  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.
