Applicability of coupling strength estimation for linear chains of restricted access
AApplicability of coupling strength estimation for linear chains of restricted access
He Feng,
1, 2
Tian-Min Yan, ∗ and Y. H. Jiang
1, 2, 3, † Shanghai Advanced Research Institute, Chinese Academy of Sciences, Shanghai 201210, China University of Chinese Academy of Sciences, Beijing 100049, China ShanghaiTech University, Shanghai 201210, China
The characterization of an unknown quantum system requires the Hamiltonian identification. Thefull access to the system, however, is usually restricted, hindering the direct retrieval of relevantparameters, and a reliable indirect estimation is usually required. In this work, the algorithmproposed by Burgarth et al. [Phys. Rev. A , 020305 (2009)], which allows estimating the couplingstrengths in a linear chain by addressing only one end site, is further investigated. The scheme isnumerically studied for states with chain structure, exploring its applicability against observationalerrors including the limited signal-noise ratio and the finite spectral width. The spectral distributionof the end state is shown to determine the applicability of the method, and reducing the loss fromtruncated spectral components is critical to realizing the robust reconstruction of coupling strengths. I. INTRODUCTION
The accurate control of the Hamiltonian of designedsystems is always a prerequisite to carry out desiredtasks, e.g., quantum computation [1, 2] quantum com-munication [3] and quantum metrology [4, 5]. The sys-tems are usually microscopic structures which are deli-cately engineered to realize specific functions. To verifyand benchmark these fabricated structures, the charac-terization of the system via Hamiltonian identification isdesired [6]. Usually, the dynamics within the system areextremely complicated, and the probing access is oftenrestricted. A delicately devised identification scheme issupposed to allow the sensible estimation of unknownparameters, reconstructing the Hamiltonian indirectlybased on partially available information, e.g., the a priori knowledge about the structure of the quantum network,the initial state, or an accessible subset of observables, etal. .Under the challenge of complicated dynamics and re-stricted addressing resources, various identification algo-rithms have been developed aiming at experimental re-alizations. In a many-body system, the dynamical de-coupling technique, which simplifies the problem by de-coupling each pair of qubits from the rest, allows for theHamiltonian identification with arbitrary long-range cou-plings between qubits [7]. Besides, the dynamics can bealtered by tuning the control pulse that is applied to theprobe spin, improving the precision of estimation scheme[8, 9]. In a network of limited access, the identificationscheme is intended to bridge the Hamiltonian parame-ters and the observables from accessible subsystem thatare relatively easy to measure. The system realizationtheory is proposed for temporal record of the observ-ables of a local subsystem [10, 11], which has been ex-perimentally demonstrated on a liquid nuclear magneticresonance quantum information processor [12]. Zeeman ∗ [email protected] † [email protected] marker protocol shows that local field-induced spectralshifts can be used to estimate parameters in spin chainsor networks [13]. Also, when the graph infection rule issatisfied, the similar parameter estimation is also avail-able utilizing the spectral information retrieved from apartially accessible spin [14, 15].In this work, we re-examine the estimation algorithmproposed in [14], where the reconstruction of couplingstrengths in a linear chain without the full access is con-sidered. The procedure, probing the global propertiesfrom a local site, is similar to the estimation of springconstants in classical-harmonic oscillator chains. By ac-cessing only the end of the linear chain, the algorithmallows deducing all coupling strengths from the data ofassociated spectral information. Nevertheless, the errorsof the input data are inevitable and may significantlyinfluence the reconstruction results. In this work, wefocus on the applicability of the algorithm when the ac-quired initial data deviate from the actual values. Thesimulations show that the spectral distribution is an im-portant indicator of the applicability of the algorithm.Examining the spectral distribution, the increasing num-ber of spectral components that approach zero are likelyto fail the reconstruction. The applicability depends onboth the properties of individual system and conditionsof measurement. The work is organized as followings.In Sec. II, the recursive relations among spectral coef-ficients and coupling strengths from adjacent sites arederived. In Sec. III, the robustness and the availabilityof the algorithm are discussed under different conditionswhen input errors are introduced. II. THEORY OF COUPLING STRENGTHESTIMATION
We consider the state transfer within a system gov-erned by the generic Schrödinger equation, ˙ c i = − i ε i c i − i (cid:88) k ∼ i J i,k c k , (1) a r X i v : . [ qu a n t - ph ] M a r with c i the amplitude of state | i (cid:105) , ε i the energy and J i,k the coupling strength. The symbol " ∼ " means the twostates are coupled. With amplitude c i = (cid:104) i | Ψ( t ) (cid:105) rep-resented spectrally, c i = (cid:80) n C i,n e − i λ n t , where C i,n = (cid:104) i | λ n (cid:105)(cid:104) λ n | ψ (0) (cid:105) is the spectral coefficient of mode n . Therelation of C i,n among coupled states reads C i,n = 1 λ n − ε i (cid:88) k ∼ i J i,k C k,n . (2)The relation allows estimating the parameters using theinformation retrieved from a subset of states under therestricted-access condition. The simplest example is alinear chain as shown in Fig. 1, where only the left-most state | (cid:105) is accessible. Given an N -state chain withenergies ε i known, all coupling strengths J i,i +1 can bereconstructed using the spectral information simply readfrom state | (cid:105) . Detector
Figure 1. Schematics of parameter estimation for a chain of N states. Assuming that the access of the chain is restricted,only state | (cid:105) , the left-most site of the chain, can be mea-sured by the detector and all the rest (states | (cid:105) , | (cid:105) , ..., | N (cid:105) )is concealed by a blackbox. Using the scheme of parameterestimation, however, all coupling strengths, J i,i +1 , can be es-timated if the spectral information of | (cid:105) is available. The feasibility of the scheme can be shown by therepetitive use of Eq. (2) and the normalization condition (cid:104) i | i (cid:105) = 1 . Given the eigenmode n , the C ,n of left-moststate | (cid:105) allows deriving C ,n of the next one, C ,n = λ n − ε J C ,n . (3)Given that only J i,k with k = i ± are nonvanishingin a linear chain, the repetitive use of Eq. (2) yieldscoefficients of subsequent states recursively, C i +1 ,n = ( λ − ε i ) C i,n − J i − ,i C i − ,n J i,i +1 . (4)The denominator J i,i +1 should be non-zero, since a "bro-ken" coupling forbids the retrieval of the information onthe further side. Eqs. (3) and (4) allow for the recursiveevaluation of C i +1 ,n from C i,n and C i − ,n .On the other hand, the normalization condition (cid:104) i | i (cid:105) =1 should be satisfied. With C i,n = (cid:104) i | λ n (cid:105)(cid:104) λ n | ψ (0) (cid:105) , wehave (cid:104) i | i (cid:105) = (cid:80) n (cid:104) i | λ n (cid:105)(cid:104) λ n | i (cid:105) = (cid:80) n | C i,n (cid:104) λ n | ψ (0) (cid:105) | = 1 . The unknown denominator (cid:104) λ n | ψ (0) (cid:105) depends on the con-crete form of initial state | ψ (0) (cid:105) . If the system is ini-tially prepared by populating state | (cid:105) only, as consid-ered in our case, | ψ (0) (cid:105) = | (cid:105) , the denominator reads |(cid:104) λ n | ψ (0) (cid:105)| = |(cid:104) λ n | (cid:105)| = C ,n and the normalizationcondition becomes (cid:104) i | i (cid:105) = (cid:88) n | C i,n | C ,n = 1 . (5)With only state | (cid:105) accessible, the information of | (cid:105) , c ( t ) = (cid:80) n C ,n e − i λ n t , is supposed to be acquired bymeasurement, providing the input values of C ,n and λ n for further parameter estimation. The measured C ,n should be normalized by Eq. (5). Next, substituting Eq.(3) into Eq. (5) with i = 2 , the normalization condition (cid:104) | (cid:105) = 1 yields J , = (cid:118)(cid:117)(cid:117)(cid:116) N (cid:88) n | λ n − ε | C ,n (6)and the normalized C ,n using Eq. (3). With the evalu-ated J i − ,i , C i,n and C i − ,n , Eq. (5) generates the J i,i +1 for i (cid:62) , J i,i +1 = (cid:118)(cid:117)(cid:117)(cid:116) N (cid:88) n | ( λ n − ε i ) C i,n − J i − ,i C i − ,n | C ,n . (7)Hence, all coupling strengths can be recursively evaluatedbased on the above derived Eqs . (3), (4), (6) and (7).In a realistic experimental setup, the above schemeworks for any system that can be reduced to the formequivalent to Eq. (1). In [15], the method is applied tothe spin chain described by Heisenberg Hamiltonian, ˆ H = N − (cid:88) i J i,i +1 ( σ + i σ − i +1 + σ − i σ + i +1 + ∆ σ zi σ zi +1 ) (8)with ∆ the anisotropy. Assuming the system is pre-pared within the single excitation section, the subsequentevolution under the Hamiltonian (8) is still restrictedto the single excitation sector. The time evolution ismapped to the generic Schrödinger equation for a sin-gle particle, Eq. (1), where the energies are given by ε i = ∆ (cid:104)(cid:80) N − j =1 J j,j +1 − J i,i +1 + J i − ,i ) (cid:105) . Therefore,the above mentioned states {| i (cid:105)} , in this case, have beenmapped to spatial sites. The values of C ,n are embeddedin the reduced density matrix of | (cid:105) , which can be exper-imentally obtained by quantum state tomography [15].Here, we are not concerned with the specific realizationof the measurement, Hence, the input variables λ n and C ,n are assumed to be readily reachable, but they arenot necessarily guaranteed to be completely precise, aswill be discussed later.As a simple example, the estimation of J i,i +1 in a chainwith six sites is demonstrated in Fig. 2. Since the influ-ence from disorder is not of concern in this work, all ener-gies ε i are zero, as will also be considered in the followingdiscussions. With only site | (cid:105) accessible, the time evolu-tion of state | (cid:105) [Fig. 2(a)] is supposed to be detectable.In the associated spectral distribution [Fig. 2(b)], theposition and the height of each spectral peak provide λ n and C ,n , respectively, as required as input values by theestimation scheme. Performing the evaluation using Eqs. (3), (4), (6) and (7) recursively, we successfully recon-struct all the five unknown coupling strengths J i,i +1 , asshown in Fig. 2(c). -3 -2 -1 0 1 2 30.00.20.40.60.81.0 0 16 32 480.80.91.01.11.2 1 2 3 54 TimeSite i Peak 1 Peak 2Truncation threshold (a) (b)(c) (d)
Figure 2. Reconstruction of J i,i +1 for a chain of N = 6 and ε i = 0 . Panel (a) shows the temporal evolution of populationon site 1, and (b) shows its spectral distribution that provides λ n and C ,n as required by the reconstruction algorithm. In(c), applying the parameter estimation, the five unknown cou-pling strengths are estimated. The estimated values J (cid:48) i,i +1 show good agreement with the actual values J i,i +1 . Panel (d)presents the possible errors with the input values λ n and C ,n that may hamper the reconstruction procedure. For peak 1,the C ,n below the threshold during the measurement is sim-ply truncated. For peak 2, the broadening of the spectralpeak results in the deviation of the measured eigenvalue λ (cid:48) n from the actual λ n . III. INFLUENCE FROM ERRORS OF INPUTVARIABLES
During the actual measurement, the finite instrumen-tal resolution, the limited signal-noise ratio and otherperturbances may hinder the precise acquisition of ini-tial input λ n and C ,n . Therefore, the robustness of thealgorithm against the deviations of these input values iscritical to the success of the parameter estimation. Theinfluence from imprecision of measured C ,n has beenmentioned in [14], showing rather robust performanceagainst small deviations. When the finite signal-noiseratio of C ,n detection is considered, the even worse situ-ation occurs when partial information are lost due to thetruncation of small values. As is shown in Fig. 2(d), thefinite signal-noise ratio sets the truncation threshold. Asthe value of C ,n for peak 1 is below the line representingthe threshold, the corresponding C ,n is missing. Besidesthe errors in C ,n which are encoded in heights of spectral peaks, the λ n that are read from positions of peaks arealso susceptible to errors during the measurement. As isshown by peak 2 in (d), the nonvanishing spectral widthcaused by either the finite instrumental resolution or thefluctuation during the measurement may contribute tothe uncertainty ∆ λ n . The imprecise λ n also deterioratethe performance of J i,i +1 -reconstruction.We take the example of the simplest parametric setup,a chain of 100 sites with identical coupling strengths J i,i +1 = 1 and energies ε i = 0 , to show how the inputvalues influence the results. Under the ideal conditionwithout any deviations of λ n and C ,n , the simulated re-sults show the J i,i +1 can be correctly recovered for a longchain (tested up to thousands sites). In a realistic exper-iment, however, the spectral peaks of small-valued C ,n may not be well resolved, or even simply be truncated.For the n th mode, when C ,n is zero, from Eqs. (3) and(4), all C i,n vanish, leading to the zero component in Eq.(7) without the associated contribution; and also, Eq.(7) is not bothered by the numerical difficulty of divi-sion by zero. However, with the less number of observedpeaks than the actual number, the lost information doesinfluence the reconstructed results. Figure 3. For a homogeneous linear chain of N = 100 , panel(a) shows the spectral distribution C ,n versus λ n . The regionbelow − C max , as indicated by purple line in (a), is zoomedas shown in panel (b). Assuming the C ,n can be resolved upto − (purple), − (green) and − (red) of C max , allthe input data ( λ n , C ,n ) with C ,n below the threshold aretruncated. The corresponding estimated coupling strengths J (cid:48) i,i +1 are shown in (c), (d) and (e), respectively, comparingwith the original coupling strengths J i,i +1 . The influence from the truncation is shown in Fig.3. For a homogeneous linear chain with all ε i = ε and J i,i +1 = J , the eigenvalues are given by λ n = ε + 2 J cos[ πn/ ( N + 1)] for n = 1 , . . . , N . The el-ement of associated eigenvector for the i th site read C i,n = sin[ nπi/ ( N + 1)] . For the probing state | (cid:105) , C ,n = sin[ nπ/ ( N +1)] , as is shown by the spectral distri-bution in Fig. 3(a). The C ,n of small values are aroundboth the far ends along the λ n -axis, where the data be-low a given threshold are truncated if a finite signal-noiseratio is specified. For different truncation thresholds asindicated in Fig. 3(b), the estimated results of J i,i +1 arecompared in Fig. 3(c)-(e).Assuming that the maximum value of C ,n is C max ,when the threshold is − C max with ten pairs of C ,n truncated, significant deviation appears from i = 50 .When the data below − C max are truncated with threepairs of C ,n missing, the J i,i +1 can be correctly recon-structed up to i = 80 . Further, when only one pair ofdata are truncated below the threshold − C max , thedeviation only appear at the rightmost sites of the chain.The results suggest the complete data of C ,n should beimportant to the success of the algorithm. More intrigu-ingly, though all C ,n contribute to the evaluation of each J i,i +1 , when some components are missing, the chain canstill be recovered to some extent instead of failing thereconstruction as a whole, which shows the robustness ofthe algorithm.Next, we consider the estimation when J i,i +1 vary with i , as we desire in practice. Without loss of generality,the J i,i +1 are randomly chosen within a given interval,i.e., the disorder induced by J i,i +1 . It is shown that theinterval is highly relevant to the performance of the es-timation. Figs. 4(a) and (b) present the dependence ofthe reconstruction on the interval. In (a), when the spanis small, J i,i +1 ∈ [0 . , . , the J i,i +1 can be correctlyestimated up to i = 28 . While when the span is large, J i,i +1 ∈ [0 . , . , the applicability deteriorates—the es-timated values J (cid:48) i,i +1 starts deviating from J i,i +1 aroundsite 16, as can be straightforwardly read from the errordefined by δJ i,i +1 = J (cid:48) i,i +1 − J i,i +1 as shown in Fig. 4(c). xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxx xxxxxxx xxx (c) x J @ [0.9 - 1.1] x J @ [0.8 - 1.2] x J @ [0.9 - 1.1] x J @ [0.8 - 1.2] -1-2 0 1 201234502468 Site
Figure 4. Estimation of J i,i +1 in a chain of N = 50 when C ,n is resolved up to − . The J i,i +1 are randomly dis-tributed in range (a) [0 . , . and (b) [0 . , . . The errors δJ i,i +1 = J (cid:48) i,i +1 − J i,i +1 for the two different intervals arecompared in panel (c). The different parametric ranges leadto distinguished spectral distributions, as shown in (d) and(e). As discussed above, the reconstruction is rather robustfor a long chain with identical J i,i +1 . When J i,i +1 arerandomly distributed, however, the effective distance ofreconstruction is significantly shortened, showing an an-ticorrelation between the distance and the distributioninterval of J i,i +1 . Examining the spectral distribution,the anticorrelation can also be explained by the trunca-tion of C ,n as discussed for the homogeneous chain. Thedistribution interval of J i,i +1 influences the spectral pat-tern and the probability to find C ,n around zero. Withthe distribution of random J i,i +1 broadened, the spec-tral distribution Figs. 4(d) and (e) is no more as regularas that in the homogeneous chain. The C ,n scatter toa larger range with the increasing distribution intervalof J i,i +1 , and more C ,n approach zero. As discussed forthe homogeneous chain, these near-zero C ,n are likely tobe truncated, and the resultant lost spectral componentsworsen the applicability of the algorithm. In addition, itis found that the decreasing J i,i +1 also lowers the value of C ,n and impedes the parameter estimation, as can be in-tuitively understood since any broken bridge hinders theprobe of further sites. The above discussion also applieswhen disorders of ε i are involved as the spectral distri-bution also presents the similar pattern and suggests theinvolvement of the localization.Besides the influence from errors of C ,n , the measure-ment of λ n also affects the J i,i +1 reconstruction. Assum-ing the actual eigenvalue is λ n , the restricted resolutionor disturbance during the experiment may deviate themeasured value from λ n , λ (cid:48) n = λ n + ∆ λ n . The fluctua-tion ∆ λ n in each measurement results in the difference ofestimated J (cid:48) i,i +1 , as illustrated in Fig. 2(d). The eventual J (cid:48) i,i +1 should be the average of reconstructed results aftermultiple measurements. We sample 2000 random ∆ λ n ofnormal distribution, ∆ λ n ∼ N (0 , σ ) , to estimated therandomly distributed J i,i +1 ∈ [0 . , . in a chain of N = 100 as shown in Fig. 5(a). Here, the influence bythe truncation of small-valued C ,n is neglected.From the errors δJ i,i +1 as shown in Fig. 5(c), (d) and(e), the J i,i +1 can be roughly estimated up to i = 20 , and , respectively, for σ = 10 − , − and − .While without ∆ λ n fluctuation [Fig. 5(a)], all J i,i +1 arecorrectly reconstructed. Therefore, the precise measure-ment λ n is shown to be critical to the precise reconstruc-tion. For a longer chain, we did not find significant de-viations. However, the denser spectral distribution withmore states involved will hinder the precise retrieval of λ n , if the instrumental resolution of λ n -acquisition is fi-nite. IV. CONCLUSION
The reconstruction of parameters within a partially ac-cessible system is an important problem when indirectprobing is the only option to acquire the desired informa-tion. In this work, the algorithm to estimate parametersin a linear chain is investigated. Starting with the generic maxmin (a) without error xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
Figure 5. The estimation of coupling strengths when ∆ λ n isconsidered for each measurement. In a chain of 100 sites with J i,i +1 ∈ [0 . , . , the reconstructed J (cid:48) i,i +1 are compared with J i,i +1 in (a) when ∆ λ n = 0 . In (b), the reconstructed J (cid:48) i,i +1 are shown when σ = 0 . . The errors δJ i,i +1 are presentedin (c), (d) and (e) for σ = 10 − , − and − , respectively. Schrödinger equation, it is confirmed that the couplingstrengths can be efficiently deduced from the recursive re-lation using the spectral information of only the end site.We focus on the applicability of the algorithm, whichis shown to be highly relevant to the spectral distribu-tion on the accessible state. Given the errors inducedby the finite signal-noise ratio, the increasing number oftruncated spectral components are shown to graduallydeteriorate the reconstruction performance. It is foundthat reducing the loss of spectral components is criticalto the success of the method. Even so, the partially suc-cessful estimation in the presence of truncations showsthe robustness of the algorithm and the estimation canbe conducted in a controllable way. Since the spectraldistribution is system dependent, the applicability of themethod also varies with systems. Accordingly, it is ad-visable to understand the nature of the system beforeapplying the method.
ACKNOWLEDGMENTS
This work is supported by Shanghai Sailing Program(16YF1412600); the National Natural Science Founda-tion of China (Grants No. 11420101003, No. 11604347,No. 11827806, No. 11874368 and No. 91636105). [1] T. D. Ladd, F. Jelezko, R. Laflamme, Y. Nakamura,C. Monroe, and J. L. O ’Brien, Nature , 45 (2010).[2] L. C. Bassett and D. D. Awschalom, Nature , 505(2012).[3] S. Bose, Physical Review Letters , 207901 (2003).[4] V. Giovannetti, S. Lloyd, and L. Maccone, Physical Re-view Letters , 010401 (2006).[5] V. Giovannetti, S. Lloyd, and L. Maccone, Nature Pho-tonics , 222 (2011).[6] J. H. Cole, New Journal of Physics , 101001 (2015).[7] S.-T. Wang, D.-L. Deng, and L.-M. Duan, New Journalof Physics , 093017 (2015).[8] J. Kiukas, K. Yuasa, and D. Burgarth, Physical ReviewA , 052132 (2017). [9] J. Liu and H. Yuan, Physical Review A , 012117(2017).[10] J. Zhang and M. Sarovar, Physical Review Letters ,080401 (2014).[11] J. Zhang and M. Sarovar, Physical Review A , 052121(2015).[12] S.-Y. Hou, H. Li, and G.-L. Long, Science Bulletin ,863 (2017).[13] D. Burgarth and A. Ajoy, Physical Review Letters ,030402 (2017).[14] D. Burgarth, K. Maruyama, and F. Nori, Physical Re-view A , 020305 (2009).[15] D. Burgarth and K. Maruyama, New Journal of Physics11