Consensus ranking by quantum annealing
Daniele Franch, Enrico Zardini, Enrico Blanzieri, Davide Pastorello
TL;DR
This paper tackles the NP-hard problem of consensus ranking via Kemeny aggregation by leveraging quantum annealing in a hybrid quantum-classical framework. It encodes the ranking as a binary upper-triangular matrix in a QUBO formulation, applying cycle-penalty terms to enforce transitivity, while the final ranking is reconstructed classically. The authors introduce an iterative method that penalizes only problem-relevant cycles and a pair-removal technique to reduce qubit usage, significantly improving performance and scalability. Across synthetic datasets and comparisons with KwikSort, the proposed approach demonstrates superior solution quality on larger candidate sets, offering a practical path toward applying Kemeny ranking to large-scale, multiagent decision scenarios. The work also extends to partial and weighted lists, confirming robustness and suggesting avenues for integration with ensemble learning and weighted decision-making in AI systems.
Abstract
Consensus ranking is a technique used to derive a single ranking that best represents the preferences of multiple individuals or systems. It aims to aggregate different rankings into one that minimizes overall disagreement or distance from each of the individual rankings. Kemeny ranking aggregation, in particular, is a widely used method in decision-making and social choice, with applications ranging from search engines to music recommendation systems. It seeks to determine a consensus ranking of a set of candidates based on the preferences of a group of individuals. However, existing quantum annealing algorithms face challenges in efficiently processing large datasets with many candidates. In this paper, we propose a method to improve the performance of quantum annealing for Kemeny rank aggregation. Our approach identifies the pairwise preference matrix that represents the solution list and subsequently reconstructs the ranking using classical methods. This method already yields better results than existing approaches. Furthermore, we present a range of enhancements that significantly improve the proposed method's performance, thereby increasing the number of candidates that can be effectively handled. Finally, we evaluate the efficiency of our approach by comparing its performance and execution time with that of KwikSort, a well-known approximate algorithm.