Non-local quantum functions and the distributed Deutsch-Jozsa algorithm

Non-local implementations of quantum gates are a vital part of quantum networks. We find an optimal non-local implementation of quantum functions, the quantum gate equivalent of a switch statement. Then, we apply this result to the Deutsch-Jozsa problem, obtaining a distributed Deutsch-Jozsa algorithm and we show the relative efficiency improvement. As an application, we find a non-cooperative game based upon the original Deutsch-Jozsa problem where a classical agent has at most a 50% probability of winning, while a quantum agent can win every time.