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Classical versus quantum communication complexity

Amnon Ta‐Shma

Published Sep 1, 1999• 23 citations
Publisher

Abstract

Classical communication complexity has been intensively studied since its conception two decades ago. Recently, its quantum analogue, the quantum communication complexity model, was defined and studied. In this paper we present some of the main results in the area.
DOI
10.1145/333623.333628

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In two-party quantum communication complexity, Alice and Bob receive some classical inputs and wish to compute some function that depends on both these inputs, while minimizing the communication. This model has found numerous applications in many areas of computer science. One notion that has received a lot of attention recently is the information cost of the protocol, namely how much information the players reveal about their inputs when they run the protocol. In the quantum world, it is not straightforward to define a notion of quantum information cost. We study two different notions and analyze their relation. We also provide new quantum protocols for the Inner Product function and for Private Information Retrieval, and show that protocols for Private Information Retrieval whose classical or quantum information cost for the user is zero can have exponentially different information cost for the server.
Sampling is an important primitive in probabilistic and quantum algorithms. In the spirit of communication complexity, given a function f:X×Y→{0,1}f: X \times Y \rightarrow \{0,1\}f:X×Y→{0,1} and a probability distribution D{\cal D}D over X×YX \times YX×Y, we define the sampling complexity of (f,D)(f, {\cal D})(f,D) as the minimum number of bits that Alice and Bob must communicate for Alice to pick x∈Xx \in Xx∈X and Bob to pick y∈Yy \in Yy∈Y as well as a value z such that the resulting distribution of (x,y,z)(x,y,z)(x,y,z) is close to the distribution (D,f(D))({\cal D}, f({\cal D}))(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.
Communication complexity has become a central completity model.In that model, we count the amount of communication bits needed between two parties in order to solve certain computational problems.We show that for certain communication complezity problems quantum communication protocols are exponentially faster than classical ores.More explicitly, we give an example for a communication complexity relation (0~ promise problem) P such that: 1) The quantum communication complexity of P is O(log m). 2) The classical probabilistic communication complexity of P is Q(m'l'/ logm).(where m is the length of the inputs).This gives an ezponential gap between quantum communication complexity and classical probabilistic communication complexity.Only a quadratic gap was previously known.Our problem P is of geometrical nature, and is a finite precision variation of the following problem: Player I gets as input a unit vector z E R" and two orthogonal subspaces MO, MI c R". Player II gets as input an orthogonal matrixT : R" + R".Their goal is to answer 0 if T(x) E MO and 1 if T(x) E M,, (and any an~luer in any other case).We give an almost tight analysis for the quantum communication complexity and for the classical-probabilistic communication complexity of this problem.