Discrete-Time Open Quantum Walks for Vertex Ranking in Graphs

TL;DR

This paper introduces a novel quantum PageRank algorithm based on discrete-time open quantum walks, demonstrating faster convergence and applicability to various complex networks, thereby enhancing network vertex importance quantification.

Contribution

The work develops a new quantum PageRank algorithm using discrete-time open quantum walks with Weyl operators, extending quantum walk models for network analysis.

Findings

Quantum PageRank converges faster than existing methods.

The algorithm effectively ranks vertices in diverse complex networks.

Quantum PageRank's convergence depends on the damping factor .

Abstract

This article presents a new quantum PageRank algorithm on graphs using discrete-time open quantum walks. Google's PageRank is a widely used algorithm for ranking the web pages on the World Wide Web in classical computation. From a broader perspective, it is also a fundamental measure for quantifying the importance of vertices in a network. Similarly, the new quantum PageRank also serves to quantify the significance of a network's vertices. In this work, we extend the concept of discrete-time open quantum walk on arbitrary directed and undirected graphs by utilizing the Weyl operators as Kraus operators. This new model of quantum walk is useful for building up the quantum PageRank algorithm, discussed in this article. We compare the classical PageRank and the newly defined quantum PageRank for different types of complex networks, such as the scale-free network, Erd\H{o}s-R\'enyi random network, Watts-Strogatz network, spatial network, Zachary Karate club network, GNC, GN, GNR networks, Barab\'asi and Albert network, etc. In addition, we study the convergence of the quantum PageRank process and its dependency on the damping factor $\alpha$. We observe that this quantum PageRank procedure is faster than many other proposals reported in the literature.