Xu Xin-Ping
Institute of Particle Physics, HuaZhong Normal University, Wuhan 430079, China.
Phys Rev E Stat Nonlin Soft Matter Phys. 2008 Jun;77(6 Pt 1):061127. doi: 10.1103/PhysRevE.77.061127. Epub 2008 Jun 19.
In this paper, we consider continuous-time quantum walks (CTQWs) on a one-dimensional ring lattice of N nodes in which every node is connected to its 2m nearest neighbors ( m on either side). In the framework of the Bloch function ansatz, we calculate the space-time transition probabilities between two nodes of the lattice. We find that the transport of CTQWs between two different nodes is faster than that of the classical continuous-time random walks (CTRWs). The transport speed, which is defined by the ratio of the shortest path length and propagating time, increases with the connectivity parameter m for both CTQWs and CTRWs. For fixed parameter m , the transport of CTRWs gets slower with the increase of the shortest distance while the transport (speed) of CTQWs turns out to be a constant value. In the long-time limit, depending on the network size N and connectivity parameter m , the limiting probability distributions of CTQWs show various patterns. When the network size N is an even number, the probability of being at the original node differs from that of being at the opposite node, which also depends on the precise value of parameter m .
在本文中，我们考虑在具有N个节点的一维环形晶格上的连续时间量子行走（CTQW），其中每个节点与其2m个最近邻节点相连（两侧各m个）。在布洛赫函数假设的框架下，我们计算晶格中两个节点之间的时空转移概率。我们发现，两个不同节点之间的CTQW传输比经典连续时间随机行走（CTRW）更快。由最短路径长度与传播时间之比定义的传输速度，对于CTQW和CTRW来说，都随连接参数m的增加而增加。对于固定参数m，CTRW的传输随着最短距离的增加而变慢，而CTQW的传输（速度）则变为一个恒定值。在长时间极限下，根据网络大小N和连接参数m，CTQW的极限概率分布呈现出各种模式。当网络大小N为偶数时，处于原始节点的概率与处于相对节点的概率不同，这也取决于参数m的精确值。
