220 research outputs found
Quantum Cellular Automata
Quantum cellular automata (QCA) are reviewed, including early and more recent
proposals. QCA are a generalization of (classical) cellular automata (CA) and
in particular of reversible CA. The latter are reviewed shortly. An overview is
given over early attempts by various authors to define one-dimensional QCA.
These turned out to have serious shortcomings which are discussed as well.
Various proposals subsequently put forward by a number of authors for a general
definition of one- and higher-dimensional QCA are reviewed and their properties
such as universality and reversibility are discussed.Comment: 12 pages, 3 figures. To appear in the Springer Encyclopedia of
Complexity and Systems Scienc
A Simple n-Dimensional Intrinsically Universal Quantum Cellular Automaton
We describe a simple n-dimensional quantum cellular automaton (QCA) capable
of simulating all others, in that the initial configuration and the forward
evolution of any n-dimensional QCA can be encoded within the initial
configuration of the intrinsically universal QCA. Several steps of the
intrinsically universal QCA then correspond to one step of the simulated QCA.
The simulation preserves the topology in the sense that each cell of the
simulated QCA is encoded as a group of adjacent cells in the universal QCA.Comment: 13 pages, 7 figures. In Proceedings of the 4th International
Conference on Language and Automata Theory and Applications (LATA 2010),
Lecture Notes in Computer Science (LNCS). Journal version: arXiv:0907.382
A Quantum Game of Life
This research describes a three dimensional quantum cellular automaton (QCA)
which can simulate all other 3D QCA. This intrinsically universal QCA belongs
to the simplest subclass of QCA: Partitioned QCA (PQCA). PQCA are QCA of a
particular form, where incoming information is scattered by a fixed unitary U
before being redistributed and rescattered. Our construction is minimal amongst
PQCA, having block size 2 x 2 x 2 and cell dimension 2. Signals, wires and
gates emerge in an elegant fashion.Comment: 13 pages, 10 figures. Final version, accepted by Journ\'ees Automates
Cellulaires (JAC 2010)
Unitarity plus causality implies localizability
We consider a graph with a single quantum system at each node. The entire
compound system evolves in discrete time steps by iterating a global evolution
. We require that this global evolution be unitary, in accordance with
quantum theory, and that this global evolution be causal, in accordance
with special relativity. By causal we mean that information can only ever be
transmitted at a bounded speed, the speed bound being quite naturally that of
one edge of the underlying graph per iteration of . We show that under these
conditions the operator can be implemented locally; i.e. it can be put into
the form of a quantum circuit made up with more elementary operators -- each
acting solely upon neighbouring nodes. We take quantum cellular automata as an
example application of this representation theorem: this analysis bridges the
gap between the axiomatic and the constructive approaches to defining QCA.
KEYWORDS: Quantum cellular automata, Unitary causal operators, Quantum walks,
Quantum computation, Axiomatic quantum field theory, Algebraic quantum field
theory, Discrete space-time.Comment: V1: 5 pages, revtex. V2: Generalizes V1. V3: More precisions and
reference
When is a quantum cellular automaton (QCA) a quantum lattice gas automaton (QLGA)?
Quantum cellular automata (QCA) are models of quantum computation of
particular interest from the point of view of quantum simulation. Quantum
lattice gas automata (QLGA - equivalently partitioned quantum cellular
automata) represent an interesting subclass of QCA. QLGA have been more deeply
analyzed than QCA, whereas general QCA are likely to capture a wider range of
quantum behavior. Discriminating between QLGA and QCA is therefore an important
question. In spite of much prior work, classifying which QCA are QLGA has
remained an open problem. In the present paper we establish necessary and
sufficient conditions for unbounded, finite Quantum Cellular Automata (QCA)
(finitely many active cells in a quiescent background) to be Quantum Lattice
Gas Automata (QLGA). We define a local condition that classifies those QCA that
are QLGA, and we show that there are QCA that are not QLGA. We use a number of
tools from functional analysis of separable Hilbert spaces and representation
theory of associative algebras that enable us to treat QCA on finite but
unbounded configurations in full detail.Comment: 37 pages, 7 figures, with changes to explanatory text and updated
figures, J. Math. Phys. versio
The Differential Scheme and Quantum Computation
It is well-known that standard models of computation are representable as simple dynamical systems that evolve in discrete time, and that systems that evolve in continuous time are often representable by dynamical systems governed by ordinary differential equations. In many applications, e.g., molecular networks and hybrid Fermi-Pasta-Ulam systems, one must work with dynamical systems comprising both discrete and continuous components.
Reasoning about and verifying the properties of the evolving state of such systems is currently a piecemeal affair that depends on the nature of major components of a system: e.g., discrete vs. continuous components of state, discrete vs. continuous time, local vs. distributed clocks, classical vs. quantum states and state evolution.
We present the Differential Scheme as a unifying framework for reasoning about and verifying the properties of the evolving state of a system, whether the system in question evolves in discrete time, as for standard models of computation, or continuous time, or a combination of both. We show how instances of the differential scheme can accommodate classical computation.
We also generalize a relatively new model of quantum computation, the quantum cellular automaton, with an eye towards extending the differential scheme to accommodate quantum computation and hybrid classical/quantum computation.
All the components of a specific instance of the differential scheme are Convergence Spaces. Convergence spaces generalize notions of continuity and convergence. The category of convergence spaces, Conv, subsumes both simple discrete structures (e.g., digraphs), and complex continuous structures (e.g., topological spaces, domains, and the standard fields of analysis: R and C). We present novel uses for convergence spaces, and extend their theory by defining differential calculi on Conv. It is to the use of convergence spaces that the differential scheme owes its generality and flexibility
Information and the reconstruction of quantum physics
The reconstruction of quantum physics has been connected with the interpretation of the quantum formalism, and has continued to be so with the recent deeper consideration of the relation of information to quantum states and processes. This recent form of reconstruction has mainly involved conceiving quantum theory on the basis of informational principles, providing new perspectives on physical correlations and entanglement that can be used to encode information. By contrast to the traditional, interpretational approach to the foundations of quantum mechanics, which attempts directly to establish the meaning of the elements of the theory and often touches on metaphysical issues, the newer, more purely reconstructive approach sometimes defers this task, focusing instead on the mathematical derivation of the theoretical apparatus from simple principles or axioms. In its most pure form, this sort of theory reconstruction is fundamentally the mathematical derivation of the elements of theory from explicitly presented, often operational principles involving a minimum of extraâmathematical content. Here, a representative series of specifically informationâbased treatmentsâfrom partial reconstructions that make connections with information to rigorous axiomatizations, including those involving the theories of generalized probability and abstract systemsâis reviewed.Accepted manuscrip
MFCS\u2798 Satellite Workshop on Cellular Automata
For the 1998 conference on Mathematical Foundations of Computer
Science (MFCS\u2798) four papers on Cellular Automata were accepted as
regular MFCS\u2798 contributions. Furthermore an MFCS\u2798 satellite
workshop on Cellular Automata was organized with ten additional talks.
The embedding of the workshop into the conference with its
participants coming from a broad spectrum of fields of work lead to
interesting discussions and a fruitful exchange of ideas.
The contributions which had been accepted for MFCS\u2798 itself may be
found in the conference proceedings, edited by L. Brim, J. Gruska and
J. Zlatuska, Springer LNCS 1450. All other (invited and regular)
papers of the workshop are contained in this technical report. (One
paper, for which no postscript file of the full paper is available, is
only included in the printed version of the report).
Contents:
F. Blanchard, E. Formenti, P. Kurka: Cellular automata in the Cantor,
Besicovitch and Weyl Spaces
K. Kobayashi: On Time Optimal Solutions of the Two-Dimensional Firing
Squad Synchronization Problem
L. Margara: Topological Mixing and Denseness of Periodic Orbits for
Linear Cellular Automata over Z_m
B. Martin: A Geometrical Hierarchy of Graph via Cellular Automata
K. Morita, K. Imai: Number-Conserving Reversible Cellular Automata and
Their Computation-Universality
C. Nichitiu, E. Remila: Simulations of graph automata
K. Svozil: Is the world a machine?
H. Umeo: Cellular Algorithms with 1-bit Inter-Cell Communications
F. Reischle, Th. Worsch: Simulations between alternating CA,
alternating TM and circuit families
K. Sutner: Computation Theory of Cellular Automat
Beyond the Spectral Theorem: Spectrally Decomposing Arbitrary Functions of Nondiagonalizable Operators
Nonlinearities in finite dimensions can be linearized by projecting them into
infinite dimensions. Unfortunately, often the linear operator techniques that
one would then use simply fail since the operators cannot be diagonalized. This
curse is well known. It also occurs for finite-dimensional linear operators. We
circumvent it by developing a meromorphic functional calculus that can
decompose arbitrary functions of nondiagonalizable linear operators in terms of
their eigenvalues and projection operators. It extends the spectral theorem of
normal operators to a much wider class, including circumstances in which poles
and zeros of the function coincide with the operator spectrum. By allowing the
direct manipulation of individual eigenspaces of nonnormal and
nondiagonalizable operators, the new theory avoids spurious divergences. As
such, it yields novel insights and closed-form expressions across several areas
of physics in which nondiagonalizable dynamics are relevant, including
memoryful stochastic processes, open non unitary quantum systems, and
far-from-equilibrium thermodynamics.
The technical contributions include the first full treatment of arbitrary
powers of an operator. In particular, we show that the Drazin inverse,
previously only defined axiomatically, can be derived as the negative-one power
of singular operators within the meromorphic functional calculus and we give a
general method to construct it. We provide new formulae for constructing
projection operators and delineate the relations between projection operators,
eigenvectors, and generalized eigenvectors.
By way of illustrating its application, we explore several, rather distinct
examples.Comment: 29 pages, 4 figures, expanded historical citations;
http://csc.ucdavis.edu/~cmg/compmech/pubs/bst.ht
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