Multi-Party Quantum Communication Complexity with Routed Messages

This paper describes a general quantum lower bounding technique for the communication complexity of a function that depends on the inputs given to two parties connected via paths, which may be shared with other parties, on a network of any topology. The technique can also be employed to obtain a lower-bound of the quantum communication complexity of some functions that depend on the inputs distributed over all parties on the network. As a typical application, we apply our technique to the distinctness problem of deciding whether there are a pair of parties with identical inputs, on a k-party ring; almost matching upper bounds are also given.