I. Kremer, "Quantum Communication", Master's Thesis, The Hebrew University of Jerusalem, 1995.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....complexity in the quantum setting. Unfortunately so far only few applicable lower bound methods for quantum protocols are known: the rank lower bound is known to hold for exact (i.e. errorless) quantum communication [4, 6] the (usually weak) discrepancy lower bound for bounded error protocols [18]. One breakthrough result of the eld of quantum computing is Grover s search algorithm that retrieves an item from an unordered list within O( questions [11] outperforming every classical algorithm for the problem. By an application of this search algorithm to communication complexity in [4] ....

....to the other, who then decides the function value. This has been investigated e.g. in [15] where a lower bound method based on the so called VC dimension is proved, which allows to prove an exponential advantage for 2 round classical compared to 1 round quantum communication complexity. Kremer [18] exhibits a Exponential gaps between quantum communication complexity and classical probabilistic communication complexity are known only for partial functions, and are possible without interaction [28, 4] complete problem for the class of problems with polylogarithmic quantum one way ....

I. Kremer. Quantum Communication. Master's thesis (Hebrew University), 1995.

....previously introduced in [CB97] in that model, Alice and Bob share an unlimited number of entangled EPRpairs before the communication even begins) The question about the complexity of protocols that allow a small error is by far more interesting. As far as lower bounds are concerned, Kremer [Kre95] based upon some ideas from the seminal paper [Yao93] proved an n) lower bound for IP n . This result was extended to the model with prior entanglement in [CDNT98] Klauck [Kla01] looked at the threshold predicates (D(s) s ) and exact predicates (D(s) s = and proved an = ....

....Of course, these norms are not invariant under unitary transformations. However, they are at least somewhat related to unitary invariant norms via the following (obvious) observation: jhA; Bij 1 (A) 1 (B) 3.3. Decomposition of quantum communication protocols Proposition 3. 3 ( Yao93, Kre95] Let P be a communication protocol of length c, and let U p be the unitary operator in the right hand side of (3) Then there exist linear operators A u on HA and B u on HB (u 2 f0; 1g ) such that for every vector a 2 HA and every vector b 2 HB , U p (jaij0ijbi) jA u (a)iju c ijB u ....

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I. Kremer. Quantum communication. Master's thesis, Hebrew University, Jerusalem, 1995. 19

.... While quite a lot of lower bound methods are known for classical communication complexity, so far only few lower bound methods for quantum protocols are known: the rank lower bound is known to hold for exact quantum communication [7] the discrepancy lower bound for bounded error protocols [25]. But the latter technique is not strong enough to prove e.g. a superlogarithmic lower bound on the bounded error quantum communication complexity of the disjointness problem DISJ (both players receive an incidence vector of a subset of f1; ng and have to decide whether the sets are not ....

....A very restricted model is one way communication, where only a monologue is transmitted from one player to the other, who decides the function value. This model has important applications for automata [14] and also allows to rephrase the Ne ciporuk lower bound on formula size nicely [22] Kremer [25] investigates quantum one way communication and exhibits a complete problem for polylog quantum communication (with bounded error) We show that the VCdimension lower bound of [26] can be extended to the (bounded error) quantum case. We get a tight bound by using the results of [3] on random ....

I. Kremer. Quantum Communication. Master's thesis (Hebrew University) , 1995.

....(see also [13] In 1993, Yao [17] introduced a variation of the above classical communication complexity scenarios, where the parties communicate with qubits,rather than with bits. Protocols in this model are at least as powerful as probabilistic protocols with independent random sources. Kremer [12] showed that, in this model, the communication complexity of IP is# (n) whenever the required correctness probability is 1 # for a constant 0 # # 1 2 (Kremer attributes the proof methodology to Yao) Cleve and Buhrman [8] see also [6] introduced another variation of the classical ....

....we consider the communication complexity of IP in two scenarios: with prior entanglement and qubit communication; and with prior entanglement and classical bit communication. As far as we know, the proof methodology of the lower bound in the qubit communication model without prior entanglement [12] does not carry over to either of these two models. Nevertheless, we show# (n) lower bounds in these models. To state our lower bounds more precisely, we introduce the following notation. Let f : 0, 1 n 0, 1 n # 0, 1 be a communication problem, and 0 # # 1 2 . Let Q # # (f) ....

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I. Kremer, "Quantum Communication", Master's Thesis, The Hebrew University of Jerusalem, 1995.

....also [13] In 1993, Yao [17] introduced a variation of the above classical communication complexity scenarios, where the parties communicate with qubits , rather than with bits. Protocols in this model are at least as powerful as probabilistic protocols with independent random sources. Kremer [12] showed that, in this model, the communication complexity of IP is Omega (n) whenever the required correctness probability is 1 Gamma for a constant 0 1 2 (Kremer attributes the proof methodology to Yao) Cleve and Buhrman [8] see also [6] introduced another variation of the classical ....

....we consider the communication complexity of IP in two scenarios: with prior entanglement and qubit communication; and with prior entanglement and classical bit communication. As far as we know, the proof methodology of the lower bound in the qubit communication model without prior entanglement [12] does not carry over to either of these two models. Nevertheless, we show Omega (n) lower bounds in these models. To state our lower bounds more precisely, we introduce the following notation. Let f : f0; 1g n Theta f0; 1g n f0; 1g be a communication problem, and 0 ffl 1 2 . Let Q ....

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I. Kremer, "Quantum Communication", Master's Thesis, The Hebrew University of Jerusalem, 1995.

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I. Kremer, "Quantum Communication", Master's Thesis, The Hebrew University of Jerusalem, 1995.

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I. Kremer, "Quantum Communication", Master's Thesis, The Hebrew University of Jerusalem, 1995.

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I. Kremer. Quantum communication. Master's thesis, Hebrew University, 1995.

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