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Limitations of Quantum Advice and OneWay Communication
 Theory of Computing
, 2004
"... 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. ..."
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Cited by 59 (15 self)
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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.
Exponential separations for oneway quantum communication complexity, with applications to cryptography
 IN PROCEEDINGS OF 39TH ACM STOC
, 2007
"... We give an exponential separation between oneway quantum and classical communication protocols for a partial Boolean function (a variant of the Boolean Hidden Matching Problem of BarYossef et al.) Earlier such an exponential separation was known only for a relational problem. The communication pr ..."
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Cited by 55 (15 self)
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We give an exponential separation between oneway quantum and classical communication protocols for a partial Boolean function (a variant of the Boolean Hidden Matching Problem of BarYossef et al.) Earlier such an exponential separation was known only for a relational problem. The communication problem corresponds to a strong extractor that fails against a small amount of quantum information about its random source. Our proof uses the Fourier coefficients inequality of Kahn, Kalai, and Linial. We also give a number of applications of this separation. In particular, we show that there are privacy amplification schemes that are secure against classical adversaries but not against quantum adversaries; and we give the first example of a keyexpansion scheme in the model of boundedstorage cryptography that is secure against classical memorybounded adversaries but not against quantum ones.
Quantum communication complexity of blockcomposed functions. Available at arXiv:0710.0095v1
, 2007
"... A major open problem in communication complexity is whether or not quantum protocols can be exponentially more efficient than classical ones for computing a total Boolean function in the twoparty interactive model. The answer appears to be “No”. In 2002, Razborov proved this conjecture for so far th ..."
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Cited by 33 (1 self)
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A major open problem in communication complexity is whether or not quantum protocols can be exponentially more efficient than classical ones for computing a total Boolean function in the twoparty interactive model. The answer appears to be “No”. In 2002, Razborov proved this conjecture for so far the most general class of functions F (x, y) = fn(x1 · y1, x2 · y2,..., xn · yn), where fn is a symmetric Boolean function on n Boolean inputs, and xi, yi are the i’th bit of x and y, respectively. His elegant proof critically depends on the symmetry of fn. We develop a lowerbound method that does not require symmetry and prove the conjecture for a broader class of functions. Each of those functions F (x, y) is the “blockcomposition ” of a “building block ” gk: {0, 1} k × {0, 1} k → {0, 1}, and an fn: {0, 1} n → {0, 1}, such that F (x, y) = fn(gk(x1, y1), gk(x2, y2),..., gk(xn, yn)), where xi and yi are the i’th kbit block of x, y ∈ {0, 1} nk, respectively. We show that as long as gk itself is “hard ” enough, its blockcomposition with an arbitrary fn has polynomially related quantum and classical communication complexities. Our approach gives an alternative proof for Razborov’s result (albeit with a slightly weaker parameter), and establishes new quantum lower bounds. For example, when gk is the Inner Product function with k = Ω(log n), the deterministic communication complexity of its blockcomposition with any fn is asymptotically at most the quantum complexity to the power of 7.
Boundederror quantum state identification and exponential separations in communication complexity
 In Proc. of the 38th Symposium on Theory of Computing (STOC
, 2006
"... We consider the problem of boundederror quantum state identification: given either state α0 or state α1, we are required to output ‘0’, ‘1 ’ or ‘? ’ (“don’t know”), such that conditioned on outputting ‘0 ’ or ‘1’, our guess is correct with high probability. The goal is to maximize the probability o ..."
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We consider the problem of boundederror quantum state identification: given either state α0 or state α1, we are required to output ‘0’, ‘1 ’ or ‘? ’ (“don’t know”), such that conditioned on outputting ‘0 ’ or ‘1’, our guess is correct with high probability. The goal is to maximize the probability of not outputting ‘?’. We prove a direct product theorem: if we are given two such problems, with optimal probabilities a and b, respectively, and the states in the first problem are pure, then the optimal probability for the joint boundederror state identification problem is O(ab). Our proof is based on semidefinite programming duality. Using this result, we present two exponential separations in the simultaneous message passing model of communication complexity. First, we describe a relation that can be computed with O(log n) classical bits of communication in the presence of shared randomness, but needs Ω(n 1/3) communication if the parties don’t share randomness, even if communication is quantum. This shows the optimality of Yao’s recent exponential simulation of sharedrandomness protocols by quantum protocols without shared randomness. Combined with an earlier separation in the other direction due to BarYossef et al., this shows that the quantum SMP model is incomparable with the classical sharedrandomness SMP model. Second, we describe a relation that can be computed with O(log n) classical bits of communication in the presence of shared entanglement, but needs Ω((n / log n) 1/3) communication if the parties share randomness but no entanglement, even if communication is quantum. This is the first example in communication complexity of a situation where entanglement buys you much more than quantum communication.
BQP and the polynomial hierarchy
 in Proceedings of the 42nd ACM symposium on Theory of computing, STOC ’10
, 2010
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Multiparty pseudotelepathy
 Proceedings of the 8th International Workshop on Algorithms and Data Structures, Volume 2748 of Lecture Notes in Computer Science
, 2003
"... Quantum information processing is at the crossroads of physics, mathematics and computer science. It is concerned with that we can and cannot do with quantum information that goes beyond the abilities of classical information processing devices. Communication complexity is an area of classical compu ..."
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Quantum information processing is at the crossroads of physics, mathematics and computer science. It is concerned with that we can and cannot do with quantum information that goes beyond the abilities of classical information processing devices. Communication complexity is an area of classical computer science that aims at quantifying the amount of communication necessary to solve distributed computational problems. Quantum communication complexity uses quantum mechanics to reduce the amount of communication that would be classically required. Pseudotelepathy is a surprising application of quantum information processing to communication complexity. Thanks to entanglement, perhaps the most nonclassical manifestation of quantum mechanics, two or more quantum players can accomplish a distributed task with no need for communication whatsoever, which would be an impossible feat for classical players. After a detailed overview of the principle and purpose of pseudotelepathy, we present a survey of recent and nosorecent work on the subject. In particular, we describe and analyse all the pseudotelepathy games currently known to the authors.
Quantum oneway communication can be exponentially stronger than classical communication
 In STOC
, 2011
"... In STOC 1999, Raz presented a (partial) function for which there is a quantum protocol communicating only O(log n) qubits, but for which any classical (randomized, boundederror) protocol requires poly(n) bits of communication. That quantum protocol requires two rounds of communication. Ever since R ..."
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In STOC 1999, Raz presented a (partial) function for which there is a quantum protocol communicating only O(log n) qubits, but for which any classical (randomized, boundederror) protocol requires poly(n) bits of communication. That quantum protocol requires two rounds of communication. Ever since Raz’s paper it was open whether the same exponential separation can be achieved with a quantum protocol that uses only one round of communication. Here we settle this question in the affirmative. 1
Quantum communication cannot simulate a public coin
, 2004
"... Abstract We study the simultaneous message passing model of communication complexity. Building on the quantum fingerprinting protocol of Buhrman et al., Yao recently showed that a large class of efficient classical publiccoin protocols can be turned into efficient quantum protocols without public ..."
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Abstract We study the simultaneous message passing model of communication complexity. Building on the quantum fingerprinting protocol of Buhrman et al., Yao recently showed that a large class of efficient classical publiccoin protocols can be turned into efficient quantum protocols without public coin. This raises the question whether this can be done always, i.e. whether quantum communication can always replace a public coin in the SMP model. We answer this question in the negative, exhibiting a communication problem where classical communication with public coin is exponentially more efficient than quantum communication. Together with a separation in the other direction due to BarYossef et al., this shows that the quantum SMP model is incomparable with the classical publiccoin SMP model. In addition we give a characterization of the power of quantum fingerprinting by means of a connection to geometrical tools from machine learning, a quadratic improvement of Yao's simulation, and a nearly tight analysis of the Hamming distance problem from Yao's paper.
Classical interaction cannot replace quantum nonlocality
, 2009
"... We present a twoplayer communication task that can be solved by a protocol of polylogarithmic cost in the simultaneous message passing model with classical communication and shared entanglement, but requires exponentially more communication in the classical interactive model. Our second result is a ..."
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Cited by 14 (1 self)
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We present a twoplayer communication task that can be solved by a protocol of polylogarithmic cost in the simultaneous message passing model with classical communication and shared entanglement, but requires exponentially more communication in the classical interactive model. Our second result is a twoplayer nonlocality game with input length n and output of polylogarithmic length, that can be won with probability 1 − o(1) by players sharing polylogarithmic amount of entanglement. On the other hand, the game is lost with probability Ω (1) by players without entanglement, even if they are allowed to exchange up to k bits in interactive communication for certain k ∈ ˜ Ω ( n 1/4). These two results give almost the strongest possible (and the strongest known) indication of nonlocal properties of twoparty entanglement.