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Quantum and Classical Strong Direct Product Theorems and Optimal TimeSpace Tradeoffs
 SIAM Journal on Computing
, 2004
"... 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 our overall success probability will be exponentially small in k. We establish such theorems for the classical as well as quantum ..."
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Cited by 53 (10 self)
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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 our overall success probability will be exponentially small in k. We establish such theorems for the classical as well as quantum query complexity of the OR function. This implies slightly weaker direct product results for all total functions. We prove a similar result for quantum communication protocols computing k instances of the Disjointness function. Our direct product theorems...
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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Cited by 26 (1 self)
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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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Cited by 15 (3 self)
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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 strict sense: (i) A simple, explicit boole ..."
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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 strict 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 for this function with bounded twosided error requires size 2 Ω(n). (ii) Quantum readonce branching programs with twosided error bounded by a constant smaller than 1/2 − 2 √ 3/7 ≈ 0.005 are shown to require size 2 Ω(n) for the setdisjointness function DISJn from communication complexity theory. 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.
SpaceBounded Communication Complexity
"... In the past thirty years, Communication Complexity has emerged as a foundational tool to proving lower bounds in many areas of computer science. Its power comes from its generality, but this generality comes at a price—no superlinear communication lower bound is possible, since a player may communic ..."
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In the past thirty years, Communication Complexity has emerged as a foundational tool to proving lower bounds in many areas of computer science. Its power comes from its generality, but this generality comes at a price—no superlinear communication lower bound is possible, since a player may communicate his entire input. However, what if the players are limited in their ability to recall parts of their interaction? We introduce memory models for 2party communication complexity. Our general model is as follows: two computationally unrestricted players, Alice and Bob, each have s(n) bits of memory. When a player receives a bit of communication, he “compresses ” his state. This compression may be an arbitrary function of his current memory contents, his input, and the bit of communication just received; the only restriction is that the compression must return at most s(n) bits. We obtain memory hierarchy theorems (also comparing this general model with its restricted variants), and show superlinear lower bounds for some explicit (nonboolean) functions. Our main conceptual and technical contribution concerns the following variant. The com
GoetheUniversität Frankfurt
, 2007
"... A basic question in complexity theory is whether the computational resources required for solving k independent instances of the same problem scale as k times the resources required for one instance. We investigate this question in various models of classical communication complexity. We introduce a ..."
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A basic question in complexity theory is whether the computational resources required for solving k independent instances of the same problem scale as k times the resources required for one instance. We investigate this question in various models of classical communication complexity. We introduce a new measure, the subdistribution bound, which is a relaxation of the wellstudied rectangle or corruption bound in communication complexity. We nonetheless show that for the communication complexity of Boolean functions with constant error, the subdistribution bound is the same as the latter measure, up to a constant factor. We prove that the oneway version of this bound tightly captures the oneway publiccoin randomized communication complexity of any relation, and the twoway version bounds the twoway publiccoin randomized communication complexity from below. More importantly, we show that the bound satisfies the strong direct product property under product distributions for both one and twoway protocols, and the weak direct product property under arbitrary distributions for twoway protocols. These results subsume and strengthen, in a unified manner, several recent results on the direct product
Classical vs. Quantum ReadOnce Branching Programs
"... Abstract. 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 si ..."
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Abstract. 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).
Research Statement
"... My research interests have been in the field of classical and quantum information theory. During my research till now, I have been primarily interested in developing information theoretic tools and applying them in areas like communication complexity, coding theory and cryptography. One of the main ..."
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My research interests have been in the field of classical and quantum information theory. During my research till now, I have been primarily interested in developing information theoretic tools and applying them in areas like communication complexity, coding theory and cryptography. One of the main themes of my work has been to show that in communication protocols for computing functions or relations, messages can be compressed close to the information content about the input. In other words, the communicating parties, in order to compute a function jointly, need not send long messages, if they only reveal a low amount of information in their messages about their respective inputs. This may be viewed as an analogue of the source coding theorems of Shannon and Schumacher in the setting of communication complexity. At the time I along with my coauthors [JRS02a], began thinking about quantum information, we found that the then available tools were not adequate to deal with large information scenarios. Most of the papers at that time dealt with situations where the information loss was much less than 1. So we came up with a new information theoretic tool called the substate theorem. This theorem roughly states that if the relative entropy between two quantum states, a quantity closely related to mutual information, is small then the first state is contained in the second state as a substate with reasonably high probability mass.
Abstract
, 2001
"... We prove new lower bounds for 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 [30] to the quantum case. Applying this ..."
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We prove new lower bounds for 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 [30] 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 Fourier based lower bound methods, notably showing that ¯s(f) / log n, for the average sensitivity ¯s(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). 1