Generalizations of the distributed Deutsch-Jozsa promise problem

In the distributed Deutsch-Jozsa promise problem, two parties are to determine whether their respective strings x, y ∈ {0, 1}n are at the Hamming distance H(x, y) = 0 or H(x, y) = n2 . Buhrman et al. (STOC’ 98) proved that the exact quantum communication complexity of this problem is O(logn) while the deterministic communication complexity is Ω(n). This was the first impressively (exponential) gap between quantum and classical communication complexity. In this paper, we generalize the above distributed Deutsch-Jozsa promise problem to determine for a fixed k ≥ n2 whether H(x, y) = 0 or H(x, y) = k. We will also discuss the promise versions of the well-known disjointness problem. Applications to finite automata of the results obtained are also shown in this paper.