Bounds on oblivious multiparty quantum communication complexity

TL;DR

This paper establishes tight lower and upper bounds on the oblivious quantum multiparty communication complexity for symmetric functions, notably proving an optimal bound for the Set-Disjointness problem.

Contribution

It introduces a method to derive strong lower bounds from two-party complexity and applies it to obtain tight bounds for symmetric functions in the oblivious quantum setting.

Findings

Proved an (k\u2212 extfinity extsubscript{n}) lower bound for Set-Disjointness.

Established tight bounds for symmetric functions with AND gadgets.

Demonstrated the bounds are nearly optimal with matching upper bounds.

Abstract

The main conceptual contribution of this paper is investigating quantum multiparty communication complexity in the setting where communication is \emph{oblivious}. This requirement, which to our knowledge is satisfied by all quantum multiparty protocols in the literature, means that the communication pattern, and in particular the amount of communication exchanged between each pair of players at each round is fixed \emph{independently of the input} before the execution of the protocol. We show, for a wide class of functions, how to prove strong lower bounds on their oblivious quantum $k$-party communication complexity using lower bounds on their \emph{two-party} communication complexity. We apply this technique to prove tight lower bounds for all symmetric functions with \textsf{AND} gadget, and in particular obtain an optimal $\Omega(k\sqrt{n})$ lower bound on the oblivious quantum $k$-party communication complexity of the $n$-bit Set-Disjointness function. We also show the tightness of these lower bounds by giving (nearly) matching upper bounds.