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93
Robust PCPs of Proximity, Shorter PCPs and Applications to Coding
 in Proc. 36th ACM Symp. on Theory of Computing
, 2004
"... We continue the study of the tradeo between the length of PCPs and their query complexity, establishing the following main results (which refer to proofs of satis ability of circuits of size n): 1. We present PCPs of length exp( ~ O(log log n) ) n that can be veri ed by making o(log log n) ..."
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Cited by 80 (25 self)
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We continue the study of the tradeo between the length of PCPs and their query complexity, establishing the following main results (which refer to proofs of satis ability of circuits of size n): 1. We present PCPs of length exp( ~ O(log log n) ) n that can be veri ed by making o(log log n) Boolean queries.
Extractors: Optimal up to Constant Factors
 STOC'03
, 2003
"... This paper provides the first explicit construction of extractors which are simultaneously optimal up to constant factors in both seed length and output length. More precisely, for every n, k, our extractor uses a random seed of length O(log n) to transform any random source on n bits with (min)ent ..."
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Cited by 51 (12 self)
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This paper provides the first explicit construction of extractors which are simultaneously optimal up to constant factors in both seed length and output length. More precisely, for every n, k, our extractor uses a random seed of length O(log n) to transform any random source on n bits with (min)entropy k, into a distribution on (1 − α)k bits that is ɛclose to uniform. Here α and ɛ can be taken to be any positive constants. (In fact, ɛ can be almost polynomially small). Our improvements are obtained via three new techniques, each of which may be of independent interest. The first is a general construction of mergers [22] from locally decodable errorcorrecting codes. The second introduces new condensers that have constant seed length (and retain a constant fraction of the minentropy in the random source). The third is a way to augment the “winwin repeated condensing” paradigm of [17] with error reduction techniques like [15] so that the our constant seedlength condensers can be used without error accumulation.
Some Applications of Coding Theory in Computational Complexity
, 2004
"... Errorcorrecting codes and related combinatorial constructs play an important role in several recent (and old) results in computational complexity theory. In this paper we survey results on locallytestable and locallydecodable errorcorrecting codes, and their applications to complexity theory ..."
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Cited by 49 (2 self)
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Errorcorrecting codes and related combinatorial constructs play an important role in several recent (and old) results in computational complexity theory. In this paper we survey results on locallytestable and locallydecodable errorcorrecting codes, and their applications to complexity theory and to cryptography.
Towards 3query locally decodable codes of subexponential length
 In Proc. of the 39th ACM Symposium on Theory of Computing (STOC
, 2007
"... A qquery Locally Decodable Code (LDC) encodes an nbit message x as an Nbit codeword C(x), such that one can probabilistically recover any bit xi of the message by querying only q bits of the codeword C(x), even after some constant fraction of codeword bits has been corrupted. We give new construc ..."
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Cited by 48 (5 self)
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A qquery Locally Decodable Code (LDC) encodes an nbit message x as an Nbit codeword C(x), such that one can probabilistically recover any bit xi of the message by querying only q bits of the codeword C(x), even after some constant fraction of codeword bits has been corrupted. We give new constructions of three query LDCs of vastly shorter length than that of previous constructions. Specifically, given any Mersenne prime p = 2 t − 1, we design three query LDCs of length N = exp � n 1/t �, for every n. Based on the largest known Mersenne prime, this translates to a length of less than exp n 10−7� compared to exp � n1/2 � in the previous constructions. It has often been conjectured that there are infinitely many Mersenne�primes. Under this conjecture, our constructions yield three query locally decodable codes of length N = exp n O
Quantum computing, postselection, and probabilistic polynomialtime
, 2004
"... I study the class of problems efficiently solvable by a quantum computer, given the ability to “postselect” on the outcomes of measurements. I prove that this class coincides with a classical complexity class called PP, or Probabilistic PolynomialTime. Using this result, I show that several simple ..."
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Cited by 39 (12 self)
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I study the class of problems efficiently solvable by a quantum computer, given the ability to “postselect” on the outcomes of measurements. I prove that this class coincides with a classical complexity class called PP, or Probabilistic PolynomialTime. Using this result, I show that several simple changes to the axioms of quantum mechanics would let us solve PPcomplete problems efficiently. The result also implies, as an easy corollary, a celebrated theorem of Beigel, Reingold, and Spielman that PP is closed under intersection, as well as a generalization of that theorem due to Fortnow and Reingold. This illustrates that quantum computing can yield new and simpler proofs of major results about classical computation.
Two Query PCP with SubConstant Error
, 2008
"... We show that the N PComplete language 3SAT has a PCP verifier that makes two queries to a proof of almostlinear size and achieves subconstant probability of error o(1). The verifier performs only projection tests, meaning that the answer to the first query determines at most one accepting answer ..."
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Cited by 35 (3 self)
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We show that the N PComplete language 3SAT has a PCP verifier that makes two queries to a proof of almostlinear size and achieves subconstant probability of error o(1). The verifier performs only projection tests, meaning that the answer to the first query determines at most one accepting answer to the second query. Previously, by the parallel repetition theorem, there were PCP Theorems with twoquery projection tests, but only (arbitrarily small) constant error and polynomial size [29]. There were also PCP Theorems with subconstant error and almostlinear size, but a constant number of queries that is larger than 2 [26]. As a corollary, we obtain a host of new results. In particular, our theorem improves many of the hardness of approximation results that are proved using the parallel repetition theorem. A partial list includes the following: 1. 3SAT cannot be efficiently approximated to within a factor of 7 8 + o(1), unless P = N P. This holds even under almostlinear reductions. Previously, the best known N Phardness
Lower Bounds for Local Search by Quantum Arguments
"... The problem of finding a local minimum of a blackbox function is central for understanding local search as well as quantum adiabatic algorithms. For functions on the Boolean hypercube {0,1} n (, we show a lower bound of Ω 2 n/4) /n on the number of queries needed by a quantum computer to solve this ..."
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Cited by 33 (2 self)
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The problem of finding a local minimum of a blackbox function is central for understanding local search as well as quantum adiabatic algorithms. For functions on the Boolean hypercube {0,1} n (, we show a lower bound of Ω 2 n/4) /n on the number of queries needed by a quantum computer to solve this problem. More surprisingly, our approach, based on Ambainis’s quantum ( adversary method, also yields a lower bound of Ω 2 n/2 /n 2 on the problem’s classical randomized query complexity. This improves and simplifies a 1983 result of Aldous. Finally, in both the randomized and quantum cases, we give the first nontrivial lower bounds for finding local minima on grids of constant dimension d ≥ 3. 1.
A survey on private information retrieval
 Bulletin of the EATCS
, 2004
"... Alice wants to query a database but she does not want the database to learn what she is querying. She can ask for the entire database. Can she get her query answered with less communication? One model of this problem is Private Information Retrieval, henceforth PIR. We survey results obtained about ..."
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Cited by 32 (1 self)
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Alice wants to query a database but she does not want the database to learn what she is querying. She can ask for the entire database. Can she get her query answered with less communication? One model of this problem is Private Information Retrieval, henceforth PIR. We survey results obtained about the PIR model including partial answers to the following questions. (1) What if there are k noncommunicating copies of the database but they are computationally unbounded? (2) What if there is only one copy of the database and it is computationally bounded? 1
Exponential separation of quantum and classical oneway communication complexity
 SIAM J. Comput
"... Abstract. We give the first exponential separation between quantum and boundederror randomized oneway communication complexity. Specifically, we define the Hidden Matching Problem HMn: Alice gets as input a string x ∈ {0, 1} n and Bob gets a perfect matching M on the n coordinates. Bob’s goal is t ..."
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Cited by 32 (2 self)
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Abstract. We give the first exponential separation between quantum and boundederror randomized oneway communication complexity. Specifically, we define the Hidden Matching Problem HMn: Alice gets as input a string x ∈ {0, 1} n and Bob gets a perfect matching M on the n coordinates. Bob’s goal is to output a tuple 〈i, j, b 〉 such that the edge (i, j) belongs to the matching M and b = xi ⊕ xj. We prove that the quantum oneway communication complexity of HMn is O(log n), yet any randomized oneway protocol with bounded error must use Ω ( √ n) bits of communication. No asymptotic gap for oneway communication was previously known. Our bounds also hold in the model of Simultaneous Messages (SM) and hence we provide the first exponential separation between quantum SM and randomized SM with public coins. For a Boolean decision version of HMn, we show that the quantum oneway communication complexity remains O(log n) and that the 0error randomized oneway communication complexity is Ω(n). We prove that any randomized linear oneway protocol with bounded error for this problem requires Ω ( 3 √ n log n) bits of communication. Key words. Communication complexity, quantum computation, separation, hidden matching AMS subject classifications. 68P30,68Q15,68Q17,81P68 1. Introduction. The
Improved Lower Bounds for Locally Decodable Codes and Private Information Retrieval
 In Proc. of 32nd International Colloquium on Automata, Languages and Programming (ICALP’05), LNCS 3580
, 2004
"... We prove new lower bounds for locally decodable codes and private information retrieval. We show that a 2query LDC encoding nbit strings over an `bit alphabet, where the decoder only uses b bits of each queried position of the codeword, needs code length m = exp i=0 ` !! Similarly, a 2 ..."
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Cited by 28 (2 self)
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We prove new lower bounds for locally decodable codes and private information retrieval. We show that a 2query LDC encoding nbit strings over an `bit alphabet, where the decoder only uses b bits of each queried position of the codeword, needs code length m = exp i=0 ` !! Similarly, a 2server PIR scheme with an nbit database and tbit queries, where the user only needs b bits from each of the two `bit answers, unknown to the servers, satis es i=0 ` This implies that several known PIR schemes are close to optimal. Our results generalize those of Goldreich et al. [6], who proved roughly the same bounds for linear LDCs and PIRs. Like earlier work by Kerenidis and de Wolf [9], our classical lower bounds are proved using quantum computational techniques. In particular, we give a tight analysis of how well a 2input function can be computed from a quantum superposition of both inputs.