From classical to quantum HodgeRank

This is a rather late summary of a cool preprint I saw a few months ago: Quantum HodgeRank: Topology-Based Rank Aggregation on Quantum Computers 

This work is inspired by and builds on quantum subroutines developed for efficiently solving high-dimensional topological data analysis problems, offering superpolynomial speedups for ranking higher-order network data by developing a quantum version of the classical HodgeRank algorithm.

What is HodgeRank? It was originally proposed in 2011 as a better way of ranking incomplete or skewed datasets, for example based on user ratings or scores.

The basic idea is to apply an analogue of the Helmholtz decomposition (used routinely in electromagnetics) to graph data, enabling one to assign a ranking based on incomplete pairwise preferences. Importantly, HodgeRank outputs not just a raw ranking, but also an estimate of the quality of the ranking via the construction of local and global cycles present in the optimal ranking. To be specific, the returned optimal ranking is unique and fully consistent if the preference matrix can be written as the gradient of some scalar ranking function. If it cannot, then there are inevitable ambiguities present in the preference data due to the existence of global or local cycles. 

An example of a local ranking cycle is the following: B is preferred over A, C is preferred over B, and yet A is preferred over C. This leads to the ranking A < C < B < A, thus forming a cycle. It is better to identify cycles such as these and acknowledge that a traditional ranking does not make sense for these items. This is what HodgeRank does! User preference data is rarely consistent, so cycles such as these routinely occur in the wild, for example in user rankings of movies on online platforms such as Netflix. 

As a generalization of HodgeRank, Quantum HodgeRank promises the ability to perform ranking tasks on preference data forming higher-order networks, avoiding the exponential scaling with network dimension faced by classical algorithms. Moreover, the authors of the preprint argue that HodgeRank cannot be dequantized (i.e. implemented efficiently using a randomized classical algorithm) in the same manner as quantum TDA algorithms for the Betti number problem. Moreover, while applications of high-dimensional Betti numbers (and even their occurrence in real datasets) remain unclear, HodgeRank represents a ranking problem with more likely concrete applications. Thus, this looks like an exciting area to keep an eye on. 

It is also interesting to speculate on whether (classical) HodgeRank or HodgeRank-inspired methods can be useful for understanding the behaviour of interacting many-body quantum systems, where it is typically intractable to sample all of the pairwise interaction elements of Hamiltonians as the system size increases, but incomplete or skewed sampling is readily available. Watch this space!