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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 882 (2 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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Cited by 482 (5 self)
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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.
Topological quantum computation
 Bull. Amer. Math. Soc. (N.S
"... Abstract. The theory of quantum computation can be constructed from the abstract study of anyonic systems. In mathematical terms, these are unitary topological modular functors. They underlie the Jones polynomial and arise in WittenChernSimons theory. The braiding and fusion of anyonic excitations ..."
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Cited by 109 (14 self)
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Abstract. The theory of quantum computation can be constructed from the abstract study of anyonic systems. In mathematical terms, these are unitary topological modular functors. They underlie the Jones polynomial and arise in WittenChernSimons theory. The braiding and fusion of anyonic excitations in quantum Hall electron liquids and 2Dmagnets are modeled by modular functors, opening a new possibility for the realization of quantum computers. The chief advantage of anyonic computation would be physical error correction: An error rate scaling like e−αℓ, where ℓ is a length scale, and α is some positive constant. In contrast, the “presumptive ” qubitmodel of quantum computation, which repairs errors combinatorically, requires a fantastically low initial error rate (about 10−4) before computation can be stabilized. Quantum computation is a catchall for several models of computation based on a theoretical ability to manufacture, manipulate and measure quantum states. In this context, there are three areas where remarkable algorithms have been found: searching a data base [15], abelian groups (factoring and discrete logarithm) [19],
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 15 (4 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
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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Cited by 1 (0 self)
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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
Clifford algebras, noncommutative tori and universal quantum computers
, 2001
"... Recently author suggested [23] an application of Clifford algebras for construction of a “compiler ” for universal binary quantum computer together with later development [24] of the similar idea for a nonbinary base. The nonbinary case is related with application of some extension of idea of Clif ..."
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Recently author suggested [23] an application of Clifford algebras for construction of a “compiler ” for universal binary quantum computer together with later development [24] of the similar idea for a nonbinary base. The nonbinary case is related with application of some extension of idea of Clifford algebras. It is noncommutative torus defined by polynomial algebraic relations of order l. For l = 2 it coincides with definition of Clifford algebra. Here is presented the joint consideration and comparison of both cases together with some discussion on possible physical consequences.
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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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.