arXiv:1202.4208v2 math-ph 10 Apr 2012 Symmetry and localization of quantum walk induced by extra link in cycles Xin-Ping Xu1, Yusuke Ide2, and Norio Konno3 1 School of Physical Science and Technology, Soochow University, Suzhou 215006, China 2 Department of Information Systems Creation, Faculty of Engineering, Kanagawa University, Yokohama, Kanagawa, 221-8686, Japan 3 Department of Applied Mathematics, Faculty of Engineering, Yokohama National University, Hodogaya, Yokohama 240-8501, Japan 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 diff erent 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 fi nd that the additional link in cycle indeed cause a very diff erent dynamical behavior compared to the dynamical behavior on the cycle. We analytically treat this problem and calculate the Laplacian spectrum for the fi rst 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 signifi cant 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 signifi cant lower bound for it even in the limit of infi nite system. Finally, we study the problem of trapping and show the survival probability also displays signifi cant localization for some special values of network parameters, and we determine the conditions for the emergence of such localization. All our fi ndings suggest that the diff erent 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. PACS numbers: 03.67.-a,05.60.Gg,89.75.Kd,71.35.-y I.INTRODUCTION The dynamic processes taking place in networks have attracted much attention in recent years 13. It is gen- erally believed that network structure fundamentally in- fl uences the dynamical processes on networks. Investiga- tion on such aspect could be done using the spectral anal- ysis and it has been shown that the dynamical behavior is related to the spectral properties of the networks 3, 4. Examples include synchronization of coupled dynamical systems 5, epidemic spreading 6, percolation 7, com- munity detection 8, and others 3. 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 dynam- ical processes taking place in networks. Quantum walks, as coherent dynamical process in net- work, have become a popular topic in the past few years 912. The continuous interest in quantum walk can be attributed to its broad applications to many dis- tinct fi elds, such as polymer physics, solid state physics, biological physics, and quantum computation 13, 14. In the literature 9, 10, there are two types of quan- tum 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 15, 16, cycle 17, hypercube 18, trees 19, 20, dendrimers 21, ultrametric spaces 22, threshold net- work 23, and other regular networks with simple topol- ogy 11, 12. The quantum dynamics displays diff erent behavior on diff erent graphs, most of the conclusions hold solely in the particular geometry and how the struc- ture infl uences the dynamics is still unknown. Because quantum walks have potential applications in teleporta- tion and cryptography in the fi eld of quantum compu- tation 14, it is clearly benefi cial to investigate how the structure infl uences 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 quan- tum walks on cycle is well known, the problem is analyti- cally solvable and directly related to quantum carpets in solid state phy