Generalizations of the distributed Deutsch–Jozsa promise problem

Abstract

In thedistributed Deutsch–Jozsa promise problem, two parties are to determine whether their respective stringsx, y∈ {0,1}nare at theHamming distanceH(x, y) = 0 orH(x, y) =$\frac{n}{2}$. Buhrmanet al.(STOC' 98) proved that the exactquantum communication complexityof this problem isO(logn) while thedeterministic communication complexityisΩ(n). This was the first impressive (exponential) gap between quantum and classical communication complexity. In this paper, we generalize the above distributed Deutsch–Jozsa promise problem to determine, for any fixed$\frac{n}{2}$⩽k⩽n, whetherH(x, y) = 0 orH(x, y) =k, and show that an exponential gap between exact quantum and deterministic communication complexity still holds ifkis an even such that$\frac{1}{2}$n⩽k< (1 − λ)n, where 0 < λ <$\frac{1}{2}$is given. We also deal with a promise version of the well-knowndisjointnessproblem and show also that for this promise problem there exists an exponential gap between quantum (and also probabilistic) communication complexity and deterministic communication complexity of the promise version of such a disjointness problem. Finally, some applications to quantum, probabilistic and deterministic finite automata of the results obtained are demonstrated.