A basic question in complexity theory is whether the computational resources required for solving k independent instances of the same problem scale as k times the resources required for one instance. We investigate this question in various models of classical communication complexity. We introduce a new measure, the subdistribution bound , which is a relaxation of the well-studied rectangle or corruption bound in communication complexity. We nonetheless show that for the communication complexity of Boolean functions with constant error, the subdistribution bound is the same as the latter measure, up to a constant factor. We prove that the one-way version of this bound tightly captures the one-way public-coin randomized communication complexity of any relation, and the two-way version bounds the two-way public-coin randomized communication complexity from below. More importantly, we show that the bound satisfies the strong direct product property under product distributions for both one- and two-way protocols, and the weak direct product property under arbitrary distributions for two-way protocols. These results subsume and strengthen, in a unified manner, several recent results on the direct product question. The simplicity and broad applicability of our technique is perhaps an indication of its potential to solve yet more challenging questions regarding the direct product problem.
Although a quantum state requires exponentially many classical bits to describe, the laws of quantum mechanics impose severe restrictions on how that state can be accessed. This paper shows in three settings that quantum messages have only limited advantages over classical ones. First, we show that BQP/qpoly⊆PP/poly, where BQP/qpoly is the class of problems solvable in quantum polynomial time, given a polynomial-size “quantum advice state” that depends only on the input length. This resolves a question of Buhrman, and means that we should not hope for an unrelativized separation between quantum and classical advice. Underlying our complexity result is a general new relation between deterministic and quantum one-way communication complexities, which applies to partial as well as total functions. Second, we construct an oracle relative to which NP⊂BQP/qpoly. To do so, we use the polynomial method to give the first correct proof of a direct product theorem for quantum search. This theorem has other applications; for example, it can be used to fix a result of Klauck about quantum time-space tradeoffs for sorting. Third, we introduce a new trace distance method for proving lower bounds on quantum one-way communication complexity. Using this method, we obtain optimal quantum lower bounds for two problems of Ambainis, for which no nontrivial lower bounds were previously known even for classical randomized protocols. A preliminary version of this paper appeared in the 2004 Conference on Computational Complexity (CCC).
We give the first exponential separation between quantum and bounded-error randomized one-way communication complexity. Specifically, we define the Hidden Matching Problem HMn: Alice gets as input a string x∈{0,1}n, and Bob gets a perfect matching M on the n coordinates. Bob's goal is to output a tuple ⟨i,j,b⟩ such that the edge (i,j) belongs to the matching M and b=xi⊕xj. We prove that the quantum one-way communication complexity of HMn is O(logn), yet any randomized one-way protocol with bounded error must use Ω(n) bits of communication. No asymptotic gap for one-way communication was previously known. Our bounds also hold in the model of Simultaneous Messages (SM), and hence we provide the first exponential separation between quantum SM and randomized SM with public coins. For a Boolean decision version of HMn, we show that the quantum one-way communication complexity remains O(logn) and that the 0-error randomized one-way communication complexity is Ω(n). We prove that any randomized linear one-way protocol with bounded error for this problem requires Ω(3nlogn) bits of communication.