Exponential Separation of Quantum Communication and Classical Information

We exhibit a Boolean function for which the quantum communication complexity is exponentially larger than the classical information complexity. An exponential separation in the other direction was already known from the work of Kerenidis et. al. [SICOMP 44, pp. 1550-1572], hence our work implies that these two complexity measures are incomparable. As classical information complexity is an upper bound on quantum information complexity, which in turn is equal to amortized quantum communication complexity, our work implies that a tight direct sum result for distributional quantum communication complexity cannot hold. The function we use to present such a separation is the Symmetric k-ary Pointer Jumping function introduced by Rao and Sinha [ECCC TR15-057], whose classical communication complexity is exponentially larger than its classical information complexity. In this paper, we show that the quantum communication complexity of this function is polynomially equivalent to its classical communication complexity. The high-level idea behind our proof is arguably the simplest so far for such an exponential separation between information and communication, driven by a sequence of round-elimination arguments, allowing us to simplify further the approach of Rao and Sinha. As another application of the techniques that we develop, we give a simple proof for an optimal trade-off between Alice's and Bob's communication while computing the related Greater-Than function on n bits: say Bob communicates at most b bits, then Alice must send n/exp(O(b)) bits to Bob. This holds even when allowing pre-shared entanglement. We also present a classical protocol achieving this bound.Comment: v1, 36 pages, 3 figure

New Separations Results for External Information

We obtain new separation results for the two-party external information complexity of boolean functions. The external information complexity of a function $f(x,y)$ is the minimum amount of information a two-party protocol computing $f$ must reveal to an outside observer about the input. We obtain the following results: 1. We prove an exponential separation between external and internal information complexity, which is the best possible; previously no separation was known. 2. We prove a near-quadratic separation between amortized zero-error communication complexity and external information complexity for total functions, disproving a conjecture of \cite{Bravermansurvey}. 3. We prove a matching upper showing that our separation result is tight

Near-Quadratic Lower Bounds for Two-Pass Graph Streaming Algorithms

We prove that any two-pass graph streaming algorithm for the $s$-$t$ reachability problem in $n$-vertex directed graphs requires near-quadratic space of $n^{2-o(1)}$ bits. As a corollary, we also obtain near-quadratic space lower bounds for several other fundamental problems including maximum bipartite matching and (approximate) shortest path in undirected graphs. Our results collectively imply that a wide range of graph problems admit essentially no non-trivial streaming algorithm even when two passes over the input is allowed. Prior to our work, such impossibility results were only known for single-pass streaming algorithms, and the best two-pass lower bounds only ruled out $o(n^{7/6})$ space algorithms, leaving open a large gap between (trivial) upper bounds and lower bounds

An Interactive Information Odometer and Applications

We introduce a novel technique which enables two players to maintain an estimate of the internal information cost of their conversation in an online fashion without revealing much extra information. We use this construction to obtain new results about communication complexity and information-theoretic privacy. As a first corollary, we prove a strong direct product theorem for communication complexity in terms of information complexity: If I bits of information are required for solving a single copy of f under μ with probability 2/3, then any protocol attempting to solve n independent copies of f under μn using o(n • I) communication, will succeed with probability 2-Ω(n). This is tight, as Braverman and Rao [BR11] previously showed that O(n • I) communication suffice to succeed with probability ~(2/3)n. We then show how the information odometer can be used to achieve the best possible information-theoretic privacy between two untrusted parties: If the players' goal is to compute a function f(x,y), and f admits a protocol with information cost is I and communication cost C, then our odometer can be used to produce a "robust" protocol which: (i) Assuming both players are honest, computes f with high probability, and (ii) Even if one party is malicious, then for any k∈N, the probability that the honest player reveals more than O(k • (I+log C)) bits of information to the other player is at most 2-Ω(k). Finally, we outline an approach which uses the odometer as a proxy for breaking state of the art interactive compression results: We show that our odometer allows to reduce interactive compression to the regime where I=O(log C), thereby opening a potential avenue for improving the compression result of [BBCR10] and to new direct sum and product theorems in communication complexity

An Interactive Information Odometer and Applications

Non UBCUnreviewedAuthor affiliation: Princeton UniversityGraduat