H. Buhrman, R. Cleve, and A. Wigderson. Quantum vs. classical communication and computation (preliminary version) . In Proceedings of 30th STOC, pages 63--68, 1998.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....task by communicating with each other. Remarkably, it has been shown that if the two parties are equipped with a quantum computer and can communicate by exchanging quantum states, they can solve certain tasks at a significantly smaller communication cost, when compared to classical protocols [6, 1, 18]. This is especially surprising since an early result due to Holevo [10] later explained in simpler terms by Nayak [16] rules out obvious methods of compressing classical information into succinct quantum messages Holevo s theorem implies that n quantum bits of communication are necessary to ....

H. Buhrman, R. Cleve, and A. Wigderson. Quantum vs. classical communication and computation. In Proceedings of the 30th Annual ACM Symposium on Theory of Computing, pages 63--68, 1998.

....fellowship and by NSF grant CCR 95 04436. Supported by NSF Career Grant 97 03399 and the Alfred P. Sloan Foundation. 1 Introduction 1. 1 Motivation In recent years many researchers have investigated the power of quantum computers which can query a black box oracle for an unknown function [4, 5, 8, 9, 10, 13, 15, 17, 18, 20, 27, 32]. The broad goal of research in this area is to understand the relationship betwen the number of quantum versus classical oracle queries which are required to answer various questions about the function computed by the oracle. For example, a well known result due to Deutsch and Jozsa [15] shows ....

.... computes a constant Boolean function or a function which is balanced between outputs 0 and 1: More recently, several researchers have studied the number of quantum oracle queries which are required to determine whether or not the function computed by a black box oracle ever assumes a nonzero value [4, 5, 8, 13, 20, 32]. A natural question which arises within this framework is the following: what is the relationship between the number of quantum versus classical oracle queries which are required in order to exactly identify the function computed by a black box oracle Here the goal is not to determine whether a ....

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H. Buhrman, R. Cleve and A. Wigderson. Quantum vs. classical communication and computation, in \Proc. 30th ACM Symp. on Theory of Computing," (

....The model of quantum communication complexity was also introduced by Yao [Yao93] Suppose that Alice and Bob can employ the laws of quantum mechanics and are allowed to exchange qubits instead of classical bits. Can it help them to reduce the amount of communication Buhrman, Cleve and Wigderson [BCW98] observed that the rank lower bound for deterministic protocols extends to the quantum case (so, after all the answer for such protocols can be NO ) In particular, both DISJ n and IP n require n) qubits to be exchanged by quantum deterministic ( zero error) protocols. The rank lower bound ....

.... this work was 12 n) AST 98, BW01] we can also mention some partial results in this direction such as bounds for constant round protocols [KNTZ01] protocols with exponentially small error [BW01] and some highly structured protocols [HW02] On the upper bounds frontier, the elegant paper [BCW98] established a strong connection between quantum search and quantum communication by showing how to convert every quantum search algorithm for any Boolean function g into a quantum communication algorithm for the associated predicate f g (x; y) g(x y) with only a logarithmic delay. Plugging ....

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H. Buhrman, R. Cleve, and A. Wigderson. Quantum vs. classical communication and computation. In Proceedings of the 30th ACM Symposium on the Theory of Computing, pages 63-86, 1998. Preliminary version available at quant-ph/9802040.

.... intriguing facts about communication using quantum states is that these states cannot be used to transmit more classical bits than the number of qubits used, yet in some scenarios there are ways of conveying information with much fewer, even exponentially fewer, qubits than possible classically [1], 2] 3] Moreover, some of these methods have a very simple structure they involve only few message exchanges between the communicating parties. We consider the question as to whether every classical protocol may be transformed to a simpler quantum protocol one that has similar eciency, but ....

....solve via classical communication. Both are examples of problems for which exponentially fewer quantum bits are required to accomplish a communication task, as compared to classical bits. A third example is the O( N log N) qubit protocol for Set Disjointness due to Buhrman, Cleve, and Wigderson [1], which represents quadratic savings in communication cost over classical protocols. The protocols presented by [2] 3] share the feature that they require minimal interaction between the communicating players. For example, in the protocol of [2] one player prepares a set of qubits in a certain ....

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H. Buhrman, R. Cleve, and A. Wigderson, \Quantum vs. classical communication and computation," in Proceedings of the 30th Annual ACM Symposium on Theory of Computing, 1998, pp. 63-68.

....between x and y is Delta(x; y) n=2. We call the first kind of (x; y) 1 inputs and the second kind 0 inputs . The aim of Alice and Bob is to determine whether x = y, using as little communication as possible. Here s a quantum protocol for EQ 0 with only log n qubits of communication from [BCW98], based on the Deutsch Jozsa algorithm: 1. Alice prepares the log n qubit state H Omega log n j0 : 0i, to which she applies the unitary transformation jii ( Gamma1) x i jii. She now has the state 1 p n X i2f0;1g log n ( Gamma1) x i jii; which she sends to Bob. 2. Bob applies ....

H. Buhrman, R. Cleve, and A. Wigderson. Quantum vs. classical communication and computation (preliminary version). In Proceedings of 30th STOC, pages 63--68,

....in a certain state and sends half of them across as the message, after which both players measure their qubits to obtain the result. In contrast, e#cient quantum protocols for problems such as checking set disjointness (DISJ) seem to require much more interaction: Buhrman, Cleve and Wigderson [4] give an O( # N log N) qubit protocol for DISJ that has O( # N) message exchanges. This represents quadratic savings in communication cost, but also an unbounded increase in the number of messages exchanged (from one message to # N ) as compared to classical protocols. Are there simpler protocols ....

H. Buhrman, R. Cleve, and A. Wigderson. Quantum vs. classical communication and computation. In Proceedings of the Thirtieth Annual ACM Symposium on Theory of Computing, New York, NY, May 23--26, 1998. ACM Press.

....and sends half of them across as the message, after which both players measure their qubits to obtain the result. In contrast, e# cient quantum protocols for computing total functions such as checking Set Disjointness (DISJ) seem to require much more interaction: Buhrman, Cleve, and Wigderson [6] give an O( # N log N) qubit protocol for DISJ that has O( # N ) message exchanges. This represents quadratic savings in communication cost, but also an unbounded increase in the number of messages exchanged (from one message to # N ) as compared to classical protocols. Are there simpler ....

....of the disjointness problem. Here Alice and Bob each receive the incidence vector of a subset of a size n universe. They reject i# the sets are disjoint. It is known the Q 1 # (DISJ) # (1 H(#) n [14, 7] Furthermore Q O( # n) 1 3 (DISJ) O( # n log n) by an application of Grover search [6]. We now prove a lower bound by reduction. Proof of Corollary 1.3. Suppose we are given a k round quantum protocol for the disjointness problem having error 1 3 and using c qubits. W.l.o.g. we can assume Bob starts the communication, because the problem is symmetrical, and that k is even. We ....

H. Buhrman, R. Cleve, and A. Wigderson. Quantum vs. classical communication and computation. In Proceedings of the Thirtieth Annual ACM Symposium on Theory of Computing, 1998.

....from an information theory perspective, they are different when viewed from a computational complexity perspective. 1. Introduction 1.1. Motivation In recent years many researchers have investigated the power of quantum computers which can query a black box oracle for an unknown function [1, 5, 6, 9, 14, 10, 11, 15, 17, 20, 21, 23, 32, 37]. The broad goal of research in this area is to understand the relationship between the number of quantum versus classical oracle queries which are required to answer various questions about the function computed by the oracle. For example, a well known result due to Deutsch and Jozsa [17] shows ....

.... oracle computes a constant Boolean function or a function which is balanced between outputs 0 and 1: More recently, several researchers have studied the number of quantum oracle queries which are required to determine whether the function computed by a black box oracle is identically zero [5, 6, 9, 15, 23, 37]. A natural question which arises in this framework is the following: what is the relationship between the number of quantum versus classical oracle queries which are required in order to exactly identify the function computed by a black box oracle Here the goal is not to determine whether a ....

[Article contains additional citation context not shown here]

H. Buhrman, R. Cleve and A. Wigderson. Quantum vs. classical communication and computation, in "Proc. 30th ACM Symp. on Theory of Computing," (

....to Bshouty and Jackson [10] who defined a quantum PAC oracle and gave an efficient algorithm for learning DNF from a uniform distribution quantum PAC oracle. In a different community, many complexity theory researchers have studied the power of quantum computation with a black box quantum oracle [3 8, 11, 13, 16, 28, 31]. This research has focused on understanding the relationship between the number of quantum versus classical black box oracle queries required to determine whether or not a black box function has some particular property such as ever taking a nonzero value [4, 6, 11, 16, 31] or being evenly ....

.... quantum oracle [3 8, 11, 13, 16, 28, 31] This research has focused on understanding the relationship between the number of quantum versus classical black box oracle queries required to determine whether or not a black box function has some particular property such as ever taking a nonzero value [4, 6, 11, 16, 31] or being evenly balanced between the outputs zero and one [13] Since a classical black box oracle query to a Boolean function is the same thing as a membership query in learning theory, in light of the work described above it is natural to consider a model of learning from quantum membership ....

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H. Buhrman, R. Cleve and A. Wigderson. Quantum vs. classical communication and computation, in "Proc. 30th Symp. on Theory of Comp." (1998), 63-68.

....and sends half of them across as the message, after which both players measure their qubits to obtain the result. In contrast, e cient quantum protocols for computing total functions such as checking Set Disjointness (DISJ) seem to require much more interaction: Buhrman, Cleve, and Wigderson [6] give an O( p N log N) qubit protocol for DISJ that has O( p N) message exchanges. This represents quadratic savings in communication cost, but also an unbounded increase in the number of messages exchanged (from one message to p N ) as compared to classical protocols. Are there simpler ....

....of the disjointness problem. Here Alice and Bob each receive the incidence vector of a subset of a size n universe. They reject i the sets are disjoint. It is known the Q 1 (DISJ) 1 H( n [14, 7] Furthermore Q O( p n) 1=3 (DISJ) O( p n log n) by an application of Grover search [6]. We now prove a lower bound by reduction. Proof of Corollary 1.3. Suppose we are given a k round quantum protocol for the disjointness problem having error 1=3 and using c qubits. W.l.o.g. we can assume Bob starts the communication, because the problem is symmetrical, and that k is even. We ....

H. Buhrman, R. Cleve, and A. Wigderson. Quantum vs. classical communication and computation. In Proceedings of the Thirtieth Annual ACM Symposium on Theory of Computing, 1998.

....related problems inverting a permutation (often used to model one way permutation) and AND of ORs only weaker lower bounds have been known. Both of these problems can be solved using Grover s algorithm with O( p N) queries for inverting a permutation and O( p N log N) queries for AND of ORs[8]. However, the best lower bounds have been 3 p N) 4] and 4 p N ) respectively. We present a new method for proving lower bounds on quantum query algorithms and use it to prove p N) lower bounds for inverting a permutation and AND of ORs. It also provides a uni ed proof for several ....

....application, consider AND of ORs: f(x 1 ; xN ) x 1 ORx 2 : ORx p N )AND (x p N 1 : x 2 p N )AND : AND(x N p N 1 OR : ORxN ) where x 1 ; xN 2 f0; 1g. f can be computed with O( p N log N) queries by a two level version of Grover s algorithm (see [8]) However, a straightforward application of lower bound methods from [4, 3] only gives an 4 p N) bound because the block sensitivity of f is ( p N) and the lower bound on the number of queries given by hybrid or polynomials method is the square root of block sensitivity[3, 23] Our method ....

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H. Buhrman, R. Cleve, A. Wigderson. Quantum vs. classical communication and computation. Proceedings of STOC'98, pages 63-68. Also quant-ph/9802046.

....solution of given problems can be cast (and possibly simplified) in the new setting. Although quantum computers do not exist, simulation of algorithms suggested by this paradigm on standard machines may offer practical alternatives for specific problems. A discussion of the ideas can be found in [1,2,3,4,6,8] and references therein. In this paper we focus on some early work in formulating a computational model for distinguishing states of a system. The outline of the paper is as follows. After a brief discussion of our model in this section we then conduct a brief review of quantum information theory ....

Buhrman H, Cleve R and Wigderson A, Quantum vs. Classical Communication and Computation, in Proc. of the 30th Annual ACM Symposium on Theory of Computing (STOC) (1998)

....in order to compute the function at hand. What happens if we generalize this setting to the quantum world and allow Alice and Bob the use of quantum computers and qubit communication It turns out that some tasks can be solved with significantly less communication if we allow such quantization [10] and this despite a fact known as Holevo s theorem stating that quantum bits cannot contain more information than classical bits It turns out there are similar advantages by sticking to classical communication, but allowing Alice and Bob the use of pre established entangled qubits. ....

H. Buhrman, R. Cleve, and A. Wigderson. Quantum vs. classical communication and computation. In Proc. 30th ACM Symp. Theor. Comput., 1998, 63--68.

....EPR pair, they are able to perform quantum teleportation, a process useful for communicating quantum information. Using protocols involving the sharing of EPR pairs, some distributed computation tasks can be achieved using fewer bits than could be achieved using only a classical channel (see e.g. [BCW98] and [Raz99] Suppose Alice and Bob share a known entangled pair of qubits j i = 00 j0ij0i 01 j0ij1i 10 j1ij0i 11 j1ij1i; where the rst qubit is in Alice s possession and the second qubit in Bob s. The Schmidt decomposition for this bipartite system allows us to express the state ....

Harry Buhrman, Richard Cleve, Avi Wigderson. \Quantum vs. Classical Communication and Computation" in Proceedings of the 30th Annual ACM Symposium on Theory of Computing (STOC98), pages 63-68. On the quant-ph archive, report no. 9705033.

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H. Buhrman, R. Cleve, and A. Wigderson. Quantum vs. classical communication and computation (preliminary version) . In Proceedings of 30th STOC, pages 63--68, 1998.

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H. Buhrman, R. Cleve, and A. Wigderson. Quantum vs. classical communication and computation (preliminary version) . In Proceedings of 30th STOC, pages 63--68, 1998.

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H. Buhrman, R. Cleve, and A. Wigderson. Quantum vs. classical communication and computation. In Proceedings of the Thirtieth Annual ACM Symposium on Theory of Computing, pages 63--68, 1998.

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H. Buhrman, R. Cleve, and A. Wigderson. Quantum vs. classical communication and computation (preliminary version) . In Proceedings of 30th STOC, pages 63--68, 1998.

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H. Buhrman, R. Cleve, and A. Wigderson. Quantum vs. classical communication and computation. In ACM Symposium on Theory of Computing (STOC), May 1998.

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H. Buhrman, R. Cleve, and A. Wigderson. Quantum vs. classical communication and computation. In ACM Symposium on Theory of Computing (STOC), May 1998.

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H. Buhrman, R. Cleve, and A. Wigderson. Quantum vs. classical communication and computation. In ACM Symposium on Theory of Computing (STOC), pages 63--68, 1998.

.... protocols for this model can be translated to the model where both parties share EPR pairs and communicate classically, since via teleportation, Theorem 2 gives us: 2C (f) Q(f ) 7 The first gap for two party qubit communication complexity was demonstrated by Buhrman, Cleve, and Wigderson [BCW98] They showed for a promise version of the equality problem 3 , EQ 0 see section 5.2 for a definition, that Q(EQ 0 ) O(log(n) and that also C(EQ 0 ) Omega Gamma n) This exhibits an exponential gap between classical and quantum communication complexity. In section 5 we will show in ....

....that there is a promise problem f such that Q(f) O(log(n) but R 2 (f) Omega Gamma p n) Theorem 3 The best known gaps between Quantum and Classical communication complexity are: 1. There exists a promise problem EQ 0 , such that Q(EQ 0 ) O(log(n) but C(EQ 0 ) Omega Gamma n) BCW98] 3 EQ(x;y) 1 if x = y and 0 otherwise. EQ requires n bits of communication. A promise version of a problem means that Alice and Bob are only required to compute the answer correctly on certain instances that fall within the promise and it doesn t matter what they compute on the other ....

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H. Buhrman, R. Cleve, and A. Wigderson. Quantum vs. classical communication and computation. In The Thirtieth Annual ACM Symposium on Theory of Computing, to appear in

....scheme with a shared key would still require fingerprints of length Omega Gamma n) On the other hand, there is a scheme with a shared quantum key of O(log n) Bell states that requires fingerprints of length only O(log n) bits. See [BCT99] for details (the results are partly based on results in [BCW98, FR87]) It should be noted that if the exactness condition is relaxed to one where the error probability must be O(1=n c ) for a constant c) then there exists also a classical scheme with classical keys and fingerprints of length O(log n) Acknowledgments We thank John Preskill for references to ....

H. Buhrman, R. Cleve, and A. Wigderson. Quantum vs. classical communication and computation. In Proceedings of 30th ACM STOC, pages 63--68,

....decrease in communication complexity was established for the Disjoint function Disjoint(x, y) # 0 if there exists an i such that x i = y i = 1, 1 otherwise. 5. 1) This well studied problem has a classical probabilistic communication complexity of ## n) 18, 28] while the authors of [5] gave a qubit protocol requiring only O( # n log n) qubits of communication. The question whether there exists a more e#cient quantum protocol for Disjoint is still an important open problem. This is especially relevant as 1840 HARRY BUHRMAN, RICHARD CLEVE, AND WIM VAN DAM Disjoint is a complete ....

....The question whether there exists a more e#cient quantum protocol for Disjoint is still an important open problem. This is especially relevant as 1840 HARRY BUHRMAN, RICHARD CLEVE, AND WIM VAN DAM Disjoint is a complete problem for the communication class co NP [2] The same article [5] also contained the first exponential separation for the exact distributed computation of a partial two party function that is related to the Deutsch Jozsa problem of [11] In [27] Raz improved on these results by establishing an exponential separation between classical and quantum communication ....

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H. Buhrman, R. Cleve, and A. Wigderson, Quantum vs. classical communication and computation, in Proceedings of the 30th Annual ACM Symposium on Theory of Computing, 1998, pp. 63--68.

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H. Buhrman, R. Cleve, and A. Wigderson. Quantum vs. classical communication and computation. In Proceedings of 30th STOC, pages 63--68, 1998.

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H. Buhrman, R.Cleve, and A. Wigderson. Quantum vs. classical communication and computation. In 30'th STOC, pages 63-68, 1998.

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H. Buhrman, R. Cleve, and A. Wigderson. Quantum vs. classical communication and computation. In Proceedings of the 30th Annual ACM Symposium on Theory of Computing, pages 63-68, 1998. 18

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H. Buhrman, R. Cleve, and A. Wigderson. Quantum vs. classical communication and computation. In Proceedings of 30th ACM STOC, pages 63-68, 1998. quantph /9802040.

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Buhrman H, Cleve R and Wigderson A, Quantum vs. Classical Communication and Computation, in Proc. of the 30th Annual ACM Symposium on Theory of Computing (STOC) (1998)

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