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Journal ArticleDOI

Reversible logic and quantum computers.

01 Dec 1985-Physical Review A (American Physical Society)-Vol. 32, Iss: 6, pp 3266-3276
TL;DR: The construction of a quantum-mechanical Hamiltonian describing a computer generates a dynamical evolution which mimics a sequence of elementary logical steps if each logical step is locally reversible (global reversibility is insufficient).
Abstract: This article is concerned with the construction of a quantum-mechanical Hamiltonian describing a computer. This Hamiltonian generates a dynamical evolution which mimics a sequence of elementary logical steps. This can be achieved if each logical step is locally reversible (global reversibility is insufficient). Computational errors due to noise can be corrected by means of redundancy. In particular, reversible error-correcting codes can be embedded in the Hamiltonian itself. An estimate is given for the minimum amount of entropy which must be dissipated at a given noise level and tolerated error rate.
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Journal ArticleDOI
TL;DR: U(2) gates are derived, which derive upper and lower bounds on the exact number of elementary gates required to build up a variety of two- and three-bit quantum gates, the asymptotic number required for n-bit Deutsch-Toffoli gates, and make some observations about the number of unitary operations on arbitrarily many bits.
Abstract: We show that a set of gates that consists of all one-bit quantum gates (U(2)) and the two-bit exclusive-or gate (that maps Boolean values (x,y) to (x,x ⊕y)) is universal in the sense that all unitary operations on arbitrarily many bits n (U(2 n )) can be expressed as compositions of these gates. We investigate the number of the above gates required to implement other gates, such as generalized Deutsch-Toffoli gates, that apply a specific U(2) transformation to one input bit if and only if the logical AND of all remaining input bits is satisfied. These gates play a central role in many proposed constructions of quantum computational networks. We derive upper and lower bounds on the exact number of elementary gates required to build up a variety of two- and three-bit quantum gates, the asymptotic number required for n-bit Deutsch-Toffoli gates, and make some observations about the number required for arbitrary n-bit unitary operations.

3,731 citations


Cites background from "Reversible logic and quantum comput..."

  • ...Quantum-mechanical Turing machines [5, 6], gate arrays [7], and cellular automata [8] have been discussed, and physical realizations of Toffoli’s[9, 10, 11] and Fredkin’s[12, 13, 14] universal three-bit gates within various quantum-mechanical physical systems have been proposed....

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Journal ArticleDOI
23 Aug 1996-Science
TL;DR: Feynman's 1982 conjecture, that quantum computers can be programmed to simulate any local quantum system, is shown to be correct.
Abstract: Feynman's 1982 conjecture, that quantum computers can be programmed to simulate any local quantum system, is shown to be correct.

2,678 citations

Journal ArticleDOI
TL;DR: This book discusses classical models of computations, quantum formalism, symplecto-classical cases, and error correction in the computation process: general principles.
Abstract: Contents §0. Introduction §1. Abelian problem on the stabilizer §2. Classical models of computations2.1. Boolean schemes and sequences of operations2.2. Reversible computations §3. Quantum formalism3.1. Basic notions and notation3.2. Transformations of mixed states3.3. Accuracy §4. Quantum models of computations4.1. Definitions and basic properties4.2. Construction of various operators from the elements of a basis4.3. Generalized quantum control and universal schemes §5. Measurement operators §6. Polynomial quantum algorithm for the stabilizer problem §7. Computations with perturbations: the choice of a model §8. Quantum codes (definitions and general properties)8.1. Basic notions and ideas8.2. One-to-one codes8.3. Many-to-one codes §9. Symplectic (additive) codes9.1. Algebraic preparation9.2. The basic construction9.3. Error correction procedure9.4. Torus codes §10. Error correction in the computation process: general principles10.1. Definitions and results10.2. Proofs §11. Error correction: concrete procedures11.1. The symplecto-classical case11.2. The case of a complete basis Bibliography

1,235 citations

Journal ArticleDOI
31 Aug 2000-Nature
TL;DR: The physical limits of computation as determined by the speed of light c, the quantum scale ℏ and the gravitational constant G are explored.
Abstract: Computers are physical systems: the laws of physics dictate what they can and cannot do. In particular, the speed with which a physical device can process information is limited by its energy and the amount of information that it can process is limited by the number of degrees of freedom it possesses. Here I explore the physical limits of computation as determined by the speed of light c, the quantum scale h and the gravitational constant G. As an example, I put quantitative bounds to the computational power of an 'ultimate laptop' with a mass of one kilogram confined to a volume of one litre.

1,020 citations

Journal Article
TL;DR: In this article, a polynomial quantum algorithm for the stabilizer problem with factoring and the discrete logarithm is presented, which is based on a procedure for measuring an eigenvalue of a unitary operator.
Abstract: We present a polynomial quantum algorithm for the Abelian stabilizer problem which includes both factoring and the discrete logarithm. Thus we extend famous Shor’s results [7]. Our method is based on a procedure for measuring an eigenvalue of a unitary operator. Another application of this procedure is a polynomial quantum Fourier transform algorithm for an arbitrary finite Abelian group. The paper also contains a rather detailed introduction to the theory of quantum computation.

766 citations