The quantum communication complexity of sampling

Sampling is an important primitive in probabilistic and quantum algorithms. In the spirit of communication complexity, given a function f : X x Y --> {0, 1} and a probability distribution D over X x Y, we de. ne the sampling complexity of ( f, D) as the minimum number of bits that Alice and Bob must communicate for Alice to pick x is an element of X and Bob to pick y is an element of Y as well as a value z such that the resulting distribution of (x, y, z) is close to the distribution (D, f(D)). In this paper we initiate the study of sampling complexity, in both the classical and quantum models. We give several variants of a definition. We completely characterize some of these variants and give upper and lower bounds on others. In particular, this allows us to establish an exponential gap between quantum and classical sampling complexity for the set-disjointness function.