A Lower Bound for Sampling Disjoint Sets

Suppose Alice and Bob each start with private randomness and no other input, and they wish to engage in a protocol in which Alice ends up with a set x subseteq[n] and Bob ends up with a set y subseteq[n], such that (x,y) is uniformly distributed over all pairs of disjoint sets. We prove that for some constant beta0 of the uniform distribution over all pairs of disjoint sets of size sqrt{n}

A Hypercontractive Inequality for Matrix-Valued Functions with Applications to Quantum Computing and LDCs

The Bonami-Beckner hypercontractive inequality is a powerful tool in Fourier analysis of real-valued functions on the Boolean cube. In this paper we present a version of this inequality for matrix-valued functions on the Boolean cube. Its proof is based on a powerful inequality by Ball, Carlen, and Lieb. We also present a number of applications. First, we analyze maps that encode $n$ classical bits into $m$ qubits, in such a way that each set of $k$ bits can be recovered with some probability by an appropriate measurement on the quantum encoding; we show that if $m<0.7 n$, then the success probability is exponentially small in $k$. This result may be viewed as a direct product version of Nayak's quantum random access code bound. It in turn implies strong direct product theorems for the one-way quantum communication complexity of Disjointness and other problems. Second, we prove that error-correcting codes that are locally decodable with 2 queries require length exponential in the length of the encoded string. This gives what is arguably the first ``non-quantum'' proof of a result originally derived by Kerenidis and de Wolf using quantum information theory, and answers a question by Trevisan.Comment: This is the full version of a paper that will appear in the proceedings of the IEEE FOCS 08 conferenc

Correlation in Hard Distributions in Communication Complexity

We study the effect that the amount of correlation in a bipartite distribution has on the communication complexity of a problem under that distribution. We introduce a new family of complexity measures that interpolates between the two previously studied extreme cases: the (standard) randomised communication complexity and the case of distributional complexity under product distributions. We give a tight characterisation of the randomised complexity of Disjointness under distributions with mutual information $k$, showing that it is $\Theta(\sqrt{n(k+1)})$ for all $0\leq k\leq n$. This smoothly interpolates between the lower bounds of Babai, Frankl and Simon for the product distribution case ($k=0$), and the bound of Razborov for the randomised case. The upper bounds improve and generalise what was known for product distributions, and imply that any tight bound for Disjointness needs $\Omega(n)$ bits of mutual information in the corresponding distribution. We study the same question in the distributional quantum setting, and show a lower bound of $\Omega((n(k+1))^{1/4})$, and an upper bound, matching up to a logarithmic factor. We show that there are total Boolean functions $f_d$ on $2n$ inputs that have distributional communication complexity $O(\log n)$ under all distributions of information up to $o(n)$, while the (interactive) distributional complexity maximised over all distributions is $\Theta(\log d)$ for $6n\leq d\leq 2^{n/100}$. We show that in the setting of one-way communication under product distributions, the dependence of communication cost on the allowed error $\epsilon$ is multiplicative in $\log(1/\epsilon)$ -- the previous upper bounds had the dependence of more than $1/\epsilon$

Quantum Information Complexity and Amortized Communication

We define a new notion of information cost for quantum protocols, and a corresponding notion of quantum information complexity for bipartite quantum channels, and then investigate the properties of such quantities. These are the fully quantum generalizations of the analogous quantities for bipartite classical functions that have found many applications recently, in particular for proving communication complexity lower bounds. Our definition is strongly tied to the quantum state redistribution task. Previous attempts have been made to define such a quantity for quantum protocols, with particular applications in mind; our notion differs from these in many respects. First, it directly provides a lower bound on the quantum communication cost, independent of the number of rounds of the underlying protocol. Secondly, we provide an operational interpretation for quantum information complexity: we show that it is exactly equal to the amortized quantum communication complexity of a bipartite channel on a given state. This generalizes a result of Braverman and Rao to quantum protocols, and even strengthens the classical result in a bounded round scenario. Also, this provides an analogue of the Schumacher source compression theorem for interactive quantum protocols, and answers a question raised by Braverman. We also discuss some potential applications to quantum communication complexity lower bounds by specializing our definition for classical functions and inputs. Building on work of Jain, Radhakrishnan and Sen, we provide new evidence suggesting that the bounded round quantum communication complexity of the disjointness function is \Omega (n/M + M), for M-message protocols. This would match the best known upper bound.Comment: v1, 38 pages, 1 figur

An Optimal Lower Bound on the Communication Complexity of Gap-Hamming-Distance

We prove an optimal $\Omega(n)$ lower bound on the randomized communication complexity of the much-studied Gap-Hamming-Distance problem. As a consequence, we obtain essentially optimal multi-pass space lower bounds in the data stream model for a number of fundamental problems, including the estimation of frequency moments. The Gap-Hamming-Distance problem is a communication problem, wherein Alice and Bob receive $n$-bit strings $x$ and $y$, respectively. They are promised that the Hamming distance between $x$ and $y$ is either at least $n/2+\sqrt{n}$ or at most $n/2-\sqrt{n}$, and their goal is to decide which of these is the case. Since the formal presentation of the problem by Indyk and Woodruff (FOCS, 2003), it had been conjectured that the naive protocol, which uses $n$ bits of communication, is asymptotically optimal. The conjecture was shown to be true in several special cases, e.g., when the communication is deterministic, or when the number of rounds of communication is limited. The proof of our aforementioned result, which settles this conjecture fully, is based on a new geometric statement regarding correlations in Gaussian space, related to a result of C. Borell (1985). To prove this geometric statement, we show that random projections of not-too-small sets in Gaussian space are close to a mixture of translated normal variables

Query-to-Communication Lifting for BPP

For any $n$-bit boolean function $f$, we show that the randomized communication complexity of the composed function $f\circ g^n$, where $g$ is an index gadget, is characterized by the randomized decision tree complexity of $f$. In particular, this means that many query complexity separations involving randomized models (e.g., classical vs. quantum) automatically imply analogous separations in communication complexity.Comment: 21 page