A Lower Bound for the Bounded Round Quantum Communication Complexity of Set Disjointness

We show lower bounds in the multi-party quantum communication complexity model. In this model, there are t parties where the ith party has input Xi ⊆ [n]. These parties communicate with each other by transmitting qubits to determine with high probability the value of some function F of their combined input (X1, . . . , Xt). We consider the class of boolean valued functions whose value depends only onX1∩· · ·∩Xt; that is, for each F in this class there is an fF : 2 → {0, 1}, such that F (X1, . . . , Xt) = fF (X1 ∩ · · · ∩ Xt). We show that the t-party k-round communication complexity of F is Ω(sm(fF )/(k)), where sm(fF ) stands for the ‘monotone sensitivity of fF ’ and is defined by sm(fF ) ∆ = maxS⊆[n] |{i : fF (S ∪ {i}) 6= fF (S)}|. For two-party quantum communication protocols for the set disjointness problem, this implies that the two parties must exchange Ω(n/k) qubits. An upper bound ofO(n/k) can be derived from the O( √ n) upper bound due to Aaronson and Ambainis [AA03]. For k = 1, our lower bound matches the Ω(n) lower bound observed by Buhrman and de Wolf [BdW01] (based on a result of Nayak [Nay99]), and for 2 ≤ k n, improves the lower bound of Ω( √ n) shown by Razborov [Raz02]. For protocols with no restrictions on the number of rounds, we can conclude that the two parties must exchange Ω(n) qubits. This, however, falls short of the optimal Ω( √ n) lower bound shown by