , so that overall k n = πnN +1 , n = 0 , , . . . , N − k n ) → − k n ) for n → N + 1 − n ( k n → π − k n ).The bulk eigenvectors should automatically match theleft-block solution (14) when ˜ w = 0. Whether using thebulk solution sin h cos − (cid:16) ˜ λ/ (cid:17) i i from Eq. (19) or setting˜ w = 0 in Eq. (14), one obtains the same result: { a , a , a , a } ∝ n , ˜ λ, ˜ λ − , ˜ λ (cid:16) ˜ λ − (cid:17)o . (20)Alternatively, one can require that the bulk solutionmatches the amplitude on the last site of the left block:sin(4 k n ) = sin(5 k n )(˜ λ − γ ) γ ˜ λ = sin(5 k n )˜ λ (˜ λ − λ − λ + 1 . (21)Using trigonometric identities it is straightforward toverify that this condition is (non-uniquely) satisfied bychoosing ˜ λ = ˜ λ ( n ) = 2 cos( k n ), as expected for the one-dimensional chain.Return again to the ˜ w ≫ λ (keeping in mind that the ˜ λ ∼ λ → − ˜ λ ).The parameter γ in Eq. (14) becomes γ → ˜ w (cid:16) − ˜ λ (cid:17) .The left-block solution for ˜ w ≫ { a , a , a , a , a N +1 } → ( , ˜ λ, ˜ λ − , ˜ w (1 − ˜ λ ) + ˜ λ − λ + 1˜ λ , ˜ w (˜ λ − λ ) , (22)neglecting unimportant prefactors. Comparison ofEqs. (20) and (22) reveals that the ˜ w = 0 and ˜ w ≫ λ ∼ a N +1 →
0. This implies thatthe ˜ w ≫ − / w , approaching unity asymptotically. The im-plication is that there must be (avoided) level crossingsat finite ˜ w for every ˜ λ ( n ) that comes into resonance withthe eigenvalue ˜ λ c of the localized state. It turns out thatefficient IST hinges on the first of these avoided crossings.Target eigenvalues are those in the vicinity of (but justbelow) unity, ˜ λ ( n ) = 2 cos( k n ) − ǫ ∼ − , so that k n & π/ k n = πn/ ( N + 1) are discrete even in thelimit of large N . There are therefore three different casesto consider: N = 3 m and 3 m + 1 with m = 2 , , . . . , and3 m − m = 3 , , . . . . Consider first the N = 3 m case.The wave vectors of interest are indexed by n = m + r : k m + r = π ( m + r )3 m + 1 , (23)where r = 1 , , . . . for k m + r > π/ λ r < r = 1 corresponds to n =3 m + r ). The r = 1 eigenvalue ˜ λ ≈ k m +1 ) mustcross the critical eigenvalue ˜ λ c in Eq. (17) at some valueof ˜ w ≫
1. The behavior of the system for large chainsis of particular interest; expanding around N = 3 m ≫ w , the levels cross when1 − π √ N ≈ −
12 ˜ w , (24)which yields the critical impurity coupling constant˜ w c ≈ ± / √ π √ N ≈ ± . √ N . (25) Alternatively, consider the left-block eigenvector (14),which becomes { a , a , a , a , a N +1 } ≈ ( √
32 + 8 π ˜ w N + O (cid:18) ˜ w N (cid:19) , √
32 + 8 π ˜ w N + O (cid:18) ˜ w N (cid:19) , − πN + O (cid:18) ˜ w N (cid:19) , − √ − π N + O (cid:18) ˜ w N (cid:19) , − π ˜ wN + O (cid:18) ˜ w N (cid:19) ) . (26)Clearly, the expansions above are analytic only if ˜ w varieswith N more slowly than √ N (or one could obtain aconvergent series by expanding the solution (14) in N/ ˜ w for ˜ w a polynomial in N with exponent greater than 1 / w c = α √ N is therefore of particularinterest. For large N one obtains { a , a , a , a , a N +1 } ≈ ( √ (cid:16) − πα √ (cid:17) , √ (cid:16) − πα √ (cid:17) , − πN (cid:16) − πα √ (cid:17) , − √ , − πα √ N (cid:16) − πα √ (cid:17) ) , (27)which matches the bulk solution at the fourth site. Theamplitudes on the first and second sites are strongly en-hanced relative to the others if one sets 1 − πα √ = 0 or α = 3 / / √ π , consistent with Eq. (25). Equivalently,the state (27) has maximal overlap with the state (18)when the first two amplitudes above equal ˜ w = α N ,which also occurs for α = 3 / / √ π .The N = 3 m + 1 and N = 3 m − N = 3 m + 1 for large N one obtains thecritical impurity coupling constant˜ w c ≈ ± / √ π √ N ≈ . √ N , (28)while for N = 3 m − w c ≈ / √ π ≈ . √ N. (29)While all critical impurity coupling constants scale as √ N , the prefactor depends on the particular choice of N (mod 3), N + 1 (mod 3), or N − N = 3 m case. As ˜ w is increasedthrough the critical value (25), the 2 cos( k m +1 ) eigen-value must exhibit and an avoided crossing with ˜ λ c , whileits associated eigenvector strongly mixes with the nextodd-parity state with eigenvalue ˜ λ . Likewise, the eigen-value ˜ λ of the first relevant even-parity state should alsofollow Eq. (17) for large ˜ w , while strongly mixing withthe ˜ λ state for intermediate ˜ w at the second avoidedcrossing in the vicinity of 2 cos( k m +3 ), etc. That said,presumably the ˜ λ state only mixes weakly with the ˜ λ state near the first avoided crossing (and of course notat all with the ˜ λ and ˜ λ states due to parity), whichshould occur for much larger values of ˜ w than the sec-ond avoided crossing. With this assumption, ˜ λ followsEq. (17) throughout the first level crossing.By inference, therefore, only three states are relevantto the first avoided crossing, corresponding to eigenval-ues indexed by r = 1 , ,
3. The same phenomenon alsoapplies to the N = 3 m + 1 and N = 3 m − k labels and there-fore at different energies. Importantly, all three statesinvolved in the avoided crossing at ˜ w c have strongly en-hanced amplitude on the first and last site of the chain(as well as the second and second-from last), of orderunity after normalization. All other states will be far off-resonant, and will have low amplitudes on the endpointsproportional to the overall normalization constant forbulk eigenvectors ∝ p /N . But the sum of outer prod-ucts of eigenvectors must resolve to the identity. Thisimplies that as N → ∞ , no off-resonant eigenvectors willcontribute to the state transfer: the sum in Eq. (2) willonly include resonant eigenvectors. Because only threeequally-spaced eigenvalues are involved (plus their nega-tives), the state transfer must be asymptotically perfect.Obtaining an analytical estimate of the energy split-ting at the critical impurity coupling constant is notas straightforward as it appears. The usual methodwould be to start with eigenfunctions | ψ ( m +1)1D i and | ψ ( m +3)1D i of the unperturbed Hamiltonian (10) and thencalculate the mixing caused by the perturbation (11),i.e. the off-diagonal term of the mixing matrix ∆ ≡h ψ ( m +1)1D | H ′ | ψ ( m +3)1D i . The impurities have no support onthe bare chain, however, so in principle the energy split-ting ∆ = 0. One can nevertheless estimate ∆ as fol-lows. The contribution to ∆ from all the chain sites willbe zero, as the unperturbed eigenfunctions are orthogo-nal. At ˜ w c , the amplitude on the impurity site for the k = m + 1 state is − / w c = −√ π/ / √ N after nor-malization (which is dominated by the amplitudes on thefirst two and last two sites of the chain). Likewise, theamplitude on the impurity site for the k = m + 3 state is − p /N (4 √ π/ / √ N ) = − √ π/ / N including thenormalization factor for sin( k n ) eigenfunctions. Becausethe action of the Hamiltonian on this site returns thesame amplitude (the energy is almost unity), one obtains∆ ≈ √ π / N √ π / √ N = 4 √ π N / . (30)While the coefficient is probably not that accurate, theanalytics suggest that the energy splitting at the avoidedcrossing scales with the chain length as N − / . The ISTtime should therefore scale as t IST ∼ N / .At this juncture the reader might well be wonderingwhat is special about adding impurities to the third andthe third-from-last sites of the chain. Suppose that theimpurities were instead located on the second and second-from-last sites of the chain. The left-block state, analo- gous to Eq. (14) is found to be { a , a , a , a N +1 } = a γ ˜ λ n , ˜ λ, ˜ λ − − ˜ w , ˜ w o , (31)where γ = 2 − ˜ λ − ˜ w . The amplitude on the first sitecan be made larger than in the bulk only if ˜ λ ∼ w ∼
0. The second condition is unfortunately equiva-lent to the unmodified chain, and is therefore not useful.Consider instead impurities on the fourth and fourth-from-last sites of the chain. The left-block solution isnow { a , a , a , a , a , a N +1 } = a ˜ λγ n , ˜ λ, ˜ λ − , ˜ λ (˜ λ − , ˜ λ − γ, ˜ w (˜ λ − o , (32)where γ = 3 + 2 w − (4 + w )˜ λ + ˜ λ . Following theanalysis above, small amplitude on the fourth and fifthsites requires ˜ w ≫ λ ≈ √ − / √ w . To leadingorder in ˜ w one obtains { a , a , a , a , a , a N +1 }≈ (cid:26) ˜ w , √ w , ˜ w , −√ , − , − ˜ w (cid:27) . (33)Just as for impurities on sites i = 3 and N − i +2, there isa strong enhancement of amplitude on the first and lastsites. This enhancement is now shared with four othersites ( i = 2, 3, N −
1, and N − V. NUMERICAL RESULTS
The strategy pursued in this work is to determine ifthere exist values of ˜ w ( N ) that allow APST to occurfor all N , but perhaps not at the absolute minimumtime allowable. Using the analytical results of the pre-vious section as a guide, for particular values of ˜ w oneexpects that for eigenvalues ˜ λ ( n ) . | ψ ( n ) i will have a largeoverlap with the first and last sites of the chain. If theeigenvalues are reverse ordered so that ˜ λ ( n ) ≥ ˜ λ ( n +1) for n = 1 , , . . . , N −
1, then there exist values of n & N/ λ ( n ) .
1. Define N ≡ m + p , where p ∈ {− , , } .Then one can define ˜ λ q = ˜ λ ( q + m ) which are all less thanunity for q = 1 , , . . . , and their associated eigenvectors | ψ q i . Then IST should result for values of ˜ w where theeigenvalues and eigenvectors satisfy the following two cri-teria: (cid:12)(cid:12)(cid:12)(cid:12) ∆ − ∆ ∆ (cid:12)(cid:12)(cid:12)(cid:12) ≤ ǫ ; (34)2 X q =1 |h ψ q | i| ≥ f, (35) FIG. 3: The first few eigenvalues ˜ λ q = λ q + m /J smaller thanunity ( q = 1 , , . . . ,
6) are shown as a function of the impuritycoupling parameter ˜ w = w/J for chain length N = 3 m = 501(black lines). Also shown is the left-block eigenvalue ˜ λ = λ/J = 1 − / w (dotted curve). The first avoided levelcrossing, closest to unity, is magnified in the inset. where the two successive eigenvalue gaps are ˜∆ = ˜ λ − ˜ λ and ˜∆ = ˜ λ − ˜ λ , and ǫ ≪ f . H was diagonalized for each chainlength N in the range 6 ≤ N ≤ w = w/J for N = 3 m = 501. The salient features predictedby the analytical treatment are readily observed here.The bare-chain eigenvalues ˜ λ q = 2 cos[ π ( q + m ) / ( N + 1)](for ˜ w = 0) are crossed by the eigenvalue ˜ λ c = 1 − / / ˜ w of the localized state at finite w , (almost) recovering theirvalues for large ˜ w . The q = 1 and q = 2 eigenvaluesboth follow ˜ λ c for large ˜ w , while the q = 3 eigenvalueapproaches 2 cos[ π (1 + m ) / ( N + 1)] in the same limit.The strong mixing between the localized and extendedbulk eigenvectors gives rise to avoided crossings, and thefirst such crossing (closest to unity) is shown in the insetof Fig. 3. As expected, the q = 2 eigenvalue closelyfollows ˜ λ c through the first avoided crossing, as it onlyweakly mixes with the q = 4 eigenvalue here. At the valueof ˜ w where ˜ λ c ≈ ˜ λ = ˜ λ , the ˜ λ and ˜ λ eigenvalues aresplit equally above and below ˜ λ . Thus, ˜ w c both definesthe point at which ˜ λ c = ˜ λ and the point at which theenergy gaps coincide, ˜∆ = ˜∆ ≡ ˜∆.Figure 3 also suggests that there are multiple values ofthe impurity hopping parameter that give rise to eigen-value resonances. The second avoided crossing occurswhen ˜ λ c = ˜ λ . For the N = 501 case displayed here,the analog of result (25) for the second avoided crossingis ˜ w ′ c = (3 / / √ π ) √ N ≈ . √ N . While the quali-tative behavior of the eigenvalue splitting is similar, onewould expect a lower IST output fidelity as the maximumamplitude on the endpoint sites is now approximately N w / J FIG. 4: The numerical value of the impurity coupling constant˜ w = w/J ensuring that the eigenvales at the first level crossingare equally spaced are shown as a function of the chain length N . Top, middle, and lower curves correspond to N = 3 m + 1,3 m , and 3 m − m integer), respectively. All curves closelyfollow ˜ w ∼ √ N for large N .
63% lower than for the first avoided crossing accordingto Eq. (18).Eq. (34) specifies that the spacing between successiveeigenvalues just less than unity be equal to within sometolerance ǫ . Figure 4 shows the value of impurity pa-rameter ˜ w = w/J satisfying Eq. (34) at the first avoidedcrossing to a tolerance ǫ = 10 − , for 6 ≤ N ≤ N . The lowest, middle, and upper curves correspondto N + 1 (mod 3), N (mod 3), and N − ≤ N ≤ > . w ≈ . N . for N = 3 m + 1,˜ w ≈ . N . for N = 3 m , and ˜ w ≈ . N . for N = 3 m −
1. In all cases, the results are compatiblewith a √ N scaling for large N . That said, the prefactorsobtained numerically are all found to be below the ana-lytical preductions by approximately 10%; it is possiblethat the correspondence between analytics and numericswould tighten up for larger N .Under the assumption that only the first three eigen-vectors | ψ q i with eigenvalues λ q just below unity (andtheir negative counterparts) contribute to the transferdynamics, the IST time can be predicted using Eq. (3).Following the discussion in Sec. II, imperfect site peri-odicity occurs at a time t = 2 π ~ / ( λ − λ ) = 2 π ~ / ∆ ,and parity conservation implies that the imperfect statetransfer time is half this, t IST ≡ π ~ / ∆ = ( π ~ /J ) / ˜∆ . Inrescaled time units t = ( ~ /J )˜ t , one obtains ˜ t IST ≡ π/ ˜∆,where ˜∆ = ˜∆ = ˜∆ . The state transfer times t IST are shown as a function of the chain length in the range N t I S T ( h _ / J ) FIG. 5: The numerical estimate of the imperfect state transfertime t IST , based on the value of the resonant energy gaps, isshown as a function of the chain length N . Top, middle, andlower curves correspond to N = 3 m + 1, 3 m , and 3 m − m integer), respectively. All curves closely follow t IST ∼ N / for large N . ≤ N ≤
501 in Fig. 5. These times are based on theenergy gaps ˜∆ = ˜∆ at the critical weight plotted inFig. 4. As was the case for the critical weights, the ISTtime scales differently depending on the value of N (mod3). For N = 3 m , the numerical results are best fit bythe function ˜ t IST ≈ . N . for large integer m; for N = 3 m + 1 and 3 m − t IST ≈ . N . and ˜ t IST ≈ . N . , respectively. The power laws areall consistent with a N / scaling of time, as predictedby the analytics discussed in the previous section. Theprefactors also appear to scale roughly with those for thecritical weights.It is worthwhile to investigate the IST dynamics gov-erned by Eq. (2) more closely. Keeping in mind that | ψ i and | ψ i have even parity while | ψ i has odd par-ity (and vice versa for λ q → − λ q ), so that for example h ψ | i = h ψ | N i while h ψ | i = −h ψ | N i , one obtains h N | e − iHt/ ~ | i ≈ − i sin [( λ + ∆) t/ ~ ] |h ψ | i| + 2 i sin [ λ t/ ~ ] |h ψ | i| − i sin [( λ − ∆) t/ ~ ] |h ψ | i| , (36)keeping only resonant eigenvectors in the sum. Choos-ing t such that the coefficients of each |h ψ q | i| termare equal, one obtains t IST = π ~ / ∆, as expected. ThenEq. (36) becomes h N | e − iHt/ ~ | i = 2 i sin (cid:18) πλ ∆ (cid:19) X q =1 |h ψ q | i| ≈ i sin (cid:18) πλ ∆ (cid:19) , (37)where the second line is obtained by assuming thatonly these eigenvectors (and their negative-eigenvalue t ( h _ / J ) F ( t ) (a) t ( h _ /J ) F ( t ) (b) FIG. 6: The IST fidelity F ( t ) is plotted as a function of timefor N = 501 at the critical impurity coupling constant ˜ w ≈ .
2. Dots correspond to numerical data while the solid linescorrespond to the envelope of the output fidelity includingonly contributions from the critical eigenvectors. The slowincrease in fidelity through the IST time t IST = π/ ˜∆ ≈ t IST is shown in (b). Note that the dots are so closely spacedhere that they resemble an oscillating line, whereas the solidline appears almost horizontal. counterparts) resolve the identity. Equation (37) ap-pears to suggest that the maximum fidelity is F ( t IST ) =sin ( πλ / ∆) ≈ sin ( π/ ˜∆). The interference arises fromthe fact that while the first three eigenvalues below unitybecome approximately evenly spaced by ∆ at resonance,the corresponding second set of approximately evenlyspaced eigenvalues above − π/ ∆. For times in thevicinity of t ∼ t IST , Eq. (3) becomes approximately h N | e − iHt/ ~ | i ≈ X q =1 (cid:16) − e − iλ t/ ~ |h ψ q | i| + e iλ t/ ~ (cid:12)(cid:12) h ψ ′ q | i (cid:12)(cid:12) (cid:17) . (38)Here, | ψ ′ q i are the eigenvectors corresponding to the firsteigenvalues above − λ q → − λ q ); note that h | ψ ′ q i = h | ψ q i while h N | ψ ′ q i = −h N | ψ q i . The right-hand side of Eq. (38) is proportional to unity if time ischosen to be t = (2 n − π ~ / λ , or ˜ t = − π/ rπ where r is an arbitrary integer. For times near t IST , theprobability on the output site varies from zero to nearunity with a period π ≪ t IST . One can therefore choosea time in the vicinity of t IST at which the fidelity shouldapproach unity.Figure 6 shows the IST fidelity F ( t ), defined in Eq. (6),as a function of time for the particular case of N = 501using the optimal impurity coupling constant ˜ w c ≈ . F ( t ) reaches a maximum value at a timenear ˜ t IST ≈ .
44, though the function oscillatesrapidly throughout this slow variation. The behaviourof the fidelity including only resonant eigenvectors in thesum (2) is shown for comparison; only the envelope ofthe fidelity is plotted for clarity, as this exhibits the same N F m a x FIG. 7: (color online) The maximum fidelity F max for thestate transfer between endpoints is shown as a function of thechain length N . Red, black, and blue curves correspond to N = 3 m +1, 3 m , and 3 m −
1, 3 ≤ m ≤ fast oscillation of the full data. Figure 6(a) clearly showsthat the time evolution of the output fidelity is governedalmost completely by the resonant eigenvectors. A close-up of the time sequence in the vicinity of t ∼ t IST re-veals the fast oscillation of F ( t ); the period is found tobe ˜ T fast ≈ .
16 which is close to the predicted value of π . While F ( t IST ) ≈ . F max ≈ .
975 at the slightly longertime ˜ t = 3867 .
70. This maximum is attained for many˜ T fast periods in the vicinity of ˜ t IST . In fact, the maxi-mum fidelity for the full dynamics slightly exceeds thevalue F ≈ .
964 obtained from including only the reso-nant eigenvectors.The maximum fidelity F max for quantum state transferis shown as a function of N in Fig. 7 for 10 ≤ N ≤ t IST − T fast ≤ ˜ t ≤ ˜ t IST + 3 ˜ T fast andrecording the maximum result for each N . The exactresults F max , where all eigenvectors are included in thesum (2), are shown as solid lines; the fidelities F ′ max when the sum is restricted only to resonant eigenvectorsis shown as dashed lines. While N -dependent oscillationsare evident in the exact results, the amplitudes decreaseand the wavelengths increase with N ; more important,though, their centers consistently follow the restricted fi-delity curves. The restricted fidelities therefore providean accurate representation of the exact fidelities in thethermodynamic limit N → ∞ . The values of restrictedfidelities for large N in the range 800 ≤ N ≤ F ′ max ≈ − . N . N = 3 m + 1;1 − . N . N = 3 m ;1 − . N . N = 3 m − , (39)with suitably chosen integer m . Thus, the IST fidelityapproaches unity in the thermodynamic limit, i.e. thequantum state transfer is asymptotically perfect. Ac-cording to the numerics, the state transfer error scales as1 − F ∝ N − for large N . VI. DISCUSSION AND CONCLUSIONS
In this work we have shown that asymptotically perfectquantum state transfer is possible in uniform chains thathave been modified by the addition of two impurites, cou-pled to the uniform chain at the third and third-from-lastsites with strength w . Choosing w ∝ √ N , the state lo-calized in the vicinity of the impurity can be tuned intoresonance with chain extended states. The associatedavoided level crossing gives rise to eigenstates with largeoverlaps with the chain endpoints and with eigenvalueswhose spacings become approximately equal. The ap-proximate linear spectrum together with reflection sym-metry yields approximately perfect state transfer, in atime that scales efficiently with length, as t IST ∝ N / .Indeed, the fidelity is found to approach unity in thethermodynamic limit N → ∞ , with error scaling as1 − F ∝ N − . To our knowledge, this is the only configu-ration with no external time-dependent or local control,where a uniform chain can be made to transfer quantuminformation perfectly in the limit of large system size,While the central insights obtained from the analyticalinvestigations are validated by the explicit calculations,the numerical results reveal additional information anddisplay some important features. First and foremost, thedetailed dependence of the energy splitting on N at res-onance (equivalently the amplitude of the t IST ∝ N / scaling) was only readily available numerically. Second,the exact time-dependence of the output probability wasfound to oscillate rapidly (with period π ~ /J ) in addi-tion to the slow evolution toward maximum fidelity inthe vicinity of t IST , independent of N . This means thatin a practical experiment (with N large but fixed) thetiming would have to be tested over a range of times | t − t IST | ≤ ( π/ ~ /J ) prior to using this device totransfer unknown quantum information. Third, the exactvalue of the maximum fidelity is found to follow the valueobtained by including only the critical eigenvectors, butfor smaller N it displays pronounced oscillations. Theamplitude and frequency of these oscillations decreasessteadily with N , so that in the thermodynamic limit themaximum fidelity is completely dominated by the reso-nant eigenvectors.Given that the high-fidelity transfer is a direct con-sequence of a resonance between the localized and ex-tended states, one might expect the model to be robust0against random small errors in the chain coupling con-stants around J . The errors would shift the frequencies ofthe extended states, so that a new value of w would needto be found to bring them back into resonance. While thisis possible in principle, in practice finding the best valueof w and time could be difficult; if the errors are time-dependent the situation is even worse. Unfortunately,numerical calculations suggest that the value of F fallsprecipitously with noise if w and t are both fixed at theiroptimal noise-free value. Given J i,i +1 = J ± δJ i,i +1 withrandom values | δJ i,i +1 | ≤ x , we find for N = 501 thatthe average fidelity drops to F ∼ . x ≈ . N impurities in two dimensions. Rather, one could envis-age arranging a sequence of impurities forming a half boxof length three centered at each corner. This would en-sure the presence of a localized state near the endpoints,which could again be tuned into resonance through a suit-able adjustment of the impurity coupling parameters. Wehope to explore this idea further in future work. Acknowledgments
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