The quantum communication complexity of sampling
Andris Ambainis, Leonard J. Schulman, Amnon Ta‐Shma, Umesh V. Vazirani, Avi Wigderson · 2002
Sampling is an important primitive in probabilistic and quantum algorithms. In the spirit of communication complexity, given a function f: X/spl times/Y/spl rarr/{0,1} and a probability distribution D over X/spl times/Y, we define the sampling complexity of (f,D) as the minimum number of bits Alice and Bob must communicate for Alice to pick x/spl isin/X and Bob to pick y/spl isin/Y as well as a valve z s.t. 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 model. We give several variants of the definition. We completely characterize some of these tasks, 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. This is the first exponential gap for any task where the classical probabilistic algorithm is allowed to err.