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arXiv:2304.12232

Variational Quantum PageRank

Sims, Christopher

Published Apr 19, 2023• 0 citations
arXivHTMLPDF

Abstract

The PageRank algorithm is used to rank web pages by their importance. Since its development, the PageRank algorithm is a critical and fundamental part of search engines today. PageRank is a graph-based algorithm that ranks pages based on how many other pages link to them. This work develops a variational quantum version of the PageRank algorithm and compares the performance of the two algorithms. It is found that quantum PageRank performs better at ranking websites than the normal PageRank algorithm

DOI
10.48550/arxiv.2304.12232

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The PageRank algorithm is used to rank web pages by their importance. Since its development, the PageRank algorithm is a critical and fundamental part of search engines today. PageRank is a graph-based algorithm that ranks pages based on how many other pages link to them. This work develops a variational quantum version of the PageRank algorithm and compares the performance of the two algorithms. It is found that quantum PageRank performs better at ranking websites than the normal PageRank algorithm
We propose an adiabatic quantum algorithm for generating a quantum pure state encoding of the PageRank vector, the most widely used tool in ranking the relative importance of internet pages. We present extensive numerical simulations which provide evidence that this algorithm can prepare the quantum PageRank state in a time which, on average, scales polylogarithmically in the number of web pages. We argue that the main topological feature of the underlying web graph allowing for such a scaling is the out-degree distribution. The top-ranked log(n) entries of the quantum PageRank state can then be estimated with a polynomial quantum speed-up. Moreover, the quantum PageRank state can be used in "q-sampling" protocols for testing properties of distributions, which require exponentially fewer measurements than all classical schemes designed for the same task. This can be used to decide whether to run a classical update of the PageRank.
The Google's PageRank algorithm is one of the most famous algorithms in data mining. A quantum version of this algorithm has been proposed in 2012. It is a meaningful step towards the quantum search engine but it can not search solutions. In this paper, we integrate a special operator of quantum search into the quantum PageRank algorithm to make a new quantum algorithm. We called it the Search Rank algorithm. This new algorithm is able to search solutions and rank results according to their importance at the same time. It is the first algorithm possessing this ability as far as we know. And it outperforms all classical algorithms when searching an unsorted database.
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.
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We introduce the characterization of a class of quantum PageRank algorithms in a scenario in which some kind of quantum network is realizable out of the current classical internet web, but no quantum computer is yet available. This class represents a quantization of the PageRank protocol currently employed to list web pages according to their importance. We have found an instance of this class of quantum protocols that outperforms its classical counterpart and may break the classical hierarchy of web pages depending on the topology of the web.
We review the main ndings on the ranking capabilities of the recently proposed Quantum PageRank algorithm (1,2) applied to large complex networks. The algorithm has been shown to identify unambiguously the underlying topology of the network and to be capable of clearly highlighting the struc- ture of secondary hubs of networks. Furthermore it can resolve the degeneracy in importance of the low lying part of the list of rankings. Examples of applications include real world instances from the WWW, which typically display a scale-free network structure and models of hierarchical networks. The quantum algorithm has been shown to display an increased stability with respect to a variation of the damping parameter, present in the Google algorithm, and a more clearly pronounced power-law behaviour in the distribution of importance among the nodes, as compared to the classical algorithm.