Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2407.20452 · Physical Review Applied

Quantum HodgeRank: Topology-based rank aggregation on quantum computers

Caesnan M. G. Leditto, Angus Southwell, Behnam Tonekaboni, Muhammad Usman, Kavan Kishore Modi

Published May 28, 2025• 0 citations
arXivHTMLPDF

Abstract

HodgeRank generalizes ranking algorithms, e.g. Google PageRank, to rank alternatives based on real-world (often incomplete) data using graphs and discrete exterior calculus. It analyzes multipartite interactions on high-dimensional networks with a complexity that scales exponentially with dimension. We develop a quantum algorithm that approximates the HodgeRank solution with complexity independent of dimension. Our algorithm extracts relevant information from the state such as the ranking consistency, which achieves a superpolynomial speedup over similar classical methods.

DOI
10.1103/m6nc-ypl7

Related papers

Topological signal processing on quantum computers for higher-order network analysis

Caesnan M. G. Leditto, Angus Southwell, Behnam Tonekaboni +3 more

Predicting and analyzing global behavior of complex systems is challenging due to the intricate nature of their component interactions. Recent work has started modeling complex systems using networks endowed with multiway interactions among nodes, known as higher-order networks. Simplicial complexes are a class of higher-order networks that have received significant attention due to their topological structure and connections to Hodge theory. Topological signal processing (TSP) utilizes these connections to analyze and manipulate signals defined on non-Euclidean domains such as simplicial complexes. Such analysis of higher-order network data is important for many real-world problems, such as detecting failure or error in communication networks, sensor coverage analysis, statistical ranking problems, finding arbitrage currency markets, etc. However, as the dimension of higher-order networks increases, the complexity of TSP scales exponentially. In this work, we present a general quantum algorithm for implementing filtering processes in TSP and describe its application to extracting network data based on the Hodge decomposition. We leverage preexisting tools introduced in recent quantum algorithms for topological data analysis and combine them with spectral filtering techniques using the quantum singular value transformation framework. While this paper serves as a proof of concept, we obtain a superpolynomial improvement over the best-known classical algorithms for TSP filtering processes, modulo some important caveats about encoding and retrieving the data from a quantum state. The proposed algorithm generalizes the applicability of tools from quantum topological data analysis to novel applications in analyzing high-dimensional complex systems.

arXiv:2312.07672•Physical Review Applied•May 21, 2025•2 citations

Discrete-time open quantum walks for vertex ranking in graphs

Supriyo Dutta

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.

arXiv:2404.14770•Physical review. E•Mar 26, 2025•6 citations

Construction and Application to Complex Networks

Francesc Comellas

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.

•Jan 1, 2014•0 citations

Adiabatic Quantum Algorithm for Search Engine Ranking

Silvano Garnerone, Paolo Zanardi A, Daniel A. Lidar

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.

arXiv:1109.6546•Physical Review Letters•Jun 4, 2012•102 citations

Quantum Navigation and Ranking in Complex Networks

Eduardo Sánchez-Burillo, Jordi Duch, Jesús Gómez‐Gardeñes +1 more

Complex networks are formal frameworks capturing the interdependencies between the elements of large systems and databases. This formalism allows to use network navigation methods to rank the importance that each constituent has on the global organization of the system. A key example is Pagerank navigation which is at the core of the most used search engine of the World Wide Web. Inspired in this classical algorithm, we define a quantum navigation method providing a unique ranking of the elements of a network. We analyze the convergence of quantum navigation to the stationary rank of networks and show that quantumness decreases the number of navigation steps before convergence. In addition, we show that quantum navigation allows to solve degeneracies found in classical ranks. By implementing the quantum algorithm in real networks, we confirm these improvements and show that quantum coherence unveils new hierarchical features about the global organization of complex systems.

arXiv:1202.3471•Scientific Reports•Aug 28, 2012•115 citations

An Enhanced Quantum PageRank Algorithm Integrated with Quantum Search

Huiquan Wang, Junjie Wu, Xuejun Yang +2 more

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.

International Conference on Innovative Mobile and Internet Services in Ubiquitous Computing (IMIS)•Jul 1, 2014•5 citations
HodgeRank generalizes ranking algorithms, e.g. Google PageRank, to rank alternatives based on real-world (often incomplete) data using graphs and discrete exterior calculus. It analyzes multipartite interactions on high-dimensional networks with a complexity that scales exponentially with dimension. We develop a quantum algorithm that approximates the HodgeRank solution with complexity independent of dimension. Our algorithm extracts relevant information from the state such as the ranking consistency, which achieves a superpolynomial speedup over similar classical methods.
Predicting and analyzing global behavior of complex systems is challenging due to the intricate nature of their component interactions. Recent work has started modeling complex systems using networks endowed with multiway interactions among nodes, known as higher-order networks. Simplicial complexes are a class of higher-order networks that have received significant attention due to their topological structure and connections to Hodge theory. Topological signal processing (TSP) utilizes these connections to analyze and manipulate signals defined on non-Euclidean domains such as simplicial complexes. Such analysis of higher-order network data is important for many real-world problems, such as detecting failure or error in communication networks, sensor coverage analysis, statistical ranking problems, finding arbitrage currency markets, etc. However, as the dimension of higher-order networks increases, the complexity of TSP scales exponentially. In this work, we present a general quantum algorithm for implementing filtering processes in TSP and describe its application to extracting network data based on the Hodge decomposition. We leverage preexisting tools introduced in recent quantum algorithms for topological data analysis and combine them with spectral filtering techniques using the quantum singular value transformation framework. While this paper serves as a proof of concept, we obtain a superpolynomial improvement over the best-known classical algorithms for TSP filtering processes, modulo some important caveats about encoding and retrieving the data from a quantum state. The proposed algorithm generalizes the applicability of tools from quantum topological data analysis to novel applications in analyzing high-dimensional complex systems.
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.
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.
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.
Complex networks are formal frameworks capturing the interdependencies between the elements of large systems and databases. This formalism allows to use network navigation methods to rank the importance that each constituent has on the global organization of the system. A key example is Pagerank navigation which is at the core of the most used search engine of the World Wide Web. Inspired in this classical algorithm, we define a quantum navigation method providing a unique ranking of the elements of a network. We analyze the convergence of quantum navigation to the stationary rank of networks and show that quantumness decreases the number of navigation steps before convergence. In addition, we show that quantum navigation allows to solve degeneracies found in classical ranks. By implementing the quantum algorithm in real networks, we confirm these improvements and show that quantum coherence unveils new hierarchical features about the global organization of complex systems.
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.