A locally decodable code encodes n-bit strings x in m-bit codewords C(x), in such a way that one can recover any bit x i from a corrupted codeword by querying only a few bits of that word. We use a quantum argument to prove that LDCs with 2 classical queries need exponential length: m = 2 Ω(n) . Previously this was known only for linear codes (Goldreich et al. 02). Our proof shows that a 2-query LDC can be decoded with only 1 quantum query, and then proves an exponential lower bound for such 1-query locally quantum-decodable codes. We also show that q quantum queries allow more succinct LDCs than the best known LDCs with q classical queries. Finally, we give new classical lower bounds and quantum upper bounds for the setting of private information retrieval. In particular, we exhibit a quantum 2-server PIR scheme with O(n 3/10 ) qubits of communication, improving upon the O(n 1/3 ) bits of communication of the best known classical 2-server PIR.

Abstract. 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 = x i ⊕ x j . We prove that the quantum one-way communication complexity of HMn is O(log n), 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(log n) 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( 3 √ n log n) bits of communication.Key words. Communication complexity, quantum computation, separation, hidden matching AMS subject classifications. 68P30,68Q15,68Q17,81P681. Introduction. The investigation of the strength and limitations of quantum computing has become an important field of study in theoretical computer science. The celebrated algorithm of Shor [22] for factoring numbers in polynomial time on a quantum computer gives strong evidence that quantum computers are more powerful than classical ones. The further study of the relationship between quantum and classical computing in models like black-box computation, communication complexity, and interactive proof systems help towards a better understanding of quantum and classical computing.In this paper we answer an open question about the relative power of quantum one-way communication protocols. We describe a problem which can be solved by a quantum one-way communication protocol exponentially faster than any classical one. No asymptotic gap was previously known. We prove a similar result in the model of Simultaneous Messages.Communication complexity, defined by Yao [23] in 1979, is a central model of computation with numerous applications. It has been used for proving lower bounds in many areas including Boolean circuits, time-space tradeoffs, data structures, automata, formulae size, etc. Examples of these applications can be found in the textbook of Kushilevitz and Nisan [15]. A communication complexity problem is defined by three sets X, Y, Z and a relation R ⊆ X ×Y ×Z. Two computationally all-powerful players, Alice and Bob, are given inputs x ∈ X and y ∈ Y , respectively. Neither of the players has any information about the other player's input. Alice and Bob exchange

We give an exponential separation between one-way quantum and classical communication protocols for two partial Boolean functions, both of which are variants of the Boolean Hidden Matching Problem of Bar-Yossef et al. Earlier such an exponential separation was known only for a relational version of the Hidden Matching Problem. Our proofs use the Fourier coefficients inequality of Kahn, Kalai, and Linial. We give a number of applications of this separation. In particular, in the bounded-storage model of cryptography we exhibit a scheme that is secure against adversaries with a certain amount of classical storage, but insecure against adversaries with a similar (or even much smaller) amount of quantum storage; in the setting of privacy amplification, we show that there are strong extractors that yield a classically secure key, but are insecure against a quantum adversary.