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13
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 1278 (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.
Quantum complexity theory
 in Proc. 25th Annual ACM Symposium on Theory of Computing, ACM
, 1993
"... Abstract. In this paper we study quantum computation from a complexity theoretic viewpoint. Our first result is the existence of an efficient universal quantum Turing machine in Deutsch’s model of a quantum Turing machine (QTM) [Proc. Roy. Soc. London Ser. A, 400 (1985), pp. 97–117]. This constructi ..."
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Abstract. In this paper we study quantum computation from a complexity theoretic viewpoint. Our first result is the existence of an efficient universal quantum Turing machine in Deutsch’s model of a quantum Turing machine (QTM) [Proc. Roy. Soc. London Ser. A, 400 (1985), pp. 97–117]. This construction is substantially more complicated than the corresponding construction for classical Turing machines (TMs); in fact, even simple primitives such as looping, branching, and composition are not straightforward in the context of quantum Turing machines. We establish how these familiar primitives can be implemented and introduce some new, purely quantum mechanical primitives, such as changing the computational basis and carrying out an arbitrary unitary transformation of polynomially bounded dimension. We also consider the precision to which the transition amplitudes of a quantum Turing machine need to be specified. We prove that O(log T) bits of precision suffice to support a T step computation. This justifies the claim that the quantum Turing machine model should be regarded as a discrete model of computation and not an analog one. We give the first formal evidence that quantum Turing machines violate the modern (complexity theoretic) formulation of the Church–Turing thesis. We show the existence of a problem, relative to an oracle, that can be solved in polynomial time on a quantum Turing machine, but requires superpolynomial time on a boundederror probabilistic Turing machine, and thus not in the class BPP. The class BQP of languages that are efficiently decidable (with small errorprobability) on a quantum Turing machine satisfies BPP ⊆ BQP ⊆ P ♯P. Therefore, there is no possibility of giving a mathematical proof that quantum Turing machines are more powerful than classical probabilistic Turing machines (in the unrelativized setting) unless there is a major breakthrough in complexity theory.
Computation and Hypercomputation
 MINDS AND MACHINES
, 2003
"... Does Nature permit the implementation of behaviours that cannot be simulated computationally? We consider the meaning of physical computationality in some detail, and present arguments in favour of physical hypercomputation: for example, modern scientific method does not allow the specification o ..."
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Cited by 23 (5 self)
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Does Nature permit the implementation of behaviours that cannot be simulated computationally? We consider the meaning of physical computationality in some detail, and present arguments in favour of physical hypercomputation: for example, modern scientific method does not allow the specification of any experiment capable of refuting hypercomputation. We consider the implications of relativistic algorithms capable of solving the (Turing) Halting Problem. We also reject as a fallacy the argument that hypercomputation has no relevance because noncomputable values are indistinguishable from sufficiently close computable approximations. In addition to
QMAcomplete problems
, 2012
"... In this paper we give an overview of the quantum computational complexity class QMA and a description of known QMAcomplete problems to date 1. Such problems are believed to be difficult to solve, even with a quantum computer, but have the property that if a purported solution to the problem is give ..."
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Cited by 12 (1 self)
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In this paper we give an overview of the quantum computational complexity class QMA and a description of known QMAcomplete problems to date 1. Such problems are believed to be difficult to solve, even with a quantum computer, but have the property that if a purported solution to the problem is given, a quantum computer would easily be able to verify whether it is correct. An attempt has been made to make this paper as selfcontained as possible so that it can be accessible to computer scientists, physicists, mathematicians, and quantum chemists. Problems of interest to all of these professions can be found here.
Quantum information is physical
 Superlatt. Micro
, 1998
"... We discuss a few current developments in the use of quantum mechanically coherent systems for information processing. In each of these developments, Rolf Landauer has played a crucial role in nudging us, and other workers in the field, into asking the right questions, some of which we have been luck ..."
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We discuss a few current developments in the use of quantum mechanically coherent systems for information processing. In each of these developments, Rolf Landauer has played a crucial role in nudging us, and other workers in the field, into asking the right questions, some of which we have been lucky enough to answer. A general overview of the key ideas of quantum error correction is given. We discuss how quantum entanglement is the key to protecting quantum states from decoherence in a manner which, in a theoretical sense, is as effective as the protection of digital data from bit noise. We also discuss five general criteria which must be satisfied to implement a quantum computer in the laboratory, and we illustrate the application of these criteria by discussing our ideas for creating a quantum computer out of the spin states of coupled quantum dots.
On Deniability in Quantum Key Exchange
"... Abstract. We show that claims of “perfect security ” for keys produced by quantum key exchange (QKE) are limited to “privacy ” and “integrity.” Unlike a onetime pad, QKE does not necessarily enable Sender and Receiver to pretend later to have established a different key. This result is puzzling in ..."
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Abstract. We show that claims of “perfect security ” for keys produced by quantum key exchange (QKE) are limited to “privacy ” and “integrity.” Unlike a onetime pad, QKE does not necessarily enable Sender and Receiver to pretend later to have established a different key. This result is puzzling in light of Mayers ’ “NoGo ” theorem showing the impossibility of quantum bit commitment. But even though a simple and intuitive application of Mayers ’ protocol transformation appears sufficient to provide deniability (else QBC would be possible), we show several reasons why such conclusions are illfounded. Mayers ’ transformation arguments, while sound for QBC, are insufficient to establish deniability in QKE. Having shed light on several unadvertised pitfalls, we then provide a candidate deniable QKE protocol. This itself indicates further shortfalls in current proof techniques, including reductions that preserve privacy but fail to preserve deniability. In sum, purchasing undeniability with an offtheshelf QKE protocol is significantly more expensive and dangerous than the mere optic fiber for which “perfect security ” is advertised. 1
Quantum/classical correspondence in the light of Bell’s inequalities
, 1990
"... Instead of the usual asymptotic passage from quantum mechanics to classical mechanics when a parameter tended to infinity, a sharp boundary is obtained for the domain of existence of classical reality. The last is treated as separable empirical reality following d’Espagnat, described by a mathemati ..."
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Instead of the usual asymptotic passage from quantum mechanics to classical mechanics when a parameter tended to infinity, a sharp boundary is obtained for the domain of existence of classical reality. The last is treated as separable empirical reality following d’Espagnat, described by a mathematical superstructure over quantum dynamics for the universal wave function. Being empirical, this reality is constructed in terms of both fundamental notions and characteristics of observers. It is presupposed that considered observers perceive the world as a system of collective degrees of freedom that are inherently dissipative because of interaction with thermal degrees of freedom. Relevant problems of foundation of statistical physics are considered. A feasible example is given of a macroscopic system not admitting such classical reality. The article contains a concise survey of some relevant domains: quantum and classical Belltype inequalities; universal wave function; approaches to quantum description of macroscopic world, with emphasis on dissipation; spontaneous reduction models; experimental tests of the universal validity of the quantum the
NQP = coC=P
, 1998
"... Adleman, Demarrais, and Huang introduced the nondeterministic quantum polynomialtime complexity class NQP as an analogue of NP. It is known that, with restricted amplitudes, NQP is characterized in terms of the classical counting complexity class C=P. In this paper we prove that, with unrestricted ..."
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Adleman, Demarrais, and Huang introduced the nondeterministic quantum polynomialtime complexity class NQP as an analogue of NP. It is known that, with restricted amplitudes, NQP is characterized in terms of the classical counting complexity class C=P. In this paper we prove that, with unrestricted amplitudes, NQP indeed coincides with the complement of C=P. As an immediate corollary, with unrestricted amplitudes BQP differs from NQP. key words: computational complexity, theory of computation 1