Continuous-time quantum walks on one-dimension regular networks

TL;DR

This paper investigates continuous-time quantum walks on one-dimensional regular networks, revealing faster transport than classical walks and analyzing how network parameters influence probability distributions and transport efficiency.

Contribution

It provides analytical calculations of transition probabilities and transport speeds, highlighting the impact of network connectivity and size on quantum walk dynamics.

Findings

Quantum walks are faster than classical random walks on the same networks.

Transport speed increases with the connectivity parameter m.

Limiting probability distributions depend on network size and connectivity, showing various patterns.

Abstract

In this paper, we consider continuous-time quantum walks (CTQWs) on one-dimension 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 spacetime 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 walk (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 the CTQWs and CTRWs. For fixed parameter m, the transport of CTRWs gets slow 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 paterns. 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.