Quantum Multiparty Communication Complexity and Circuit Lower Bounds

François Le Gall, Daiki Suruga

Iordanis Kerenidis

We define a quantum model for multiparty communication complexity and prove a simulation theorem between the classical and quantum models. As a result, we show that if the quantum k-party communication complexity of a function f is Ω(n/2k), its classical k-party communication is Ω(n/2k/2). Finding such an f would allow us to prove strong classical lower bounds for k ≥ log n players and make progress towards solving a major open question about symmetric circuits.

Takashi Mihara

Iordanis Kerenidis

We define a quantum model for multiparty communication complexity and prove a simulation theorem between the classical and quantum models. As a result of our simulation, we show that if the quantum k-party communication complexity of a function f is $Ω(n/2^k)$, then its classical k-party communication is $Ω(n/2^{k/2})$. Finding such an f would allow us to prove strong classical lower bounds for (k>log n) players and hence resolve a main open question about symmetric circuits. Furthermore, we prove that for the Generalized Inner Product (GIP) function, the quantum model is exponentially more efficient than the classical one. This provides the first exponential separation for a total function between any quantum and public coin randomized communication model.

Ronald de Wolf

Troy Lee, Gideon Schechtman, Adi Shraibman

A major open question in communication complexity is if randomized and quantum communication are polynomially related for all total functions. So far, no gap larger than a power of two is known, despite significant efforts. We examine this question in the number-on-the-forehead model of multiparty communication complexity. We show that essentially all lower bounds known on randomized complexity in this model also hold for quantum communication. This includes bounds of size Omega(n/2k) for the k-party complexity of explicit functions, bounds for the generalized inner product function, and recent work on the multiparty complexity of disjointness. To the best of our knowledge, these are the first lower bounds of any kind on quantum communication in the general number-on-the-forehead model. We show this result in the following way. In the two-party case, there is a lower bound on quantum communication complexity in terms of a norm gamma2, which is known to subsume nearly all other techniques in the literature. For randomized complexity there is another natural bound in terms of a different norm mu which is also one of the strongest techniques available. A deep theorem in functional analysis, Grothendieck's inequality, implies that gamma2and mu are equivalent up to a constant factor. This connection is one of the major obstacles to showing a larger gap between randomized and quantum communication complexity in the two-party case. The lower bound technique in terms of the norm mu was recently extended to the multiparty number-on-the-forehead model. Here we show how the gamma2norm can be also extended to lower bound quantum multiparty complexity. Surprisingly, even in this general setting the two lower bounds, on quantum and classical communication, are still very closely related. This implies that separating quantum and classical communication in this setting will require the development of new techniques. The relation between these extensions of mu and gamma2is proved by a multi-dimensional version of Grothendieck's inequality.