Quantum HodgeRank: Topology-Based Rank Aggregation on Quantum Computers

Rank aggregation techniques play a crucial role in identifying the best alternatives, but the inherent imbalance and incompleteness of real-world data pose significant challenges. The HodgeRank algorithm, which utilizes discrete exterior calculus on a graph, can overcome these issues and output a consistent global ranking of alternatives. Recently, higher-order networks have been employed to analyze complex multipartite interactions in network data. Extending HodgeRank to ranking on higher-order networks, however, requires computational costs that are exponential in the network dimension. To address this challenge, we develop a quantum algorithm that outputs a quantum state proportional to the HodgeRank solution. By incorporating quantum singular value transformation and tools from quantum topological data analysis, our approach achieves time complexity independent of the network dimension, avoiding the need for quantum random access memory or sparse access input models. We also present efficient algorithms and heuristics to extract meaningful information from the output quantum state, including an algorithm to compute the consistency of the ranking. Using our algorithm, the consistency measure estimation has the potential to achieve superpolynomial quantum speedups for higher-order network data with specific structures. Beyond ranking tasks, our methods suggest that quantum computing could find fruitful applications in studying higher-order networks and discrete exterior calculus.