Discrete-Time Open Quantum Walks for Vertex Ranking in Graphs

TL;DR

This work introduces a discrete-time open quantum walk framework for vertex ranking (qPageRank) on arbitrary graphs by encoding node transitions with Weyl-based Kraus operators. A key advance is reducing the effective Hilbert space from $n^2$ to $n$ while preserving the Markovian, convergent dynamics, and incorporating teleportation via a CPTP restart channel. The resulting qPageRank converges without time-averaging and is shown to match classical PageRank on many undirected graphs while revealing different rankings on several directed graphs; convergence is generally fast relative to prior quantum-pageRank formulations. The approach provides a quantum-augmented centrality measure with potential applications in quantum networks and graph analytics, and it offers practical guidance on parameter choices such as the damping factor $oldsymbol{b1}$ for stability.

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, Erdős-Rényi random network, Watts-Strogatz network, spatial network, Zachary Karate club network, GNC, GN, GNR networks, Barabási and Albert network, etc. In addition, we study the convergence of the quantum PageRank process and its dependency on the damping factor $α$. We observe that this quantum PageRank procedure is faster than many other proposals reported in the literature.

Figures (13)

• Figure 1: Comparison between the PageRank and qPageRank of vertices on path graph with $60$ vertices. (Color online)
• Figure 2: Undirected balanced tree as well as a bar diagram representing the PageRank and qPageRank of its vertices. (Color online)
• Figure 3: We consider a directed graph with $7$ vertices and $11$ edges which are depicted in subfigure \ref{['Paparo_graph']}. For the vertices we plot the PageRank, qPageRank based on paparo2013quantum and qPageRank based on DTOQW with bar diagrams in subfigure \ref{['Paparo_graph_bar']}. (Color online.)
• Figure 4: We consider a directed trinary tree with $121$ vertices and $120$ edges which are depicted in subfigure \ref{['trinary_tree']}. For the vertices we plot the PageRank and qPageRank as bar diagram in subfigure \ref{['trinary_tree_bar']}. Note that, the qPageRank is more sensitive to find out the vertices in the outermost layer. (Color online.)
• Figure 5: A GNC graph with $120$ vertices and $478$ edges is depicted in figure \ref{['GNC_graph']}. The Kendall's rank correlation coefficient between the rank of the vertices generated by PageRank and qPageRank is $0.777311$. (Color online.)

Theorems & Definitions (2)

• Definition 1
• Definition 2