Lower Bounds on Quantum Multiparty Communication Complexity

Abstract

Abstract-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 Ω(n/2 k ) 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 twoparty case, there is a lower bound on quantum communication complexity in terms of a norm γ2, 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 µ which is also one of the strongest techniques available. A deep theorem in functional analysis, Grothendieck's inequality, implies that γ2 and µ 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 µ was recently extended to the multiparty number-on-the-forehead model. Here we show how the γ2 norm 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 µ and γ2 is proved by a multi-dimensional version of Grothendieck's inequality.

Tensor Rank and Strong Quantum Nondeterminism in Multiparty Communication

“…We also define the nondeterministic rank of f , denoted nrank( f ), to be the minimum rank over the complex field among all nondeterministic tensors for f . The following lemma, given in Lee, Schechtman, and Shraibman [14], generalizes a previous observation made by Yao [16] and Kremer [17] on 2-party protocols.…”

“…To model NOF and NIH in the quantum setting, we follow the work of Lee, Schechtman, and Shraibman [14], originally defined by Kerenidis [15]. 1. in NIH, an arbitrary unitary that only depends on x i is applied on H i ⊗ C, and acts as the identity anywhere else; 2. in NOF, an arbitrary unitary that depends on all inputs except x i is applied on H i ⊗ C, and acts as the identity anywhere else.…”

“…We can use the same lower bound to prove the separation for BQP cc . In order to do that we make use of the following two results by Lee, Schechtman, and Shraibman [14]. Let γ α be the approximate quantum norm as defined in [14].…”

On Black-Box Reductions between Predicate Encryption Schemes

“…Various models such as deterministic, randomized (public or private coins), nondeterministic, etc., communication complexity can be defined naturally. For details on such models, we refer to the classic book by Kushilevitz and Nisan [22], the paper by Babai et al [1], and the surveys by Lokam [26] and Lee and Shraibman [23].…”

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“…Roughly speaking, the idea is to combine two classical parties into one quantum party, prove lower bounds in the resulting quantum model, and then translate these back to strong lower bounds for the classical model. (Unfortunately, it seems hard to prove good lower bounds for the resulting quantum model [85]. )…”