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A lower bound for the bounded round quantum communication complexity of set disjointness

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3 Author(s)
Jain, R. ; Sch. of Technol. & Comput. Sci., Tata Inst. of Fundamental Res., Mumbai, India ; Radhakrishnan, J. ; Sen, P.

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 on X1 ∩...∩ Xt; that is, for each F in this class there is an fF : 2[n] → {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)/(k2)), where sm(fF) stands for the monotone sensitivity of fF' and is defined by sm(fF) = maxS⊆[n] |{i : fF(S ∪ {i}) ≠ fF(S)}|. For two-party quantum communication protocols for the set disjointness problem, this implies that the two parties must exchange Ω(n/k2) qubits. An upper bound of O(n/k) can be derived from the O(√n) upper bound due to S. Aaronson and A. Ambainis (2003). For k = 1, our lower bound matches the Ω(n) lower bound observed by H. Buhrman and R. de Wolf (2001) (based on a result of A. Nayak (1999)), and for 2 ≤ k ≪ n14 /, improves the lower bound of Ω(√n) shown by A. Razborov (2002). For protocols with no restrictions on the number of rounds, we can conclude that the two parties must exchange Ω(n13/) qubits. This, however, falls short of the optimal Ω (√n) lower bound shown by A. Razborov (2002). Our result is obtained by adapting to the quantum setting the elegant information-theoretic arguments of Z. Bar-Yossef et al. (2002). Using this method we can show similar lower bounds for the L function considered in Z. Bar-Yossef et al. (2002).

Published in:

Foundations of Computer Science, 2003. Proceedings. 44th Annual IEEE Symposium on

Date of Conference:

11-14 Oct. 2003