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1,065
Strengths and weaknesses of quantum computing
, 1996
"... Recently a great deal of attention has focused on quantum computation following a sequence of results [4, 16, 15] suggesting that quantum computers are more powerful than classical probabilistic computers. Following Shor’s result that factoring and the extraction of discrete logarithms are both solv ..."
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Cited by 381 (10 self)
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solvable in quantum polynomial time, it is natural to ask whether all of NP can be efficiently solved in quantum polynomial time. In this paper, we address this question by proving that relative to an oracle chosen uniformly at random, with probability 1, the class NP cannot be solved on a quantum Turing
Analysis Of Two Simple Heuristics On A Random Instance Of kSAT
 Journal of Algorithms
, 1996
"... We consider the performance of two algorithms, GUC and SC studied by Chao and Franco [2], [3], and Chv'atal and Reed [4], when applied to a random instance ! of a boolean formula in conjunctive normal form with n variables and bcnc clauses of size k each. For the case where k = 3, we obtain th ..."
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Cited by 147 (4 self)
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We consider the performance of two algorithms, GUC and SC studied by Chao and Franco [2], [3], and Chv'atal and Reed [4], when applied to a random instance ! of a boolean formula in conjunctive normal form with n variables and bcnc clauses of size k each. For the case where k = 3, we obtain
On Lattices, Learning with Errors, Random Linear Codes, and Cryptography
 In STOC
, 2005
"... Our main result is a reduction from worstcase lattice problems such as SVP and SIVP to a certain learning problem. This learning problem is a natural extension of the ‘learning from parity with error’ problem to higher moduli. It can also be viewed as the problem of decoding from a random linear co ..."
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Cited by 364 (6 self)
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code. This, we believe, gives a strong indication that these problems are hard. Our reduction, however, is quantum. Hence, an efficient solution to the learning problem implies a quantum algorithm for SVP and SIVP. A main open question is whether this reduction can be made classical. We also present a
A spaceefficient randomized DNA algorithm for kSAT
 SIXTH INTERNATIONAL WORKSHOP ON DNABASED COMPUTERS, VOLUME 2054 OF LNCS
, 2001
"... We present a randomized DNA algorithm for kSAT based on the classical algorithm of Paturi et al. [8]. For an nvariable, mclause instance of kSAT (m>n), our algorithm finds a satisfying assignment, assuming one exists, with probability 1 − e −α, in worstcase time O(k 2 mn) and space O(2 (1 − ..."
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Cited by 2 (0 self)
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We present a randomized DNA algorithm for kSAT based on the classical algorithm of Paturi et al. [8]. For an nvariable, mclause instance of kSAT (m>n), our algorithm finds a satisfying assignment, assuming one exists, with probability 1 − e −α, in worstcase time O(k 2 mn) and space O(2 (1
Certifying unsatisfiability of random 2ksat formulas using approximation techniques
 FCT
, 2003
"... Abstract. It is known that random kSAT formulas with at least (2 k · ln 2) · n random clauses are unsatisfiable with high probability. This result is simply obtained by bounding the expected number of satisfying assignments of a random kSAT instance by an expression tending to 0 when n, the numbe ..."
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Cited by 7 (2 self)
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, the number of variables tends to infinity. This argument does not give us an efficient algorithm certifying the unsatisfiability of a given random instance. For even k it is known that random kSAT instances with at least Poly(log n) · n k/2 clauses can be efficiently certified as unsatisfiable. For k = 3
Exactly solvable potentials and quantum algebras
 Physical Review Letters
, 1992
"... A set of exactly solvable onedimensional quantum mechanical potentials is described. It is defined by a finitedifferencedifferential equation generating in the limiting cases the RosenMorse, harmonic, and PöschlTeller potentials. General solution includes Shabat’s infinite number soliton system ..."
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Cited by 8 (0 self)
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A set of exactly solvable onedimensional quantum mechanical potentials is described. It is defined by a finitedifferencedifferential equation generating in the limiting cases the RosenMorse, harmonic, and PöschlTeller potentials. General solution includes Shabat’s infinite number soliton
Quantum property testing of group solvability
 In Proceedings of 8th LATIN
, 2008
"... Abstract. Testing efficiently whether a finite set Γ with a binary operation · over it, given as an oracle, is a group is a wellknown open problem in the field of property testing. Recently, Friedl, Ivanyos and Santha have made a significant step in the direction of solving this problem by showing ..."
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Cited by 4 (0 self)
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that it it possible to test efficiently whether the input (Γ, ·) is an Abelian group or is far, with respect to some distance, from any Abelian group. In this paper, we make a step further and construct an efficient quantum algorithm that tests whether (Γ, ·) is a solvable group, or is far from any solvable group
Onedimensional quantum chaos: Explicitly solvable cases
, 2008
"... We present quantum graphs with remarkably regular spectral characteristics. We call them regular quantum graphs. Although regular quantum graphs are strongly chaotic in the classical limit, their quantum spectra are explicitly solvable in terms of periodic orbits. We present analytical solutions for ..."
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Cited by 4 (0 self)
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We present quantum graphs with remarkably regular spectral characteristics. We call them regular quantum graphs. Although regular quantum graphs are strongly chaotic in the classical limit, their quantum spectra are explicitly solvable in terms of periodic orbits. We present analytical solutions
Explicitly solvable cases of onedimensional quantum chaos
, 2008
"... We identify a set of quantum graphs with unique and precisely defined spectral properties called regular quantum graphs. Although chaotic in their classical limit with positive topological entropy, regular quantum graphs are explicitly solvable. The proof is constructive: we present exact periodic o ..."
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Cited by 2 (0 self)
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We identify a set of quantum graphs with unique and precisely defined spectral properties called regular quantum graphs. Although chaotic in their classical limit with positive topological entropy, regular quantum graphs are explicitly solvable. The proof is constructive: we present exact periodic
Grover’s Quantum Searching Algorithm is Optimal” Phys
 Rev. A
, 1999
"... I improve the tight bound on quantum searching [4] to a matching bound, thus showing that for any probability of success Grover’s quantum searching algorithm is optimal. E.g. for near certain success we have to query the oracle π/4 √ N times, where N is the size of the search space. I also show that ..."
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Cited by 102 (0 self)
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that unfortunately quantum searching cannot be parallelized better than by assigning different parts of the search space to independent quantum computers. Earlier results left open the possibility of a more efficient parallelization. 1 Quantum searching Imagine we have N cases of which only one fulfills our
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