321 research outputs found
5-State Rotation-Symmetric Number-Conserving Cellular Automata are not Strongly Universal
We study two-dimensional rotation-symmetric number-conserving cellular
automata working on the von Neumann neighborhood (RNCA). It is known that such
automata with 4 states or less are trivial, so we investigate the possible
rules with 5 states. We give a full characterization of these automata and show
that they cannot be strongly Turing universal. However, we give example of
constructions that allow to embed some boolean circuit elements in a 5-states
RNCA
Number-conserving cellular automata with a von Neumann neighborhood of range one
We present necessary and sufficient conditions for a cellular automaton with
a von Neumann neighborhood of range one to be number-conserving. The conditions
are formulated for any dimension and for any set of states containing zero. The
use of the geometric structure of the von Neumann neighborhood allows for
computationally tractable conditions even in higher dimensions.Comment: 15 pages, 3 figure
A split-and-perturb decomposition of number-conserving cellular automata
This paper concerns -dimensional cellular automata with the von Neumann
neighborhood that conserve the sum of the states of all their cells. These
automata, called number-conserving or density-conserving cellular automata, are
of particular interest to mathematicians, computer scientists and physicists,
as they can serve as models of physical phenomena obeying some conservation
law. We propose a new approach to study such cellular automata that works in
any dimension and for any set of states . Essentially, the local rule of
a cellular automaton is decomposed into two parts: a split function and a
perturbation. This decomposition is unique and, moreover, the set of all
possible split functions has a very simple structure, while the set of all
perturbations forms a linear space and is therefore very easy to describe in
terms of its basis. We show how this approach allows to find all
number-conserving cellular automata in many cases of and . In
particular, we find all three-dimensional number-conserving CAs with three
states, which until now was beyond the capabilities of computers
A guided tour of asynchronous cellular automata
Research on asynchronous cellular automata has received a great amount of
attention these last years and has turned to a thriving field. We survey the
recent research that has been carried out on this topic and present a wide
state of the art where computing and modelling issues are both represented.Comment: To appear in the Journal of Cellular Automat
Nonequilibrium Critical Phenomena and Phase Transitions into Absorbing States
This review addresses recent developments in nonequilibrium statistical
physics. Focusing on phase transitions from fluctuating phases into absorbing
states, the universality class of directed percolation is investigated in
detail. The survey gives a general introduction to various lattice models of
directed percolation and studies their scaling properties, field-theoretic
aspects, numerical techniques, as well as possible experimental realizations.
In addition, several examples of absorbing-state transitions which do not
belong to the directed percolation universality class will be discussed. As a
closely related technique, we investigate the concept of damage spreading. It
is shown that this technique is ambiguous to some extent, making it impossible
to define chaotic and regular phases in stochastic nonequilibrium systems.
Finally, we discuss various classes of depinning transitions in models for
interface growth which are related to phase transitions into absorbing states.Comment: Review article, revised version, LaTeX, 153 pages, 63 encapsulated
postscript figure
Quanten-Gittersysteme mit diskreten Zeitschritten
Discrete time quantum lattice systems recently have come into the focus of quantum computation because they provide a versatile tool for many different applications and they are potentially implementable in current experimental realizations. In this thesis we study the fundamental structures of such quantum lattice systems as well as consequences of experimental imperfections.
Essentially, there are two models of discrete time quantum lattice systems, namely quantum cellular automata and quantum walks, which are quantum versions of their classical counterparts, i.e., cellular automata and random walks. In both cases, the dynamics acts locally on the lattice and is usually also translationally invariant. The main difference between these structures is that quantum cellular automata can describe the dynamics of many interacting particles, where quantum walks describe the evolution of a single particle.
The first part of this thesis is devoted to quantum cellular automata. We characterize one-dimensional quantum cellular automata in terms of an index theory up to local deformations. Further, we characterize in detail a subclass of quantum cellular automata by requiring that Pauli operators are mapped to Pauli operators. This structure can be understood in terms of certain classical cellular automata.
The second part of this thesis is concerned with quantum walks. We identify a quantum walk with the one-particle sector of a quantum cellular automaton. We also establish an index theory for quantum walks and we discuss decoherent quantum walks, i.e., the behavior of quantum walks with experimental imperfections.Quanten-Gittersysteme haben in den letzten Jahren zunehmend an Bedeutung im Bereich des Quantenrechnens gewonnen, weil sie ein vielseitiges Instrument fĂĽr unterschiedliche Anwendungen darstellen und in derzeitigen experimentellen Realisierungen potentiell implementierbar sind. Wir untersuchen in dieser Arbeit sowohl grundlegende Strukturen solcher Quanten-Gittersysteme als auch experimentelle Imperfektionen.
Im wesentlichen gibt es zwei Modelle von Quanten-Gittersystemen in diskreter Zeit: Quanten-Zellularautomaten und Quanten Walks. In beiden Fällen ist die Dynamik lokal und translationsinvariant. Der Hauptunterschied besteht darin, dass Quanten-Zellularautomaten viele miteinander wechselwirkende Teilchen beschreiben können, wohingegen Quanten Walks die Zeitentwicklung eines einzelnes Teilchen darstellen. Zu beiden Modellen gibt es entsprechende klassischen Strukturen, nämlich Zellularautomaten, bzw. Random Walks.
Im ersten Teil dieser Arbeit werden Quanten-Zellularautomaten behandelt. Wir charakterisieren eindimensionale Automaten mithilfe einer Index Theorie bis auf lokale Deformation. AuĂźerdem untersuchen wir im Detail die Struktur einer Unterklasse von Quanten-Zellularautomaten, die dadurch festgelegt ist, dass Pauli Operatoren auf Pauli Operatoren abgebildet werden. Wir zeigen, dass sich solche Automaten durch spezielle klassische Zellularautomaten verstehen lassen.
Im zweiten Teil dieser Arbeit behandeln wir Quanten Walks, welche wir mit Ein-Teilchen-Sektoren von Quanten-Zellularautomaten identifizieren. Wir führen ebenso eine Index Theorie für Quanten Walks ein und wir diskutieren dekohärente Quanten Walks, d.h., das Verhalten von Quanten Walks mit experimentellen Imperfektionen
Entanglement dynamics and phase transitions of the Floquet cluster spin chain
Cluster states were introduced in the context of measurement based quantum
computing. In one dimension, the cluster Hamiltonian possesses topologically
protected states. We investigate the Floquet dynamics of the cluster spin chain
in an external field, interacting with a particle. We explore the entanglement
properties of the topological and magnetic phases, first in the integrable spin
lattice case, and then in the interacting quantum walk case. We find, in
addition to thermalization, dynamical phase transitions separating low and high
entangled nonthermal states, reminiscent of the ones present in the integrable
case, but differing in their magnetic properties.Comment: 12 pages, 9 figures; comments are welcom
Hilbert space fragmentation and slow dynamics in particle-conserving quantum East models
Quantum kinetically constrained models have recently attracted significant
attention due to their anomalous dynamics and thermalization. In this work, we
introduce a hitherto unexplored family of kinetically constrained models
featuring a conserved particle number and strong inversion-symmetry breaking
due to facilitated hopping. We demonstrate that these models provide a generic
example of so-called quantum Hilbert space fragmentation, that is manifested in
disconnected sectors in the Hilbert space that are not apparent in the
computational basis. Quantum Hilbert space fragmentation leads to an
exponential in system size number of eigenstates with exactly zero entanglement
entropy across several bipartite cuts. These eigenstates can be probed
dynamically using quenches from simple initial product states. In addition, we
study the particle spreading under unitary dynamics launched from the domain
wall state, and find faster than diffusive dynamics at high particle densities,
that crosses over into logarithmically slow relaxation at smaller densities.
Using a classically simulable cellular automaton, we reproduce the logarithmic
dynamics observed in the quantum case. Our work suggests that particle
conserving constrained models with inversion symmetry breaking realize so far
unexplored universality classes of dynamics and invite their further
theoretical and experimental studies
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