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A strong direct product theorem for disjointness
 In 42nd ACM Symposium on Theory of Computing (STOC
, 2010
"... A strong direct product theorem says that if we want to compute k independent instances of a function, using less than k times the resources needed for one instance, then the overall success probability will be exponentially small in k. We establish such a theorem for the randomized communication co ..."
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A strong direct product theorem says that if we want to compute k independent instances of a function, using less than k times the resources needed for one instance, then the overall success probability will be exponentially small in k. We establish such a theorem for the randomized communication complexity of the Disjointness problem, i.e., with communication const · kn the success probability of solving k instances can only be exponentially small in k. We show that this bound even holds in an AM communication protocol with limited ambiguity. The main result implies a new lower bound for Disjointness in a restricted 3player NOF protocol, and optimal communicationspace tradeoffs for Boolean matrix product. Our main result follows from a solution to the dual of a linear programming problem, whose feasibility comes from a socalled Intersection Sampling Lemma that generalizes a result by Razborov [Raz92]. We also discuss a new lower bound technique for randomized communication complexity called the generalized rectangle bound that we use in our proof. 1
STRONG DIRECT PRODUCT THEOREMS FOR QUANTUM COMMUNICATION AND QUERY COMPLEXITY
"... ABSTRACT. A strong direct product theorem (SDPT) states that solving n instances of a problem requires ˝.n / times the resources for a single instance, even to achieve success probability 2 ˝.n / : We prove that quantum communication complexity obeys an SDPT whenever the communication lower bound fo ..."
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ABSTRACT. A strong direct product theorem (SDPT) states that solving n instances of a problem requires ˝.n / times the resources for a single instance, even to achieve success probability 2 ˝.n / : We prove that quantum communication complexity obeys an SDPT whenever the communication lower bound for a single instance is proved by the generalized discrepancy method, the strongest technique in that model. We prove that quantum query complexity obeys an SDPT whenever the query lower bound for a single instance is proved by the polynomial method, one of the two main techniques in that model. In both models, we prove the corresponding XOR lemmas and threshold direct product theorems. 1.
Quantum vs. classical readonce branching programs
, 504
"... Abstract. The paper presents the first nontrivial upper and lower bounds for (nonoblivious) quantum readonce branching programs. It is shown that the computational power of quantum and classical readonce branching programs is incomparable in the following sense: (i) A simple, explicit boolean func ..."
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Abstract. The paper presents the first nontrivial upper and lower bounds for (nonoblivious) quantum readonce branching programs. It is shown that the computational power of quantum and classical readonce branching programs is incomparable in the following sense: (i) A simple, explicit boolean function on 2n input bits is presented that is computable by errorfree quantum readonce branching programs of size O � n 3 � , while each classical randomized readonce branching program and each quantum OBDD for this function with bounded twosided error requires size 2 Ω(n). (ii) Quantum branching programs reading each input variable exactly once are shown to require size 2 Ω(n) for computing the setdisjointness function DISJn from communication complexity theory with twosided error bounded by a constant smaller than 1/2−2 √ 3/7. This function is trivially computable even by deterministic OBDDs of linear size. The technically most involved part is the proof of the lower bound in (ii). For this, a new model of quantum multipartition communication protocols is introduced and a suitable extension of the information cost technique of Jain, Radhakrishnan, and Sen (2003) to this model is presented. 1.