The accurate control of the Hamiltonian of the designed systems is always a prerequisite to carry out desired tasks, e.g., quantum computation,^[1,2] quantum communication,^[3] and quantum metrology.^[4–7] The systems are usually microscopic structures which are delicately engineered to realize specific functions. To verify and benchmark these fabricated structures, the characterization of the system via Hamiltonian identification is desired.^[8] Usually, the dynamics within the system are extremely complicated, and the probing access is often restricted. A delicately devised identification scheme is supposed to allow the sensible estimation of the unknown parameters, reconstructing the Hamiltonian indirectly based 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, et al.

Under the challenge of complicated dynamics and restricted addressing resources, various identification algorithms have been developed aiming at experimental realizations. In a many-body system, the dynamical decoupling technique, which simplifies the problem by decoupling each pair of qubits from the rest, allows for the Hamiltonian identification with arbitrary long-range couplings between qubits.^[9] Besides, the dynamics can be altered by tuning the control pulse that is applied to the probe spin, improving the precision of the estimation scheme.^[10,11] In a network of limited access, the identification scheme is intended to bridge the Hamiltonian parameters and the observables from accessible subsystems that are relatively easy to measure. The system realization theory is proposed for the temporal record of the observables of a local subsystem,^[12,13] which has been experimentally demonstrated on a liquid nuclear magnetic resonance quantum information processor.^[14] The Zeeman marker protocol shows that local field-induced spectral shifts can be used to estimate parameters in spin chains or networks.^[15] Also, when the graph infection rule is satisfied, the similar parameter estimation is also available by utilizing the spectral information retrieved from a partially accessible spin.^[16,17] Recently, a framework for inferring local Hamiltonians is proposed that recovers a short-ranged Hamiltonian on a large system by measuring the observables in a Gibbs state or a single eigenstate.^[18]

In this work, we revisit the estimation algorithm proposed in Ref. [16] and focus on its applicability when the acquired initial data deviate from the actual values. The procedure, probing the global properties from a local site, is similar to the estimation of spring constants in classical-harmonic oscillator chains. By accessing only the end of the linear chain, the algorithm allows deducing all coupling strengths from the data of the associated spectral information. Nevertheless, the errors of the input data are inevitable and may significantly influence the reconstruction results. The simulations show that the spectral distribution is an important indicator of the applicability of the algorithm. Examining the spectral distribution, the increasing number of spectral components that approach zero are likely to fail the reconstruction. The applicability depends on both the properties of the individual system and conditions of measurement. Though there exist various Hamiltonian identification techniques (e.g., the ERA), the chain structure investigated here being simple allows for the direct presentation of the reconstruction procedure in our work to estimate the Hamiltonian parameter and elucidate possible sources of errors in the restricted system. Moreover, the simplicity of the original model and the explicit recursive formula allow us to track intuitively how the initial measurement errors propagate along the chain and to quantitatively assess the robustness of the reconstruction algorithm. The work is organized as follows. In Section 2, the recursive relations among spectral coefficients and coupling strengths from adjacent sites are derived. In Section 3, the robustness and the availability of the algorithm are discussed under different conditions when the input errors are introduced.

We consider the state transfer within a system governed by the generic Schrödinger equation . With the wave function represented in basis set {|i⟩}, |Ψ(t)⟩ = ∑_i c_i(t)|i⟩, the Hamiltonian is with ε_i the on-site energy and J_{i, j} the coupling strength. Thus, the Schrödinger equation in the matrix form reads

Fig. 1.

Figure Option ViewDownloadNew Window

Fig. 1. Schematics of parameter estimation for a chain of N sites. Assuming that the access of the chain is restricted, only site 1, the left-most site of the chain, can be measured by the detector and all the rest (sites 2, 3,…, N) are concealed by a blackbox. Using the scheme of parameter estimation, however, all coupling strengths, Ji, i + 1, can be estimated if the spectral information of site 1 is available.

The feasibility of the scheme can be shown by the repetitive use of Eq. (2) and the normalization condition 〈i|i⟩ = 1. Given the eigenmode n, the C_{1, n} of the leftmost site 1 allows deriving C_{2, n} of the next one,

On the other hand, the normalization condition 〈i|i⟩ = 1 should be satisfied. With C_{i, n} = 〈i|λ_n ⟩ 〈λ_n|ψ (0)⟩, we have The unknown denominator 〈λ_n|ψ(0) ⟩ depends on the concrete form of the initial state |ψ(0)⟩. If the system is initially prepared by populating state |1⟩ only, as considered in our case, |ψ(0)⟩ = |1⟩, the denominator reads |〈λ_n|ψ (0) ⟩ |² = |〈 λ_n |1 ⟩ |² = C_{1, n} and the normalization condition becomes

In a realistic experimental setup, the above scheme works for any system that can be reduced to the form equivalent to Eq. (1). In Ref. [17], the method is applied to the N-spin chain described by Heisenberg Hamiltonian

The initial state of the entire spin chain system for the probing in the reconstruction algorithm can be chosen as |Ψ(0)⟩ = α|0₁ 00 … 0 ⟩ + c |1₁ 00 … 0 ⟩ = α | 0 ⟩ + c |1⟩, which means that only the leftmost spin has been excited to the |1⟩ state. For our purpose of unknown parameters estimation, it suffices to restrict to the single excitation subspace . As the ground state |0⟩ is the eigenstate of , α dose not change during the evolution. Thus, at time t, the initial state |Ψ (0) ⟩ evolves to with complex coefficients α and c_i(t), where . The initial conditions are then given by c₁(0) = c and c_i(0) = 0 (for all i ≠ 1). The spin chain system can be initialized in α |0⟩ + c |1⟩ by acting on the leftmost spin 1 only.^[16] Then, by performing quantum-state tomography on spin 1 after time t, we can measure the element of the reduced density matrix of spin 1, . And the eigenvalues λ_n and the spectral coefficients {C_{1, n} = |〈1|λ_n⟩ |²} can be obtained through Fourier transform, which are the input variables of the reconstruction algorithm.

As a simple example, the estimation of J_{i, i + 1} in a chain with 6 sites is demonstrated in Fig. 2. With only site 1 accessible, the time evolution of state |1⟩ [Fig. 2(a)] is supposed to be detectable. In the associated spectral distribution [Fig. 2(b)], the position and the height of each spectral peak provide λ_n and C_{1, n}, respectively, as required input values for the estimation scheme. Performing the evaluation using Eqs. (3), (4), (6), and (7) recursively, we successfully reconstruct all the five unknown coupling strengths J_{i, i + 1}, as shown in Fig. 2(c).

Fig. 2.

Figure Option ViewDownloadNew Window

Fig. 2. Reconstruction of Ji, i + 1 for a chain of N = 6 and εi = 0. Panel (a) shows the temporal evolution of population on site 1, and panel (b) shows its spectral distribution that provides λn and C1, n as required by the reconstruction algorithm. In panel (c), applying the parameter estimation, the five unknown coupling strengths are estimated. The estimated values show good agreement with the actual values Ji, i + 1. Panel (d) presents the possible errors with the input values λn and C1, n that may hamper the reconstruction procedure. For peak 1, the C1, n below the threshold during the measurement is simply truncated. For peak 2, the broadening of the spectral peak results in the deviation of the measured eigenvalue from the actual λn.

During the actual measurement, the finite instrumental resolution, the limited signal-noise ratio, and other perturbations may hinder the precise acquisition of initial input λ_n and C_{1, n}. Therefore, the robustness of the algorithm against the deviations of these input values is critical to the success of the parameter estimation. The influence from imprecision of measured C_{1, n} has been mentioned in Ref. [16], showing rather robust performance against small deviations. When the finite signal-noise ratio of the C_{1, n} detection is considered, the even worse situation occurs when partial information is lost due to the truncation of small values. As shown in Fig. 2(d), the finite signal-noise ratio sets the truncation threshold. When the value of C_{1, n} for peak 1 is below the line representing the threshold, the corresponding C_{1, n} is missing. Besides the errors in C_{1, n} which are encoded in the heights of the spectral peaks, the λ_n that are read from the positions of the peaks are also susceptible to errors during the measurement. As shown by peak 2 in Fig. 2(d), the nonvanishing spectral width caused by either the finite instrumental resolution or the fluctuation during the measurement may contribute to the uncertainty Δλ_n. The imprecise λ_n also deteriorates the 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 input values influence the results. Under the ideal condition without any deviations of λ_n and C_{1, n}, the simulated results show that J_{i, i + 1} can be correctly recovered for a long chain (tested up to thousands sites). In a realistic experiment, however, the spectral peaks of small-valued C_{1, n} may not be well resolved, or even simply be truncated. With the less number of observed peaks than the actual one, the lost information does influence the reconstructed results.

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 element of the associated eigenvector for the i-th site reads C_{i, n} = sin [nπi/(N + 1)]. For the accessible site 1, C_{1, n} = sin [nπ/(N + 1)], as shown by the spectral distribution in Fig. 3(a). The C_{1, n} of small values are around both the far ends along the λ_n-axis, where the data below a given threshold are truncated if a finite signal-noise ratio is specified. For different truncation thresholds as indicated in Fig. 3(b), the estimated results of J_{i, i + 1} are compared in Figs. 3(c)–3(e).

Fig. 3.

Figure Option ViewDownloadNew Window

Fig. 3. For a homogeneous linear chain of N = 100, panel (a) shows the spectral distribution C1, n versus λn. The region below 10−1 Cmax, as indicated by the purple line in (a), is zoomed as shown in panel (b). Assuming the C1, n can be resolved up to 10− 1 (purple), 10− 2 (green), and 10− 3 (red) of Cmax, all the input data (λn, C1, n) with C1, n below the threshold are truncated. The corresponding estimated coupling strengths are shown in (c), (d), and (e), respectively, comparing with the original coupling strengths Ji, i + 1.

Assuming that the maximum value of C_{1, n} is C_max, when the threshold is 10^{− 1} C_max with ten pairs of C_{1, n} truncated, significant deviation appears from i = 50. When the data below 10^{− 2} C_max are truncated with three pairs of C_{1, n} missing, the J_{i, i + 1} can be correctly reconstructed up to i = 80. Further, when only one pair of data are truncated below the threshold 10^{− 3} C_max, the deviations only appear at the rightmost sites of the chain. The results suggest that the complete data of C_1,n should be important to the success of the algorithm. More intriguingly, though all C_{1, n} contribute to the evaluation of each J_{i, i + 1}, when some components are missing, the chain can still be recovered to some extent instead of failing in the reconstruction as a whole, which shows the robustness of the algorithm.

The error δJ_{i, i + 1} induced by the truncation appears to accumulate along the chain. To clarify this, we assume that a pair of small-valued input data C_{1, m1} = C_{1, m2} = δ for eigenvalues λ_m1 = −λ_m2 are truncated during the measurement and perform perturbation analysis of error propagation with respect to δ. From Eq. (4), it shows that all C_{i, m} are absent during the reconstruction as long as C_{1, m} = δ of mode m is ignored from the input data. Taking variables up to the first order of δ, the linear approximations assume , C_{i, m} = ξ_{i, m} δ, and , where and are the reconstructed variables tainted by the truncation, and C_{i, n} and J_{i, i + 1} are the true values. Perturbation parameter θ_i determines the deviation of the reconstructed coupling strength from the true value J_{i, i + 1}. Next, we need to find σ_i,n, ξ_{i, m}, and θ_i from the recursive relations (4) and (7). Substituting the approximations into Eq. (7) and separating the contribution of the truncated components from the rest, the recursive relation of J_{i, i + 1} takes the form

Similarly, substituting the linear approximations into recursive relation (4), we can find the recursive formulas for ς_{i, n} and ξ_{i, m},

In Fig. 4, we present the comparisons between error estimation under the linear approximation of Eqs. (9), (10), and (11) with the results from the reconstruction algorithm under different circumstances. Figures 4(a) and 4(b) show the results for a homogeneous chain of identical coupling strengths J_{i, i + 1} = 1 when N = 50 and 100, respectively. Figures 4(c) and 4(d) present the results for a non-homogeneous chain of randomly distributed J ∈ [0.9, 1.1] when N = 50 and 100, respectively. In all four cases, when the error propagates along the chain, the deviations appear earlier under the linear approximation than the original algorithm. It suggests that some nonlinearity associated to the algorithm could protect the reconstruction from deterioration. In this case, the linear approximation based on the perturbation analysis may provide the lower bound of the error introduced by the truncation of the input data during the measurement. However, the role of nonlinearity still requires further investigation.

Fig. 4.

Figure Option ViewDownloadNew Window

Fig. 4. Comparison among true values of Ji, i + 1 (circle symbol), reconstructed (cross symbol), and predicted (triangle symbol) from perturbation analysis under the linear approximation when (a) N = 50 and Ji, i + 1 = 1, (b) N = 100 and Ji, i + 1 = 1, (c) N = 50 and random Ji, i + 1 ∈ [0.9, 1.1], and (d) N = 100 and random Ji, i + 1 ∈ [0.9, 1.1]. The reconstruction starts with a pair of C1, n truncated.

Next, we consider the estimation when J_{i, i + 1} vary with i, as we desired 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 the interval is highly relevant to the performance of the estimation. Figures 5(a) and 5(b) present the dependence of the reconstruction on the interval. In Fig. 5(a), when the span is small, J_{i, i + 1} ∈ [0.9, 1.1], the J_{i, i + 1} can be correctly estimated up to i = 28. While when the span is large, J_{i, i + 1} ∈ [0.8, 1.2], the applicability deteriorates, the estimated values start deviating from J_{i, i + 1} around site 16, as can be straightforwardly read from the error defined by as shown in Fig. 5(c).

Fig. 5.

Figure Option ViewDownloadNew Window

Fig. 5. Estimation of Ji, i + 1 in a chain of N = 50 when C1, n is resolved up to 10− 3. The Ji, i + 1 are randomly distributed in range (a) [0.9, 1.1] and (b) [0.8, 1.2]. The errors for the two different intervals are compared in panel (c). Panels (d) and (e) show distributions of input data C1, n = | 〈λn |1⟩ |2 on site 1 for Ji, i + 1 ∈ [0.9, 1.1] and Ji, i + 1 ∈ [0.8, 1.2], respectively, to reconstruct Ji, i + 1 as shown in (a) and (b).

As discussed above, the reconstruction is rather robust for a long chain with identical J_{i, i + 1}. When J_{i, i + 1} are randomly distributed, however, the effective distance of reconstruction is significantly shortened, showing an anticorrelation between the distance and the distribution interval of J_{i, i + 1}. Examining the spectral distribution, the anticorrelation can also be explained by the truncation of C_{1, n} as discussed for the homogeneous chain. The distribution interval of J_{i, i + 1} influences the spectral pattern and the probability to find C_{1, n} around zero. With the distribution of random J_{i, i + 1} broadened, the spectral distribution in Figs. 5(d) and 5(e) is no more as regular as that in the homogeneous chain. The C_{1, n} scatter to a larger range with the increasing distribution interval of J_{i, i + 1}, and more C_{1, n} approach zero. As discussed for the homogeneous chain, these near-zero C_{1, n} are likely to be truncated, and the resultant lost spectral components worsen the applicability of the algorithm. In addition, it is found that the decreasing J_{i, i + 1} also lowers the value of C_{1, n} and impedes the parameter estimation, as can be intuitively understood since any broken bridge hinders the probe of further sites. The above discussion also applies when disorders of ε_i are involved as the spectral distribution also presents the similar pattern and suggests the involvement of the localization.

Besides the influence from the errors of C_{1, n}, the measurement of λ_n also affects the J_{i, i + 1} reconstruction. Assuming the actual eigenvalue is λ_n, the restricted resolution or disturbance during the experiment may deviate the measured value from λ_n, . The fluctuation Δλ_n in each measurement results in the difference of the estimated , as illustrated in Fig. 2(d). The eventual should be the average of the reconstructed results after multiple measurements. We examine the parameter reconstruction when fluctuations Δλ_n around the true value of λ_n are involved. An example for a chain of N = 100 with coupling strengths J_{i, i + 1} ∈ [0.9, 1, 1] is shown in Fig. 6(a), where all coupling strengths can be precisely reconstructed if no random error is introduced. To account for the influence from nonvanishing random errors, each of Δλ_n is chosen randomly for a single measurement, and the samples are generated from a normal distribution, . Then we perform multiple simulated measurements and evaluate the averaged . Figure 6(b) shows the averaged from 2000 simulated measurements when σ = 0.001. Here, the influence of the truncation of small-valued C_{1, n} is neglected.

Fig. 6.

	Figure Option ViewDownloadNew Window
	Fig. 6. The estimation of coupling strengths when Δλn is considered for each measurement. In a chain of 100 sites with Ji, i + 1 ∈ [0.9, 1.1], the reconstructed are compared with Ji, i + 1 in (a) when Δλn = 0. In (b), the reconstructed are shown when σ = 0.001. The errors δ Ji, i + 1 are presented in (c), (d), and (e) for σ = 10− 1, 10− 2, and 10− 3, respectively.

From errors δ J_{i, i + 1} as shown in Figs. 6(c)–6(e), the J_{i, i + 1} can be roughly estimated up to i = 20, 40, and 60, respectively, for σ = 10^{− 1}, 10^{− 2}, and 10^{− 3}. While without Δλ_n fluctuation [Fig. 6(a)], all J_{i, i + 1} are correctly reconstructed. For the chain of N = 100, the average and minimum gaps of λ_n are 0.04 and 0.002, respectively. The average gap is larger than the fluctuations considered in Figs. 6(e) (σ = 0.001) and 6(d) (σ = 0.01) so that the spectral peaks are supposed to be resolved. For comparison, figure 6(c) takes σ = 0.1 which is large, showing the expected failure of the reconstruction scheme when the spectral peaks cannot be well resolved. Therefore, the precise measurement λ_n is shown to be critical to the precise reconstruction. For a longer chain, we do not find significant deviations. However, the denser spectral distribution with more states involved will hinder the precise retrieval of λ_n, if the instrumental resolution of λ_n-acquisition is finite.

The reconstruction of parameters within a partially accessible system is an important problem when indirect probing is the only option to acquire the desired information. In this work, the algorithm to estimate parameters in a linear chain with restricted access is extensively investigated, providing the necessary information to assess the algorithm robustness. Starting with the generic Schrödinger equation, it is confirmed that the coupling strengths can be efficiently deduced from the recursive relation using the spectral information of only the end site. We focus on the applicability of the algorithm, which is shown to be highly relevant to the spectral distribution on the accessible site. Given the errors induced by the finite signal-noise ratio, the increasing number of truncated spectral components is shown to gradually deteriorate the reconstruction performance. It is found that reducing the loss of spectral components is critical to the success of the method. Even so, the partially successful estimation in the presence of truncations shows the robustness of the algorithm and the estimation can be conducted in a controllable way. Since the spectral distribution is system dependent, the applicability of the method also varies with the systems. Accordingly, it is advisable to understand the nature of the system before applying the method.