Symmetry and localization of quantum walk induced by extra link in cycles
It is generally believed that the network structure has a profound impact on diverse dynamical processes taking place on networks. Tiny change in the structure may cause completely different dynamics. In this paper, we study the impact of single extra link on the coherent dynamics modeled by continuous-time quantum walks. For this purpose, we consider the continuous-time quantum walk on the cycle with an additional link. We find that the additional link in cycle indeed cause a very different dynamical behavior compared to the dynamical behavior on the cycle. We analytically treat this problem and calculate the Laplacian spectrum for the first time, and approximate the eigenvalues and eigenstates using the Chebyshev polynomial technique and perturbation theory. It is found that the probability evolution exhibits a similar behavior like the cycle if the exciton starts far away from the two ends of the added link. We explain this phenomenon by the eigenstate of the largest eigenvalue. We prove symmetry of the long-time averaged probabilities using the exact determinant equation for the eigenvalues expressed by Chebyshev polynomials. In addition, there is a significant localization when the exciton starts at one of the two ends of the extra link, we show that the localized probability is determined by the largest eigenvalue and there is a significant lower bound for it even in the limit of infinite system. Finally, we study the problem of trapping and show the survival probability also displays significant localization for some special values of network parameters, and we determine the conditions for the emergence of such localization. All our findings suggest that the different dynamics caused by the extra link in cycle is mainly determined by the largest eigenvalue and its corresponding eigenstate. We hope the Laplacian spectral analysis in this work provides a deeper understanding for the dynamics of quantum walks on networks.
The dynamic processes taking place in networks have attracted much attention in recent years rn1 (); rn2 (); rn3 (). It is generally believed that network structure fundamentally influences the dynamical processes on networks. Investigation on such aspect could be done using the spectral analysis and it has been shown that the dynamical behavior is related to the spectral properties of the networks rn3 (); rn4 (). Examples include synchronization of coupled dynamical systems rn5 (), epidemic spreading rn6 (), percolation rn7 (), community detection rn8 (), and others rn3 (). In several of these examples, the largest eigenvalue plays an important role in relevant dynamics. All these examples suggest that spectral property is crucial to understanding the dynamical processes taking place in networks.
Quantum walks, as coherent dynamical process in network, have become a popular topic in the past few years rn9 (); rn10 (); rn11 (); add0 (). The continuous interest in quantum walk can be attributed to its broad applications to many distinct fields, such as polymer physics, solid state physics, biological physics, and quantum computation rn12 (); rn13 (). In the literature rn9 (); rn10 (), there are two types of quantum walks: continuous-time and discrete-time quantum walks. It is shown that both type of quantum walks is closely related to the spectral properties of the Laplacian matrix of the network. Most of previous studies have studied quantum walks on some simple graphs, such as the line rn14 (); add1 (), cycle rn15 (), hypercube rn16 (), trees rn17 (); add2 (), dendrimers rn18 (), ultrametric spaces add3 (), threshold network add4 (), and other regular networks with simple topology rn11 (); add0 (). The quantum dynamics displays different behavior on different graphs, most of the conclusions hold solely in the particular geometry and how the structure influences the dynamics is still unknown. Because quantum walks have potential applications in teleportation and cryptography in the field of quantum computation rn13 (), it is clearly beneficial to investigate how the structure influences the dynamics of quantum walks.
In this paper, we focus on continuous-time quantum walks (CTQWs) and study the impact of single extra link on its coherent dynamics. For this purpose, we consider the continuous-time quantum walk on the cycle with an additional link. The dynamics of continuous-time quantum walks on cycle is well known, the problem is analytically solvable and directly related to quantum carpets in solid state physics rn15 (). The topology of cycle is highly symmetric and the quantum dynamics reflects such topological symmetry. If one extra link is added into the cycle, the topological symmetry is broken and the structure has a small difference compared to the cycle. This enables us to treat the problem analytically and investigate how an additional link added to the network affects the dynamics. Similar study of the impact of single links on dynamical processes can be found in Refs. rn19 (); rn20 (), where the authors study the impact of single link addition in percolation rn19 () and impact of single link failure in quantum walks rn20 () respectively. In our case, in view of experimental implementation of CTQWs on cycles has been realized, we hope our study of the impact of single link (extra coupling) provides useful insight in quantum computation.
The rest of the paper is organized as follows. Section II introduces the model of continuous-time quantum walks and defines the network structure. Section III investigates the eigenvalues and eigenstates of the Laplacian matrix (Hamiltonian). We obtain the determinant equation for the eigenvalues expressed by Chebyshev polynomial, and calculate the largest eigenvalue on certain limit condition. The other eigenvalues and eigenstates are obtained by the perturbation theory. As we will show, like other dynamical processes, the largest eigenvalue and its eigenstate play an important role in the relevant dynamics. In Section IV, we study the characteristics of probability evolution and distribution. The impact of the extra link depends on the initial exciton position, which can be well understood by the contribution of the eigenstate of the largest eigenvalue. The long time averaged probabilities shows significant symmetry and localization, and we give an explanation to these features using the determinant equation and largest eigenvalue. In Section V, we study the trapping process and the survival probabilities show significant localization for some special values of network parameters. We discuss this finding using the perturbation theory again. Conclusions and discussions are given in the last part, Sec. VI.
Ii Continuous-time quatum walks and the addition of link in cycle
ii.1 Continuous-time quatum walks
The coherent exciton transport on a connected network is modeled by the continuous-time quantum walks (CTQWs), which is obtained by replacing the classical transfer matrix by the Laplacian matrix , i.e., rn11 (); rn15 (). The transfer matrix relates to the Laplacian matrix by , where for simplicity we assume the transmission rates of all bonds to be equal and set in the following rn11 (); rn15 (). The Laplacian matrix has nondiagonal elements equal to if nodes and are connected and otherwise. The diagonal elements equal to degree of node , i.e., . The states endowed with the node of the network form a complete, ortho-normalised basis set, which span the whole accessible Hilbert space. The time evolution of a state starting at time is given by , where is the quantum mechanical time evolution operator. The transition amplitude from state at time to state at time reads and obeys Schrödinger¡¯s equation rn11 (); rn15 (). The classical and quantum transition probabilities to go from the state at time to the state at time are given by and rn11 (); rn15 (), respectively. Using and to represent the th eigenvalue and orthonormalized eigenstate of , the quantum amplitudes between two nodes can be written as rn11 ()
For finite networks, do not decay ad infinitum but at some time fluctuates about a constant value. This value is determined by the long time average of
where takes value 1 if equals to and 0 otherwise. To calculate the exact analytical expressions for and , all the eigenvalues and eigenstates of the Laplacian matrix are required. If all the eigenvalues are distinct, i.e., all the eigenvalues are not degenerated, Eq. (3) can be simplified as,
The eigenvalues for the cycle graph is twofold degenerated. However, as we will show, the addition of single link causes the degeneracy to disappear. Therefore we can use Eq. (4) to calculate the probability distribution for our model. The transition prababilities (See Eqs. (2)-(4)) are closely related to and , it is crucial to calculate the Laplacian eigenspectrum and we will do this in Sec. III.
ii.2 Addition of link in cycle
Now we specify the topology of the cycle with an extra link. First we construct a cycle of size where each node connected to its two nearest-neighbor nodes, then we connect two node of certain distance with an additional link. The topology is completely determined by the distance and size of the cycle. For the sake of simplicity, we assign a consecutive number from to for each neighbored node in the cycle. The additional link connects node and node in the cycle. Thus the structure, denoted by , are characterized by the network size and connecting node (). Here, we use to denote the (shortest) distance between node and node in the cycle (without an extra link), the graph can also be characterized by . The structure of is illustrated in Fig. 1.
It is interesting to note that the network considered also has a symmetric structure, and the symmetry axis lies at the central position between node and . The structure can be mapped into a 1D chain of size (the notation denotes the integer part of ) if we make a horizontal projection of the graph (See Fig. 1). Such projection sheds some light on the implicit relationship between the two structures. For example, as we show in Appendix C, the eigenvalues of the Laplacian matrix of the projected 1D chain also belong to eigenvalues of our model.
Iii Laplacian Eigenvalues and Eigenstates
Since the transition probabilities are determined by the Laplacian eigenvalues and eigenstates (see Eqs. (2)-(4)), a detailed analysis to the eigenvalues and eigenstates will be helpful for the problem. In this section, we will study the Laplacian eigenvalues and eigenstates in detail.
Figure 2(a) shows the eigenvalues obtained by numerical diagonalizing the Laplacian matrix using the software Mathematica 7.0. The eigenvalues are ranked in ascending order for networks of size with , and . It is interesting to note that the largest eigenvalue is larger than and isolated from the other eigenvalues (the other eigenvalues are less than ), this means the gap between the largest eigenvalue and the second largest eigenvalue does not converge to zero when the size of the system goes to infinity, whereas the other nearest eigenvalue gaps tend to zero in the limit of infinite system. This suggests the eigenvalue spectra is continuous except the largest eigenvalue. Such characteristic feature of eigenvalue spectra is useful for understanding the coherent dynamical behavior. As we will show, the largest eigenvalue plays an important role in the dynamics and gives significant contribution to the localized probabilities. Hence in the following, we will try to obtain Laplacian eigenspectrum and determine the largest eigenvalue and its corresponding eigenstate.
iii.1 Determinant equation for the eigenvalues
We start our analysis on the eigen equation of the Hamilton(Laplacian matrix). The Laplacian matrix of () takes the following form:
According to the eigen equation , suppose the eigenstate can be expressed as
The eigen equation can be decomposed into the following linear equations,
Eq. (8) can be rewritten as . This is similar to the recursive definition of the Chebyshev polynomials of the second kind (See Appendix A). Noting the recursive relations and setting in the definition of Chebyshev polynomials, the variables in Eq. (8) can be expressed as a function of and ,
where and , and
In the Appendix B, we show the substraction of the products can be simplified to be a much simple form in Eq. (61), thus we have obtained determinant equation for the eigenvalues,
where and are Chebyshev polynomials of the first kind and the second kind, respectively. Solving the above equation, we can get all the Laplacian eigenvalues. This equation is useful to determine the largest eigenvalue and interpret the symmetric structure of the transition probabilities.
iii.2 The largest eigenvalue
The largest eigenvalue can be determined using Eq. (22) under certain limit conditions. As we have shown, the largest eigenvalue is larger than , this corresponds to the solution in Eq. (22). Note that the Chebyshev polynomials is divergent for in the limit of infinite order, Eq. (22) divided by leads to,
For large size of system and long range coupling, i.e., is large and the distance between node and node is not small, if we apply the asymptotic solution of the Chebyshev polynomials (See Eq. (45) in Appendix A), the first two terms of the above equation equal to , , which leads to (the other solutions do not satisfy less than when ) and . Thus the largest eigenvalue equals to a constant value under this limit condition.
To test the above prediction, we plot the largest eigenvalue as a function of and in Fig. 2(b). As we can see, the largest eigenvalue converges to a constant value as or increases. For moderate range coupling, e.g., , the largest eigenvalue is close to the analytical prediction . The constant value of the largest eigenvalue suggests that its corresponding eigenstate (eigenvector) approaches to certain stationary distribution. Here, the obtained largest eigenvalue is useful to consider its corresponding eigenstate, which we will show in the following.
iii.3 Eigenstate of the largest eigenvalue
The components of eigenstate corresponding to the largest eigenvalue are shown in Fig. 2(c). The straight line in the linear-log plot indicates that the components display an exponential decay, , . The distribution of the components of eigenstate is symmetric, ; . As we will prove in Sec. IV (subsection B) and Appendix C, this symmetric behavior is true for all the eigenstates.
To obtain an approximate formula for the eigenstate of the largest eigenvalue, we use Eq. (18) to present as a function of , and substitute it into Eq. (16) to write as a function of . Using the Chebyshev polynomial identity (47) in Appendix A, Eq. (16) can be written as . According to Eq. (18), and noting the simplified , in (55) and (56) in Appendix A, we obtain (See the detailed derivation in Eq. (62) in Appendix B). Substituting and into Eqs. (12) and (13) and noting for the largest eigenvalue (See Eq. (74) in Appendix C), Eqs. (12) and (13) can be written as,
For the largest eigenvalue, , the ratios and approach to the constant value for large system and long range coupling. Therefore, Eq. (24) can be recasted as,
The above formula only describes the right part of the symmetric distribution of the eigenstate (See the points on the dashed lines in Fig. 2(c)). The decay exponents in Eq. (25) only depend on the (shortest) distance between the node pairs and , i.e., . Eq. (25) can be summarized in a much simple form using the shortest distance ,
In the calculation, the eigenstate should be normalized, . For large system and long-range couple, the summation has a week dependence on and , thus the summation of finite geometric series approaches to the summation of the infinite geometric series, i.e.,
which leads to . For large systems and long range coupling, the distribution of the eigenstate of the largest eigenvalue approaches to a stationary exponential decay with the maximal component . In Fig. 2(c), we plot the analytical predictions in Eq. (26), which agrees well with the exact numerical result.
We would like to point out that the exponential distribution of the components of the eigenstate is an interesting characteristic for the largest eigenvalue. For the other eigenstates, the components do not display this feature. This could be interpreted by the striking different behavior of the Chebyshev polynomials for and . For , the Chebyshev polynomials show finite regular oscillations, but for , Chebyshev polynomials show exponential growth. The largest eigenvalue and its eigenstate play a significant role in the coherent dynamics, we will discuss this in the next section.
iii.4 Other eigenvalues and eigenstates
We have determined the largest eigenvalue and its corresponding eigenstate using the determinant equation (22). However, it is not intuitionistic to determine the other eigenvalues and eigenstates by Eq. (22). Here, we use the perturbation theory to get the approximate results for the other eigenvalues and eigenstates.
According to perturbation theory, the Hamilton (Laplacian matrix) of the cycle with an additional link can be divided into two parts: the unperturbed Hamilton and perturbed Hamilton (, ). The unperturbed Hamilton is the Laplacian matrix for the cycle, whose eigenvalues and eigenstates are well known: , , (, ). The perturbed Hamilton can be written as . The eigenvalues of the cycle are twofold degenerated except the minimal eigenvalue and maximal eigenvalue . The first order corrections for eigenvalues and are and respectively (maximal eigenvalue only exists for even ). The zero order approximation of the eigenstates corresponding to and are equal to the original unperturbed eigenstates of , i.e., , . For the twofold degenerated states, the degeneracy disappears when an extra link is added. To approximate these eigenvalues and eigenstates, we apply degenerate perturbation theory to calculate the first order corrections of the eigenvalues and zero order approximation of the eigenstates.
We diagonalize the the perturbed Hamilton using the unperturbed degenerated eigenstates and . The perturbed Hamilton can be written as,
where , . The first order corrections of the eigenvalues are given by the two eigenvalues in Eq. (28),
The corresponding eigenstates for and are,
The zero order approximation sof the eigenstates are given by
Substitute the Bloch states into the above equation, we obtain,
Thus we have obtained the first order corrections of the degenerated eigenvalues and the zero order approximation of the corresponding eigenstates(See Eqs. (29) and (32)). Using these eigenvalues and eigenstates, we can calculate the transition probabilities in Eqs. (2)-(4).
It is worth mentioning that the perturbation theory can not be used to calculate the largest eigenvalue, since the eigenvalue gaps do not converge to zero in this case. However, the analytical predicted eigenvalues by perturbation theory (See Eq. (29)) can also be obtained by Taylor series expansion of the Chebyshev polynomials. In Appendix D, we approximate the other eigenvalues using the determinant equation (22), which are consistent with the predictions in Eq. (29).
Iv Probability evolution and distribution
In this section, we consider the probability evolution and probability distribution. We will explain the observed phenomenon using the obtained eigenvalues and eigenstates.
iv.1 Probability Evolution
The time evolution of the probability is given by Eq. (2). Here, we focus on the return probability (set in Eq. (2)). The decay behavior is used to quantify the efficiency of the transport rn11 (). In our case the return probability depends on the starting position . We study the time evolution of the return probability on different starting position . The behavior of versus are plotted in Fig. 3. It is found that shows different time range for scaling behavior : For the starting positions far from the connected nodes and , the scaling range of time is much larger than that of the walk starting near the nodes and (Compare scaling in the four figures). It is well known that for the cycle, . This suggests that the additional link gives small influence on the dynamics when the walk starts at outlying positions, and gives larger influence for exciton near nodes and . If the walk starts at the center nodes (nodes at symmetry axis) ( is odd) or ( is odd) (See Fig. 3 (a)), the probabilities are exact the same as the probabilities in cycle of the same size, in this case the additional link has no impact on the dynamics.
The observed phenomenon can be well understood using the eigenstates obtained in the previous section. If the exciton starts at positions far away from and , the components of the eigenstate of the largest eigenvalue are very small due to the exponential decay of the components. Thus the contribution from the largest eigenvalue in Eq. (1) is very small, the main contribution to the amplitude comes from the other eigenvalues, which leads to the scaling behavior resembling to the case in cycles. If the exciton starts at the center nodes (nodes at symmetry axis) ( is odd) or ( is odd), the corresponding components of the largest eigenvalue’s eigenstate are (See Eqs. (75)-(76) in Appendix C). In this particular case, the largest eigenvalue gives no contribution to the amplitude, the probabilities are exact the same as in the cycles, thus the additional link has no impact on the dynamics. If the exciton starts at the positions near the connected nodes and , the components of the largest eigenvalue’s eigenstate become larger and give contribution to the amplitude, thus the scaling behavior disappears gradually (See Fig. 3(b) (c)). If the exciton starts at nodes or , the components of the eigenstate of the largest eigenstate ( or ) give large contribution to the amplitude, the time range of the scaling behavior is very short (See Fig. 3(d)). In this case, the return probabilities do not decay ad infinitum but fluctuate about a constant value, suggesting significant localization induced by the largest eigenvalue. We will study this feature in detail in the following.
iv.2 Probability Distribution
In this section, we consider the long time averaged probabilities. We find that these limiting probabilities display a symmetric structure, symmetric nodes and ( or , see the two nodes in the horizontal projection in Fig. 1) have the same limiting probabilities. This suggests that the probability of finding the exciton at a certain node is equal to the probability of finding the exciton at the symmetric position (or ), i.e., (or ). This symmetric structure can be understood as a result of the axis symmetry of graph. Here, we give a rigorous mathematical proof for this symmetric structure using Eq. (4). In Eq. (4), the limiting probabilities are only dependent on the eigenstates. Suppose the th eigenstate is expanded as , the probabilities in Eq. (4) can be written as . In Appendix C, we prove that the components of eigenstate are symmetric, ; . Here, no superscript for means the identity holds for all the eigenstates (). The symmetric structure of the eigenstates leads to the symmetric limiting probabilities. This is one of the main conclusions in our paper.
Now we try to use the eigenstates to find analytical approximate results for the limiting probabilities. The contribution in Eq. (4) comes from three parts: the eigenstate of the largest eigenvalue, non-degenerated eigenstates ( and ) and degenerated eigenstates. Therefore Eq. (4) transforms into
Fig. 4 shows the probability distribution for different starting positions. It is evident that the exact numerical results agree well with analytical predictions in Eq. (34). For exciton starting at and , there is a high probability to find the exciton at and , suggesting significant localization in the dynamics. For excitons starting at center node ( is odd) and ( is odd), the probability distribution is exact the same as the cycles. This could be explained by Eq. (34) as follows: For or , (See Eqs. (75)-(76) in Appendix C), the contribution from the largest eigenstate’s eigenstate is (See the first term in Eq. (34)), the localized probability mainly depends on the trigonometric function in Eq. (34). For return probabilities with ( is odd) or ( is odd), limits of the trigonometric function ratio in the big bracket equal to , thus Eq. (34) becomes as,
which are consistent with the results in Ref. rn21 (). For this particular case, , the probability distribution is exact the same as the cycles, this indicates the adding of link has no impact on the dynamics for center node excitations ( or at the symmetry axis, See Fig. 1). Here we provide a theoretical interpretation for this phenomena. However, if and are not odd numbers, this particular case does not exist.
As we have shown, the return probabilities has a high value at and . Such a strong localization may vanish as the size of the system becomes large. In order to investigate the dependence of localized probability (or ) on the size of the system, we plot (or ) as a function of for different values of in Fig. 5. We find that the return probability converges to a constant value when tends to infinity. This constant value can be calculated using Eq. (34). For infinite systems, (or ) equals to the first term of Eq. (34) (the other terms are in the limit of ). Therefore, for infinite systems, the limiting probabilities are determined by the components of the eigenstate of the largest eigenvalue. Noting that , we get
Therefore (or ) converges to the , indicating there is a significant lower bound for the return probabilities. This result has not been previously reported and suggests that quantum walks display nontrivial localization on nonsymmetric system. Since all the components of the largest eigenvalue’s eigenstate are known (See Eqs. (25) and (26)), the limiting probabilities have a lower bound given by . Using the expression of in Eq. (26),