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arXiv:2312.07672 · Physical Review Applied

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

Caesnan M. G. Leditto, Angus Southwell, Behnam Tonekaboni, Gregory A. L. White, Muhammad Usman, Kavan Kishore Modi

Published May 21, 2025• 2 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevapplied.23.054054

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Graph Neural Networks (GNNs) excel at learning from graph-structured data but are limited to modeling pairwise interactions, insufficient for capturing higher-order relationships present in many real-world systems. Topological Deep Learning (TDL) has allowed for systematic modeling of hierarchical higher-order interactions by relying on combinatorial topological spaces such as simplicial complexes. In parallel, Quantum Neural Networks (QNNs) have been introduced to leverage quantum mechanics for enhanced computational and learning power. In this work, we present the first Quantum Topological Deep Learning Model: Quantum Simplicial Networks (QSNs), being QNNs operating on simplicial complexes. QSNs are a stack of Quantum Simplicial Layers, which are inspired by the Ising model to encode higher-order structures into quantum states. Experiments on synthetic classification tasks show that QSNs can outperform classical simplicial TDL models in accuracy and efficiency, demonstrating the potential of combining quantum computing with TDL for processing data on combinatorial topological spaces.
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.
Quantum algorithms for topological data analysis provide significant advantages over the best known classical algorithms. Unlike previous work on simplicial complexes built from point clouds, path homology on digraphs is defined for directed graphs and provides a natural topological framework for analyzing data with intrinsic directional structures. Path homology has become an emerging area in Topological Data Analysis (TDA), attracting increasing attention in recent years. We propose a quantum algorithm for path homology on digraphs that offers a significant advantage over the best known classical algorithms. We design a universal encoding protocol for the paths and boundary operators of digraphs on quantum systems. We prove a property of path homology that provides the theoretical guarantee for the algorithm. The speedup of the quantum algorithm for path homology depends on input-access assumptions. The exponential speedup arises when the path space can be efficiently accessed, while for standard digraph input the algorithm provides polynomial speedup.
Abstract Quantum algorithms for topological data analysis (TDA) provide significant advantages over the best known classical algorithms. Unlike previous work on simplicial complexes built from point clouds, path homology on digraphs is defined for directed graphs and provides a natural topological framework for analyzing data with intrinsic directional structures. Path homology has become an emerging area in TDA, attracting increasing attention in recent years. We propose a quantum algorithm for path homology on digraphs that offers a significant advantage over the best known classical algorithms. We design a universal encoding protocol for the paths and boundary operators of digraphs on quantum systems. We prove a property of path homology that provides the theoretical guarantee for the algorithm. The speedup of the quantum algorithm for path homology depends on input-access assumptions. The exponential speedup arises when the path space can be efficiently accessed, while for standard digraph input the algorithm provides polynomial speedup.