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PolynomialTime Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer
 SIAM J. on Computing
, 1997
"... A digital computer is generally believed to be an efficient universal computing device; that is, it is believed able to simulate any physical computing device with an increase in computation time by at most a polynomial factor. This may not be true when quantum mechanics is taken into consideration. ..."
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Cited by 1277 (4 self)
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A digital computer is generally believed to be an efficient universal computing device; that is, it is believed able to simulate any physical computing device with an increase in computation time by at most a polynomial factor. This may not be true when quantum mechanics is taken into consideration. This paper considers factoring integers and finding discrete logarithms, two problems which are generally thought to be hard on a classical computer and which have been used as the basis of several proposed cryptosystems. Efficient randomized algorithms are given for these two problems on a hypothetical quantum computer. These algorithms take a number of steps polynomial in the input size, e.g., the number of digits of the integer to be factored.
Oracle quantum computing
 Brassard & U.Vazirani, Strengths and weaknesses of quantum computing
, 1994
"... \Because nature isn't classical, dammit..." ..."
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Cited by 115 (8 self)
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\Because nature isn't classical, dammit..."
Quantum Computing and Phase Transitions in Combinatorial Search
 J. of Artificial Intelligence Research
, 1996
"... We introduce an algorithm for combinatorial search on quantum computers that is capable of significantly concentrating amplitude into solutions for some NP search problems, on average. This is done by exploiting the same aspects of problem structure as used by classical backtrack methods to avoid un ..."
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Cited by 23 (7 self)
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We introduce an algorithm for combinatorial search on quantum computers that is capable of significantly concentrating amplitude into solutions for some NP search problems, on average. This is done by exploiting the same aspects of problem structure as used by classical backtrack methods to avoid unproductive search choices. This quantum algorithm is much more likely to find solutions than the simple direct use of quantum parallelism. Furthermore, empirical evaluation on small problems shows this quantum algorithm displays the same phase transition behavior, and at the same location, as seen in many previously studied classical search methods. Specifically, difficult problem instances are concentrated near the abrupt change from underconstrained to overconstrained problems. August
Quantum computing with trapped ions
, 2008
"... Quantum computers hold the promise to solve certain computational task much more efficiently than classical computers. We review the recent experimental advancements towards a quantum computer with trapped ions. In particular, various implementations of qubits, quantum gates and some key experiments ..."
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Cited by 17 (2 self)
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Quantum computers hold the promise to solve certain computational task much more efficiently than classical computers. We review the recent experimental advancements towards a quantum computer with trapped ions. In particular, various implementations of qubits, quantum gates and some key experiments are discussed. Furthermore, we review some implementations of quantum algorithms such as a deterministic teleportation of quantum information and an error correction scheme.
New Connections between Mathematics and Computer Science
, 1996
"... A workshop on "New Connections between Mathematics and Computer Science" was held at the Isaac Newton Institute for Mathematical Sciences in Cambridge, England from 2024 November 1995. The workshop was supported by the Engineering and Physical Science Research Council of the United Kingdo ..."
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Cited by 2 (2 self)
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A workshop on "New Connections between Mathematics and Computer Science" was held at the Isaac Newton Institute for Mathematical Sciences in Cambridge, England from 2024 November 1995. The workshop was supported by the Engineering and Physical Science Research Council of the United Kingdom, the London Mathematical Society and HewlettPackard's Basic Research Institute in the Mathematical Sciences. This document contains a report on the workshop, the abstracts of the talks and the accompanying bibliography.
M. B. Plenio and P. L. Knight Optics Section, Blackett Laboratory, Imperial College, London SW7 2BZ, England. submitted to Phys. Rev. A 14. 11. 1995
"... We investigate the time T a quantum computer requires to factorize a given number dependent on the number of bits L required to represent this number. We stress the fact that in most cases one has to take into account that the execution time of a single quantum gate is related to the decoherence ..."
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We investigate the time T a quantum computer requires to factorize a given number dependent on the number of bits L required to represent this number. We stress the fact that in most cases one has to take into account that the execution time of a single quantum gate is related to the decoherence time of the qubits that are involved in the computation. Although exhibited here only for special systems, this interdependence of decoherence and computation time seems to be a restriction in many current models for quantum computers and leads to the result that the computation time T scales much stronger with L than previously expected. PACS: 42.50.Lc I. Introduction Since Shor's discovery [1, 2] of an algorithm that allows thefactorization of a large number by a quantum computer in polynomial time instead of an exponential time as in classical computing, interest in the practical realization of a quantum computer has been much enhanced. Recent advances in the preparation and manip...
Quantum Computation Primitives
, 1996
"... This paper surveys the state of the field of quantum computation, with an emphasis on the recent effort to devise simple quantum computational primitives upon which more complex quantum calculations can be built. This effort is exemplified by the recent papers [2, 49], which we review. We also discu ..."
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This paper surveys the state of the field of quantum computation, with an emphasis on the recent effort to devise simple quantum computational primitives upon which more complex quantum calculations can be built. This effort is exemplified by the recent papers [2, 49], which we review. We also discuss Shor's [38] paper, which describes the quantumpolynomialtime factoring algorithm that has provided a major motivation for much of the recent activity. 1 Introduction Quantitative Church's Thesis. The "Quantitative Church's Thesis" [50, 48] claims that Turing machines are as efficient as any realistic computer, within a polynomial factor. However, Feynman [25] has pointed out that Turing machines seem to be unable to efficiently simulate quantum physics; that is, they seem to require an exponential slowdown to simulate it (although this has not been proven). This leads naturally to the supposition that a computer that was designed to take full advantage of quantum physical principles m...