Link prediction is a fundamental task in network science that involves estimating the likelihood of missing or future connections between nodes based on the structural patterns of a graph. While Graph Neural Networks (GNNs) have become a dominant approach, Random Walk (RW)-based methods remain highly relevant due to their lower computational demands in large-scale networks. Although recent hybrid quantum-classical walks have shown promise in exploring graph structures, existing continuous-time models are computationally expensive as they require solving high-dimensional systems of differential equations. In this paper, we propose a discrete version of a recent hybrid quantum-classical random walk algorithm that significantly improves computational efficiency by utilizing state vector representations instead of density matrices. Our method, inspired by the Quantum Jumps approach, alternates between coherent evolution and measurement-driven state collapse to generate trajectories for node embeddings. We integrate this algorithm into a link prediction pipeline and evaluate its performance across various synthetic topologies, including Grid, Power-Law, Erdös-Rényi, Barabási-Albert, and Community graphs. The experimental results demonstrate that the proposed discrete hybrid walks consistently outperform classical random walk approaches in Community and Grid topologies, showing that quantum-inspired dynamics can capture complex structural information more effectively than classical dynamics for specific topologies.
Link prediction is a fundamental task in network science that involves estimating the likelihood of missing or future connections between nodes based on the structural patterns of a graph. While Graph Neural Networks (GNNs) have become a dominant approach, Random Walk (RW)-based methods remain highly relevant due to their lower computational demands in large-scale networks. Although recent hybrid quantum-classical walks have shown promise in exploring graph structures, existing continuous-time models are computationally expensive as they require solving high-dimensional systems of differential equations. In this paper, we propose a discrete version of a recent hybrid quantum-classical random walk algorithm that significantly improves computational efficiency by utilizing state vector representations instead of density matrices. Our method, inspired by the Quantum Jumps approach, alternates between coherent evolution and measurement-driven state collapse to generate trajectories for node embeddings. We integrate this algorithm into a link prediction pipeline and evaluate its performance across various synthetic topologies, including Grid, Power-Law, Erdös-Rényi, Barabási-Albert, and Community graphs. The experimental results demonstrate that the proposed discrete hybrid walks consistently outperform classical random walk approaches in Community and Grid topologies, showing that quantum-inspired dynamics can capture complex structural information more effectively than classical dynamics for specific topologies.
Graph Representation Learning (GRL) has emerged as a cornerstone technique for analysing complex, networked data across diverse domains, including biological systems, social networks, and data analysis. Traditional GRL methods often struggle to capture intricate relationships within complex graphs, particularly those exhibiting non-trivial structural properties such as power-law distributions or hierarchical structures. This paper introduces a novel quantum-inspired algorithm for GRL, utilizing hybrid Quantum-Classical Walks to overcome these limitations. Our approach combines the benefits of both quantum and classical dynamics, allowing the walker to simultaneously explore both highly local and far-reaching connections within the graph. Preliminary results for a case study in network community detection shows that this hybrid dynamic enables the algorithm to adapt effectively to complex graph topologies, offering a robust and versatile solution for GRL tasks.
Quantum walks, both discrete and continuous, serve as fundamental tools in quantum information processing with diverse applications. This work introduces a hybrid quantum walk model that integrates the coin mechanism of discrete walks with the Hamiltonian-driven time evolution of continuous walks. Through systematic analysis of probability distributions, standard deviations, and entanglement entropy on fundamental graph structures, we reveal distinctive dynamical characteristics that differentiate our model from conventional quantum walk paradigms. The proposed framework demonstrates unifying capabilities by naturally encompassing existing quantum walk models as special cases. Two significant applications emerge from this hybrid architecture: (1) We develop a novel protocol for perfect state transfer(PST) in general connected graphs, overcoming the limitations of previous graph-specific approaches. A PST on a tree graph has been implemented on a quantum superconducting processor. (2) We devise a quantum algorithm for multiplying K adjacency matrices of n -vertex regular graphs with time complexity O ( n 2 d 1 ⋯ d K ), outperforming classical matrix multiplication ( O ( n 2.371552 )) when vertex degrees d i are bounded. The algorithm’s efficacy for triangle counting is experimentally validated through the quantum simulation on PennyLane. These results establish the hybrid quantum walk as a versatile framework bridging discrete and continuous paradigms while enabling practical quantum advantage in graph computation tasks.
Compared with the classical random walk, the quantum walk has the advantages of less time to find the target node and less time scattered from the source node to all nodes. Therefore, in order to further improve the accuracy of link prediction algorithms, a link prediction algorithm based on discrete-time quantum walk (DTQW) on weighted graphs is proposed by integrating node attributes and introducing the idea of assigning weights to edges in network dynamics. In this algorithm, a weighted graph is first constructed based on the attribute information of nodes in the graph, and then a DTQW model is constructed based on the weighted graph. Then, under the control of the evolution operator, the walker performs a DTQW on the weighted graph; Finally, the link prediction algorithm is implemented by defining the similarity between nodes based on the average probability of the walker reaching each node. The experimental results show that in three real social networks, Douban, Weibo and Zhihu, this algorithm has different degrees of improvement in prediction accuracy and the algorithm stability is among the top in comparison with the optimal benchmark prediction algorithm.
This book explores the intersection of quantum computing and network science. It bridges the theoretical foundations of quantum walk algorithms with their applications in the structural exploration and representation learning of complex networks. Quantum walks, a technology that is pivotal to universal quantum computational models, examines the movement of particles on a graph composed of nodes and links. Quantum superposition enables these particles to traverse these graphs more quickly, while measurement-induced collapse introduces fluctuations, making the identification of critical nodes challenging yet intriguing. At its core, the book explores how quantum walk algorithms can transform the structural and representational analysis of complex networks. It begins by introducing the fundamental concepts of quantum computing and quantum walks, including generalized definitions and the properties of low-dimensional quantum walks. Then, it discusses the implementation of discrete- and continuous-time quantum walks for mining network nodes, links, and subgraphs, as well as their use in network representation learning and graph neural networks. The book will serve as a valuable reference for researchers, students, and educators interested in quantum walks, complex networks, quantum mechanics, and information engineering.
Original scientific paperLink prediction is one of the key issues of complex networks, which attracts much research interest currently.Many link prediction methods have been proposed so far.The classical random walk as an effective tool has been widely used to study the link prediction problems.Quantum walk is the quantum analogue of classical random walk.Numerous research results show that quantum algorithms using quantum walk outperform their classical counterparts in many applications, such as graph matching and searching.But there have been few studies of the link prediction based on quantum walk, especially on discrete-time quantum walk.This paper proposes a new link prediction method based on discrete-time quantum walk.Experiment results show that prediction accuracy of our method is better than the typical methods.The time complexity of our method running on classical computers, compared with the methods based on classical random walk, is slightly higher.But our method can be greatly accelerated by executing on quantum computers.
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.