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Lower Bounds for Quantum Communication Complexity
 42nd IEEE Symposium on Foundations of Computer Science
"... Abstract. We prove lower bounds on the bounded error quantum communication complexity. Our methods are based on the Fourier transform of the considered functions. First we generalize a method for proving classical communication complexity lower bounds developed by Raz [35] to the quantum case. Apply ..."
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Cited by 54 (7 self)
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Abstract. We prove lower bounds on the bounded error quantum communication complexity. Our methods are based on the Fourier transform of the considered functions. First we generalize a method for proving classical communication complexity lower bounds developed by Raz [35] to the quantum case. Applying this method we give an exponential separation between bounded error quantum communication complexity and nondeterministic quantum communication complexity. We develop several other lower bound methods based on the Fourier transform, notably showing that√ s̄(f) / logn, for the average sensitivity s̄(f) of a function f, yields a lower bound on the bounded error quantum communication complexity of f((x ∧ y) ⊕ z), where x is a Boolean word held by Alice and y, z are Boolean words held by Bob. We then prove the first large lower bounds on the bounded error quantum communication complexity of functions, for which a polynomial quantum speedup is possible. For all the functions we investigate, the only previously applied general lower bound method based on discrepancy yields bounds that are O(log n).
On quantum and probabilistic communication: Las Vegas and oneway protocols
 in Proceedings of the 32nd Annual ACM Symposium on Theory of Computing, 2000
, 2000
"... We investigate the power of quantum communication protocols compared to classical probabilistic protocols. In our first result we describe a total Boolean function that has a quantum Las Vegas protocol communicating at most O(N^{10/11+ epsilon}) qubits for all epsilon > 0, while any classical pro ..."
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Cited by 43 (6 self)
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We investigate the power of quantum communication protocols compared to classical probabilistic protocols. In our first result we describe a total Boolean function that has a quantum Las Vegas protocol communicating at most O(N^{10/11+ epsilon}) qubits for all epsilon > 0, while any classical probabilistic protocol (with bounded error) needs Omega(N/log N) bits. Then we investigate quantum oneway communication complexity. First we show that the VCdimension lower bound on oneway probabilistic communication of [26] holds for quantum protocols, too. Then we prove that for oneway protocols computing total functions quantum Las Vegas communication is asymptotically as efficient as exact quantum communication, which is exactly as efficient as deterministic communication. We describe applications of the lower bounds for oneway communication complexity to quantum finite automata and quantum formulae.
Quantum Communication and Complexity
 Theoretical Computer Science
, 2000
"... In the setting of communication complexity, two distributed parties want to compute a function depending on both their inputs, using as little communication as possible. The required communication can sometimes be significantly lowered if we allow the parties the use of quantum communication. We sur ..."
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Cited by 36 (15 self)
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In the setting of communication complexity, two distributed parties want to compute a function depending on both their inputs, using as little communication as possible. The required communication can sometimes be significantly lowered if we allow the parties the use of quantum communication. We survey the main results of the young area of quantum communication complexity: its relation to teleportation and dense coding, the main examples of fast quantum communication protocols, lower bounds, and some applications. 1 Introduction The area of communication complexity deals with the following type of problem. There are two separated parties, called Alice and Bob. Alice receives some input x 2 X, Bob receives some y 2 Y , and together they want to compute some function f(x; y). As the value f(x; y) will generally depend on both x and y, neither Alice nor Bob will have sufficient information to do the computation by themselves, so they will have to communicate in order to achieve their go...
Improved quantum communication complexity bounds for disjointness and equality
 In Proc. Intl. Symp. on Theoretical Aspects of Computer Science (STACS
, 2002
"... Abstract. We prove new bounds on the quantum communication complexity of the disjointness and equality problems. For the case of exact and nondeterministic protocols we show that these complexities are all equal to n+1, the previous best lower bound being n/2. We show this by improving a general bo ..."
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Cited by 30 (5 self)
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Abstract. We prove new bounds on the quantum communication complexity of the disjointness and equality problems. For the case of exact and nondeterministic protocols we show that these complexities are all equal to n+1, the previous best lower bound being n/2. We show this by improving a general bound for nondeterministic protocols of de Wolf. We also give an O ( √ n·c log ∗ n)qubit boundederror protocol for disjointness, modifying and improving the earlier O ( √ n log n) protocol of Buhrman, Cleve, and Wigderson, and prove an Ω ( √ n) lower bound for a class of protocols that includes the BCWprotocol as well as our new protocol. 1
Quantum communication complexity
 In Proc. Intl. Colloquium on Automata, Languages, and Programming (ICALP
, 2000
"... This paper surveys the field of quantum communication complexity. Some interesting recent results are collected concerning relations to classical communication, lower bound methods, oneway communication, and applications of quantum communication complexity. 1 ..."
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Cited by 19 (3 self)
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This paper surveys the field of quantum communication complexity. Some interesting recent results are collected concerning relations to classical communication, lower bound methods, oneway communication, and applications of quantum communication complexity. 1
Quantum communication complexity
 Foundations of Physics
"... Can quantum communication be more efficient than its classical counterpart? Holevo’s theorem rules out the possibility of communicating more than n bits of classical information by the transmission of n quantum bits—unless the two parties are entangled, in which case twice as many classical bits can ..."
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Cited by 19 (6 self)
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Can quantum communication be more efficient than its classical counterpart? Holevo’s theorem rules out the possibility of communicating more than n bits of classical information by the transmission of n quantum bits—unless the two parties are entangled, in which case twice as many classical bits can be communicated but no more. In apparent contradiction, there are distributed computational tasks for which quantum communication cannot be simulated efficiently by classical means. In some cases, the effect of transmitting quantum bits cannot be achieved classically short of transmitting an exponentially larger number of bits. In a similar vein, can entanglement be used to save on classical communication? It is well known that entanglement on its own is useless for the transmission of information. Yet, there are distributed tasks that cannot be accomplished at all in a classical world when communication is not allowed, but that become possible if the noncommunicating parties share prior entanglement. This leads to the question of how expensive it is, in terms of classical communication, to provide an exact simulation of the spooky power of entanglement. KEY WORDS: Bell’s theorem; communication complexity; distributed computation; entanglement simulation; pseudotelepathy; spooky communication.
On Rounds in Quantum Communication
 IEEE Transactions on Information Theory
, 2000
"... We investigate the power of interaction in two player quantum communication protocols. Our main result is a roundscommunication hierarchy for the pointer jumping function f k . We show that f k needs quantum communication n) if Bob starts the communication and the number of rounds is limited to k ( ..."
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Cited by 16 (3 self)
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We investigate the power of interaction in two player quantum communication protocols. Our main result is a roundscommunication hierarchy for the pointer jumping function f k . We show that f k needs quantum communication n) if Bob starts the communication and the number of rounds is limited to k (for any constant k). Trivially, if Alice starts, O(k log n) communication in k rounds suces. The lower bound employs a result relating the relative von Neumann entropy between density matrices to their trace distance and uses a new measure of information.
Nondeterministic quantum query and communication complexities
 SIAM JOURNAL ON COMPUTING
, 2003
"... We study nondeterministic quantum algorithms for Boolean functions f. Such algorithms have positive acceptance probability on input x iff f(x) = 1. In the setting of query complexity, we show that the nondeterministic quantum complexity of a Boolean function is equal to its “nondeterministic poly ..."
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Cited by 13 (0 self)
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We study nondeterministic quantum algorithms for Boolean functions f. Such algorithms have positive acceptance probability on input x iff f(x) = 1. In the setting of query complexity, we show that the nondeterministic quantum complexity of a Boolean function is equal to its “nondeterministic polynomial ” degree. We also prove a quantumvs.classical gap of 1 vs. n for nondeterministic query complexity for a total function. In the setting of communication complexity, we show that the nondeterministic quantum complexity of a twoparty function is equal to the logarithm of the rank of a nondeterministic version of the communication matrix. This implies that the quantum communication complexities of the equality and disjointness functions are n + 1 if we do not allow any error probability. We also exhibit a total function in which the nondeterministic quantum communication complexity is exponentially smaller than its classical counterpart.
What Can Be Observed Locally? RoundBased Models for Quantum Distributed Computing
 COMPUTING, IN "23RD INTERNATIONAL SYMPOSIUM ON DISTRIBUTED COMPUTING (DISC) DISC, ESPAGNE ELCHE/ELX
, 2009
"... It is a wellknown fact that, by resorting to quantum processing in addition to manipulating classical information, it is possible to reduce the time complexity of some centralized algorithms, and also to decrease the bit size of messages exchanged in tasks requiring communication among several agen ..."
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Cited by 3 (0 self)
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It is a wellknown fact that, by resorting to quantum processing in addition to manipulating classical information, it is possible to reduce the time complexity of some centralized algorithms, and also to decrease the bit size of messages exchanged in tasks requiring communication among several agents. Recently, several claims have been made that certain fundamental problems of distributed computing, including Leader Election and Distributed Consensus, begin to admit feasible and efficient solutions when the model of distributed computation is extended so as to apply quantum processing. This has been achieved in one of two distinct ways: (1) by initializing the system in a quantum entangled state, and/or (2) by applying quantum communication channels. In this paper, we explain why some of these prior claims are misleading, in the sense that they rely on changes to the model unrelated to quantum processing. On the positive side, we consider the aforementioned quantum extensions when applied to Linial’s wellestablished LOCAL model of distributed computing. For both types of extensions, we put forward valid proofofconcept examples of distributed problems whose round complexity