Quantum Query Complexity of Tournament Solutions

A tournament solution records a set of winners from a tournament, that is, from the outcomes of all pairwise comparisons among N alternatives. We study the quantum query complexity of computing tournament solutions when the outcome of each comparison has to be learned by a query. Classically, the picture is uniform: the Copeland set, the uncovered set and the top cycle all require Θ(N²) queries, and so does finding a single vertex of maximum out-degree. We show that the quantum query complexity of the top cycle can be computed with Θ(N) queries, and a Condorcet winner can be found with Θ(√ N) queries. Finding a strong king requires Θ(N3/2) queries, the same as finding a vertex of maximum out-degree. We also give a self-contained alternative proof of the known Θ(N3/2) bound for finding a vertex of maximum degree, established by Gilani, Mande, Patro, and Raja. Our lower bounds reduce counting problems to tournament problems using Ambainis's adversary method. For the top cycle, we find a Hamiltonian path and locate the prefix forming the top cycle by divide and conquer.