3,351 research outputs found
Quantum Field as a quantum cellular automaton: the Dirac free evolution in one dimension
We present a quantum cellular automaton model in one space-dimension which
has the Dirac equation as emergent. This model, a discrete-time and causal
unitary evolution of a lattice of quantum systems, is derived from the
assumptions of homogeneity, parity and time-reversal invariance. The comparison
between the automaton and the Dirac evolutions is rigorously set as a
discrimination problem between unitary channels. We derive an exact lower bound
for the probability of error in the discrimination as an explicit function of
the mass, the number and the momentum of the particles, and the duration of the
evolution. Computing this bound with experimentally achievable values, we see
that in that regime the QCA model cannot be discriminated from the usual Dirac
evolution. Finally, we show that the evolution of one-particle states with
narrow-band in momentum can be effi- ciently simulated by a dispersive
differential equation for any regime. This analysis allows for a comparison
with the dynamics of wave-packets as it is described by the usual Dirac
equation. This paper is a first step in exploring the idea that quantum field
theory could be grounded on a more fundamental quantum cellular automaton model
and that physical dynamics could emerge from quantum information processing. In
this framework, the discretization is a central ingredient and not only a tool
for performing non-perturbative calculation as in lattice gauge theory. The
automaton model, endowed with a precise notion of local observables and a full
probabilistic interpretation, could lead to a coherent unification of an
hypothetical discrete Planck scale with the usual Fermi scale of high-energy
physics.Comment: 21 pages, 4 figure
Quantum, Stochastic, and Pseudo Stochastic Languages with Few States
Stochastic languages are the languages recognized by probabilistic finite
automata (PFAs) with cutpoint over the field of real numbers. More general
computational models over the same field such as generalized finite automata
(GFAs) and quantum finite automata (QFAs) define the same class. In 1963, Rabin
proved the set of stochastic languages to be uncountable presenting a single
2-state PFA over the binary alphabet recognizing uncountably many languages
depending on the cutpoint. In this paper, we show the same result for unary
stochastic languages. Namely, we exhibit a 2-state unary GFA, a 2-state unary
QFA, and a family of 3-state unary PFAs recognizing uncountably many languages;
all these numbers of states are optimal. After this, we completely characterize
the class of languages recognized by 1-state GFAs, which is the only nontrivial
class of languages recognized by 1-state automata. Finally, we consider the
variations of PFAs, QFAs, and GFAs based on the notion of inclusive/exclusive
cutpoint, and present some results on their expressive power.Comment: A new version with new results. Previous version: Arseny M. Shur,
Abuzer Yakaryilmaz: Quantum, Stochastic, and Pseudo Stochastic Languages with
Few States. UCNC 2014: 327-33
Quantum cellular automata and free quantum field theory
In a series of recent papers it has been shown how free quantum field theory
can be derived without using mechanical primitives (including space-time,
special relativity, quantization rules, etc.), but only considering the easiest
quantum algorithm encompassing a countable set of quantum systems whose network
of interactions satisfies the simple principles of unitarity, homogeneity,
locality, and isotropy. This has opened the route to extending the axiomatic
information-theoretic derivation of the quantum theory of abstract systems to
include quantum field theory. The inherent discrete nature of the informational
axiomatization leads to an extension of quantum field theory to a quantum
cellular automata theory, where the usual field theory is recovered in a regime
where the discrete structure of the automata cannot be probed. A simple
heuristic argument sets the scale of discreteness to the Planck scale, and the
customary physical regime where discreteness is not visible is the relativistic
one of small wavevectors. In this paper we provide a thorough derivation from
principles that in the most general case the graph of the quantum cellular
automaton is the Cayley graph of a finitely presented group, and showing how
for the case corresponding to Euclidean emergent space (where the group resorts
to an Abelian one) the automata leads to Weyl, Dirac and Maxwell field dynamics
in the relativistic limit. We conclude with some perspectives towards the more
general scenario of non-linear automata for interacting quantum field theory.Comment: 10 pages, 2 figures, revtex style. arXiv admin note: substantial text
overlap with arXiv:1601.0483
An evolutionary model for simple ecosystems
In this review some simple models of asexual populations evolving on smooth
landscapes are studied. The basic model is based on a cellular automaton, which
is analyzed here in the spatial mean-field limit. Firstly, the evolution on a
fixed fitness landscape is considered. The correspondence between the time
evolution of the population and equilibrium properties of a statistical
mechanics system is investigated, finding the limits for which this mapping
holds. The mutational meltdown, Eigen's error threshold and Muller's ratchet
phenomena are studied in the framework of a simplified model. Finally, the
shape of a quasi-species and the condition of coexistence of multiple species
in a static fitness landscape are analyzed. In the second part, these results
are applied to the study of the coexistence of quasi-species in the presence of
competition, obtaining the conditions for a robust speciation effect in asexual
populations.Comment: 36 pages, including 16 figures, to appear in Annual Review of
Computational Physics, D. Stauffer (ed.), World Scientific, Singapor
Minimizing local automata
International audienceWe design an algorithm that minimizes irreducible deterministic local automata by a sequence of state mergings. Two states can be merged if they have exactly the same outputs. The running time of the algorithm is O(min(m(n − r + 1), m log n)), where m is the number of edges, n the number of states of the automaton, and r the number of states of the minimized automaton. In particular, the algorithm is linear when the automaton is already minimal and contrary to Hopcroft's minimisation algorithm that has a O(kn log n) running time in this case, where k is the size of the alphabet, and that applies only to complete automata. (Note that kn ≥ m.) While Hopcroft's algorithm relies on a "negative strategy", starting from a partition with a single class of all states, and partitioning classes when it is discovered that two states cannot belong to the sam class, our algorithm relies on a "positive strategy", starting from the trivial partition for which each class is a singleton. Two classes are then merged when their leaders have the same outputs. The algorithm applies to irreducible deterministic local automata, where all states are considered both initial and final. These automata, also called covers, recognize symbolic dynamical shifts of finite type. They serve to present a large class of constrained channels, the class of finite memory systems, used for channel coding purposes. The algorithm also applies to irreducible deterministic automata that are left-closing and have a synchronizing word. These automata present shifts that are called almost of finite type. Almost-of-finite-type shifts make a meaningful class of shifts, intermediate between finite type shifts and sofic shifts
Automata theoretic aspects of temporal behaviour and computability in logical neural networks
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