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 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.