Multiparty Quantum Communication Complexity of Triangle Finding

Triangle finding (deciding if a graph contains a triangle or not) is a central problem in quantum query complexity. The quantum communication complexity of this problem, where the edges of the graph are distributed among the players, was considered recently by Ivanyos et al. in the two- party setting. In this paper we consider its k-party quantum communication complexity with k >= 3. Our main result is a ~O(m^(7/12))-qubit protocol, for any constant number of players k, deciding with high probability if a graph with m edges contains a triangle or not. Our approach makes connections between the multiparty quantum communication complexity of triangle finding and the quantum query complexity of graph collision, a well-studied problem in quantum query complexity