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Reliable quantum computers
 Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
, 1998
"... The new field of quantum error correction has developed spectacularly since its origin less than two years ago. Encoded quantum information can be protected from errors that arise due to uncontrolled interactions with the environment. Recovery from errors can work effectively even if occasional mist ..."
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Cited by 126 (3 self)
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The new field of quantum error correction has developed spectacularly since its origin less than two years ago. Encoded quantum information can be protected from errors that arise due to uncontrolled interactions with the environment. Recovery from errors can work effectively even if occasional mistakes occur during the recovery procedure. Furthermore, encoded quantum information can be processed without serious propagation of errors. Hence, an arbitrarily long quantum computation can be performed reliably, provided that the average probability of error per quantum gate is less than a certain critical value, the accuracy threshold. A quantum computer storing about 106 qubits, with a probability of error per quantum gate of order 106, would be a formidable factoring engine. Even a smaller lessaccurate quantum computer would be able to perform many useful tasks. This paper is based on a talk presented at the ITP Conference on Quantum Coherence
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],
From quantum cellular automata to quantum lattice gases
 Journal of Statistical Physics
, 1996
"... A natural architecture for nanoscale quantum computation is that of a quantum cellular automaton. Motivated by this observation, in this paper we begin an investigation of exactly unitary cellular automata. After proving that there can be no nontrivial, homogeneous, local, unitary, scalar cellular a ..."
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Cited by 99 (17 self)
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A natural architecture for nanoscale quantum computation is that of a quantum cellular automaton. Motivated by this observation, in this paper we begin an investigation of exactly unitary cellular automata. After proving that there can be no nontrivial, homogeneous, local, unitary, scalar cellular automaton in one dimension, we weaken the homogeneity condition and show that there are nontrivial, exactly unitary, partitioning cellular automata. We find a one parameter family of evolution rules which are best interpreted as those for a one particle quantum automaton. This model is naturally reformulated as a two component cellular automaton which we demonstrate to limit to the Dirac equation. We describe two generalizations of this automaton, the second of which, to multiple interacting particles, is the correct definition of a quantum lattice gas. KEY WORDS: quantum cellular automaton; quantum lattice gas; quantum computation. to appear in J. Stat. Phys.
Simulation of topological field theories by quantum computers
 Comm.Math.Phys.227
"... Abstract: Quantum computers will work by evolving a high tensor power of a small (e.g. two) dimensional Hilbert space by local gates, which can be implemented by applying a local Hamiltonian H for a time t. In contrast to this quantum engineering, the most abstract reaches of theoretical physics has ..."
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Cited by 77 (12 self)
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Abstract: Quantum computers will work by evolving a high tensor power of a small (e.g. two) dimensional Hilbert space by local gates, which can be implemented by applying a local Hamiltonian H for a time t. In contrast to this quantum engineering, the most abstract reaches of theoretical physics has spawned “topological models ” having a finite dimensional internal state space with no natural tensor product structure and in which the evolution of the state is discrete, H ≡ 0. These are called topological quantum field theories (TQFTs). These exotic physical systems are proved to be efficiently simulated on a quantum computer. The conclusion is twofold: 1. TQFTs cannot be used to define a model of computation stronger than the usual quantum model “BQP”. 2. TQFTs provide a radically different way of looking at quantum computation. The rich mathematical structure of TQFTs might suggest a new quantum algorithm. 1.
Parallel quantum computation
 Complexity, Entropy, and the Physics of Information,SFI Studies in the Sciences of Complexity
, 1990
"... A computer is a physical system which has a very general ability to simulate other physical systems (and in particular, other computers). In this paper we investigate the question of whether microscopic quantum systems can be computers. Using a reversible cellular automaton model of computation we i ..."
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Cited by 53 (10 self)
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A computer is a physical system which has a very general ability to simulate other physical systems (and in particular, other computers). In this paper we investigate the question of whether microscopic quantum systems can be computers. Using a reversible cellular automaton model of computation we illustrate several approaches to this question. We then attempt to extend Feynman’s construction of a quantum computer in order to arrive at a quantum model of parallel processing. 1
Simulating quantum mechanics on a quantum computer
 PHYSICA D
, 1998
"... Algorithms are described for efficiently simulating quantum mechanical systems on quantum computers. A class of algorithms for simulating the Schrödinger equation for interacting manybody systems are presented in some detail. These algorithms would make it possible to simulate nonrelativistic quant ..."
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Cited by 41 (3 self)
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Algorithms are described for efficiently simulating quantum mechanical systems on quantum computers. A class of algorithms for simulating the Schrödinger equation for interacting manybody systems are presented in some detail. These algorithms would make it possible to simulate nonrelativistic quantum systems on a quantum computer with an exponential speedup compared to simulations on classical computers. Issues involved in simulating relativistic systems of Dirac or gauge particles are discussed.
Quantum Mechanics Of Lattice Gas Automata I. One Particle Plane Waves And Potentials
, 1996
"... Classical lattice gas automata effectively simulate physical processes such as diffusion and fluid flow (in certain parameter regimes) despite their simplicity at the microscale. Motivated by current interest in quantum computation we recently defined quantum lattice gas automata; in this paper we i ..."
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Cited by 35 (10 self)
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Classical lattice gas automata effectively simulate physical processes such as diffusion and fluid flow (in certain parameter regimes) despite their simplicity at the microscale. Motivated by current interest in quantum computation we recently defined quantum lattice gas automata; in this paper we initiate a project to analyze which physical processes these models can effectively simulate. Studying the single particle sector of a one dimensional quantum lattice gas we find discrete analogues of plane waves and wave packets, and then investigate their behaviour in the presence of inhomogeneous potentials.
Incomputability in Nature
"... To what extent is incomputability relevant to the material Universe? We look at ways in which this question might be answered, and the extent to which the theory of computability, which grew out of the work of Godel, Church, Kleene and Turing, can contribute to a clear resolution of the current conf ..."
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Cited by 14 (8 self)
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To what extent is incomputability relevant to the material Universe? We look at ways in which this question might be answered, and the extent to which the theory of computability, which grew out of the work of Godel, Church, Kleene and Turing, can contribute to a clear resolution of the current confusion. It is hoped that the presentation will be accessible to the nonspecialist reader.
Differential calculus and gauge theory on finite sets, Göttingen preprint GOETP 33/93
"... We develop differential calculus and gauge theory on a finite set G. An elegant formulation is obtained when G is supplied with a group structure and in particular for a cyclic group. Connes ’ twopoint model (which is an essential ingredient of his reformulation of the standard model of elementary ..."
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Cited by 14 (6 self)
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We develop differential calculus and gauge theory on a finite set G. An elegant formulation is obtained when G is supplied with a group structure and in particular for a cyclic group. Connes ’ twopoint model (which is an essential ingredient of his reformulation of the standard model of elementary particle physics) is recovered in our approach. Reductions of the universal differential calculus to ‘lowerdimensional’ differential calculi are considered. The ‘complete reduction ’ leads to a differential calculus on a periodic lattice.