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134
Improved Lower Bounds for Locally Decodable Codes and Private Information Retrieval
, 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 ℓbit alphabet, where the decoder only uses b bits of each queried position of the codeword, needs code length m = exp Ω ..."
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Cited by 40 (3 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 ℓbit alphabet, where the decoder only uses b bits of each queried position of the codeword, needs code length m = exp Ω
A hypercontractive inequality for matrixvalued functions with applications to quantum computing and LDCs
"... The BonamiBeckner hypercontractive inequality is a powerful tool in Fourier analysis of realvalued functions on the Boolean cube. In this paper we present a version of this inequality for matrixvalued functions on the Boolean cube. Its proof is based on a powerful inequality by Ball, Carlen, and ..."
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Cited by 39 (3 self)
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The BonamiBeckner hypercontractive inequality is a powerful tool in Fourier analysis of realvalued functions on the Boolean cube. In this paper we present a version of this inequality for matrixvalued functions on the Boolean cube. Its proof is based on a powerful inequality by Ball, Carlen, and Lieb. We also present a number of applications. First, we analyze maps that encode n classical bits into m qubits, in such a way that each set of k bits can be recovered with some probability by an appropriate measurement on the quantum encoding; we show that if m<0.7n, then the success probability is exponentially small in k. This result may be viewed as a direct product version of Nayak’s quantum random access code bound. It in turn implies strong direct product theorems for the oneway quantum communication complexity of Disjointness and other problems. Second, we prove that errorcorrecting codes that are locally decodable with 2 queries require length exponential in the length of the encoded string. This gives what is arguably the first “nonquantum” proof of a result originally derived by Kerenidis and de Wolf using quantum information theory.
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 32 (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.
Continuoustime quantum walks on the symmetric group
 Proc. RANDOMAPPROX (Sanjeev
, 2003
"... In this paper we study continuoustime quantum walks on Cayley graphs of the symmetric group, and prove various facts concerning such walks that demonstrate significant differences from their classical analogues. In particular, we show that for several natural choices for generating sets, these quan ..."
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Cited by 28 (0 self)
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In this paper we study continuoustime quantum walks on Cayley graphs of the symmetric group, and prove various facts concerning such walks that demonstrate significant differences from their classical analogues. In particular, we show that for several natural choices for generating sets, these quantum walks do not have uniform limiting distributions, and are effectively blind to large areas of the graphs due to destructive interference. 1
Lattice problems in NP ∩ coNP
 Journal of the ACM
"... We show that the problems of approximating the shortest and closest vector in a lattice to within a factor of √ n lie in NP intersect coNP. The result (almost) subsumes the three mutuallyincomparable previous results regarding these lattice problems: Banaszczyk [7], Goldreich and Goldwasser [14], a ..."
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Cited by 27 (1 self)
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We show that the problems of approximating the shortest and closest vector in a lattice to within a factor of √ n lie in NP intersect coNP. The result (almost) subsumes the three mutuallyincomparable previous results regarding these lattice problems: Banaszczyk [7], Goldreich and Goldwasser [14], and Aharonov and Regev [2]. Our technique is based on a simple fact regarding succinct approximation of functions using their Fourier series over the lattice. This technique might be useful elsewhere – we demonstrate this by giving a simple and efficient algorithm for one other lattice problem (CVPP) improving on a previous result of Regev [26]. An interesting fact is that our result emerged from a “dequantization ” of our previous quantum result in [2]. This route to proving purely classical results might be beneficial elsewhere. 1
General Constructions for InformationTheoretic Private Information Retrieval
, 2003
"... A Private Information Retrieval (PIR) protocol enables a user to retrieve a data item from a database while hiding the identity of the item being retrieved; specifically, in a tprivate, kserver PIR protocol the database is replicated among k servers, and the user's privacy is protected from a ..."
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Cited by 23 (0 self)
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A Private Information Retrieval (PIR) protocol enables a user to retrieve a data item from a database while hiding the identity of the item being retrieved; specifically, in a tprivate, kserver PIR protocol the database is replicated among k servers, and the user's privacy is protected from any collusion of up to t servers. The main costmeasure of such protocols is the communication complexity of retrieving asingle bit of data. This work addresses the informationtheoretic setting for PIR, where the user's privacy should be unconditionally protected against computationally unbounded servers. We present a general construction, whose abstract components can be instantiated to yield both old and new families of PIR protocols. Amain ingredient in the new protocols is a generalization of a solution by Babai, Kimmel, and Lokam for a communication complexity problem in the multiparty simultaneous messages model.Our protocols simplify and improve upon previous ones, and resolve some previous anomalies. In particular, we get: (1) 1private kserver PIR protocols with O(k3n1=(2k\Gamma 1)) communication bits, where n is the database size; (2) tprivate kserver protocols with O(n1=b(2k\Gamma 1)=tc) communication bits, for anyconstant integers k? t * 1; and (3) tprivate kserver protocols in which the user sends O(log n) bitsto each server and receives O(nt=k+ffl) bits in return, for any constant integers k? t * 1 and constant ffl? 0. The latter protocols have applications to the construction of efficient families of locally decodablecodes over large alphabets and to PIR protocols with reduced work by the servers.
New Limits to Classical and Quantum Instance Compression
 ELECTRONIC COLLOQUIUM ON COMPUTATIONAL COMPLEXITY, REPORT NO. 112
, 2012
"... Given an instance of a hard decision problem, a limited goal is to compress that instance into a smaller, equivalent instance of a second problem. As one example, consider the problem where, given Boolean formulas ψ 1,...,ψ t, we must determine if at least one ψ j is satisfiable. An ORcompression s ..."
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Cited by 20 (1 self)
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Given an instance of a hard decision problem, a limited goal is to compress that instance into a smaller, equivalent instance of a second problem. As one example, consider the problem where, given Boolean formulas ψ 1,...,ψ t, we must determine if at least one ψ j is satisfiable. An ORcompression scheme for SAT is a polynomialtime reduction R that maps (ψ 1,...,ψ t) to a string z, such that z lies in some “target ” language L ′ if and only if ∨ j [ψj ∈ SAT] holds. (Here, L ′ can be arbitrarily complex.) ANDcompression schemes are defined similarly. A compression scheme is strong if z  is polynomially bounded in n = maxj ψ j , independent of t. Strong compression for SAT seems unlikely. Work of Harnik and Naor (FOCS ’06/SICOMP ’10) and Bodlaender, Downey, Fellows, and Hermelin (ICALP ’08/JCSS ’09) showed that the infeasibility of strong ORcompression for SAT would show limits to instance compression for a large number of natural problems. Bodlaender et al. also showed that the infeasibility of strong ANDcompression for SAT would have consequences for a different list of problems. Motivated by this, Fortnow and Santhanam (STOC ’08/JCSS ’11) showed that if SAT is strongly ORcompressible,
A Note on Yekhanin’s Locally Decodable Codes
 ELECTRONIC COLLOQUIUM ON COMPUTATIONAL COMPLEXITY, REPORT NO. 16 (2007)
, 2007
"... Locally Decodable codes(LDC) support decoding of any particular symbol of the input message by reading constant number of symbols of the codeword, even in presence of constant fraction of errors. In a recent breakthrough [9], Yekhanin constructedquery LDCs that hugely improve over earlier construct ..."
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Locally Decodable codes(LDC) support decoding of any particular symbol of the input message by reading constant number of symbols of the codeword, even in presence of constant fraction of errors. In a recent breakthrough [9], Yekhanin constructedquery LDCs that hugely improve over earlier constructions. Specifically, for a Mersenne prime, binary LDCs of length for infinitely many were obtained. Using the largest known Mersenne prime, this implies LDCs of length less than. Assuming infinitude of Mersenne primes, the construction yields LDCs of length for infinitely many. Inspired by [9], we constructquery binary LDCs with same parameters from Mersenne primes. While all the main technical tools are borrowed from [9], we give a selfcontained simple construction of LDCs. Our bounds do not improve over [9], and have worse soundness of the decoder. However the LDCs are simpler and generalize naturally to prime fields other than. The LDCs presented also translate directly in to three server Private Information Retrieval(PIR) protocols with communication! complexities for a database of size, starting with a Mersenne prime.