One-dimensional Cellular Automata in Quantum and Fermionic Theories

The thesis deals with quantum cellular automata (QCAs) and Fermionic quantum cellular automata (FQCAs) on one-dimensional lattices. With the term cellular automaton, we refer to a class of algorithms that can process information distributed on a regular grid in a local fashion. Its quantum counterpart—where at each site of the grid we can find a quantum system—represents a model for massive parallel quantum computation on finitely generated grids. The model is particularly well-suited for describing and simulating a vast class of physical phenomena. The work presented in the thesis is threefold. We first introduce a new definition of QCA in terms of super maps, namely functions from quantum operations to quantum operations, that preserves locality and composition of transformations. Thereby, we define the so-called T-operator, i.e. a local operator that incorporates all the necessary information for univocally defining a QCA. The T-operator plays here the role of the Choi operator of the automaton. Secondly, we classify all nearest-neighbor FQCAs over the one-dimensional lattice where each site contains one local Fermionic mode. We observe that the Fermionic automata are divided into two classes. In the first one, we find some FQCAs that are equivalent to a subset of quantum cellular automata. On the other hand, the second class of FQCAs has been found to have no quantum counterparts. Finally, we report the experimental realization of a photonic platform to simulate the evolution of a one-dimensional quantum walk, i.e. a quantum cellular automaton whose action is linear in the field operators. Specifically, we observe the Zitterbewegung of a particle satisfying the Dirac dispersion relation. The theoretical background, numerical simulation, and optimization of the parameter space are discussed with special attention

Topology regulates pattern formation capacity of binary cellular automata on graphs

We study the effect of topology variation on the dynamic behavior of a system with local update rules. We implement one-dimensional binary cellular automata on graphs with various topologies by formulating two sets of degree-dependent rules, each containing a single parameter. We observe that changes in graph topology induce transitions between different dynamic domains (Wolfram classes) without a formal change in the update rule. Along with topological variations, we study the pattern formation capacities of regular, random, small-world and scale-free graphs. Pattern formation capacity is quantified in terms of two entropy measures, which for standard cellular automata allow a qualitative distinction between the four Wolfram classes. A mean-field model explains the dynamic behavior of random graphs. Implications for our understanding of information transport through complex, network-based systems are discussed.Comment: 16 text pages, 13 figures. To be published in Physica

The inverse behavior of a reversible one-dimensional cellular automaton obtained by a single welch diagram

Reversible cellular automata are discrete dynamical systems based on local interactions which are able to produce an invertible global behavior. Reversible automata have been carefully analyzed by means of graph and matrix tools, in particular the extensions of the ancestors in these systems have a complete representation by Welch diagrams. This paper illustrates how the whole information of a reversible one-dimensional cellular automaton is conserved at both sides of the ancestors for sequences with an adequate length. We give this result implementing a procedure to obtain the inverse behavior by means of calculating and studying a single Welch diagram corresponding with the extensions of only one side of the ancestors. This work is a continuation of our study about reversible automata both in the local and global sense. An illustrative example is also presented

A tight linear bound on the synchronization delay of bijective automata

AbstractReversible cellular automata (RCA) are models of massively parallel computation that preserve information. We generalize these systems by introducing the class of ωωbijective finite automata. It consists of those finite automata where for any bi-infinite word there exists a unique path labelled by that word. These systems are strictly included in the class of local automata. Although the synchronization delay of an n-state local automaton is known to be Θ(n2) in the worst case, we prove that in the case of ωωbijective finite automata the synchronization delay is at most n−1. Based on this we prove that for a one-dimensional n-state RCA where the neighborhood consists of m consecutive cells, the neighbourhood of the inverse automaton consists of at most nm−1−(m−1) cells. Similar bounds are obtained also in [E. Czeizler, J. Kari, A tight linear bound on the neighborhood of inverse cellular automata, in: Proceedings of ICALP 2005, in: LNCS, vol. 3580, 2005, pp. 410–420] but here the result comes as a direct consequence of the more general result. We also construct examples of RCA with large inverse neighbourhoods proving that the upper bounds provided here are the best possible in the case m=2

Fungal Automata

We study a cellular automaton (CA) model of information dynamics on a single hypha of a fungal mycelium. Such a filament is divided in compartments (here also called cells) by septa. These septa are invaginations of the cell wall and their pores allow for flow of cytoplasm between compartments and hyphae. The septal pores of the fungal phylum of the Ascomycota can be closed by organelles called Woronin bodies. Septal closure is increased when the septa become older and when exposed to stress conditions. Thus, Woronin bodies act as informational flow valves. The one dimensional fungal automata is a binary state ternary neighbourhood CA, where every compartment follows one of the elementary cellular automata (ECA) rules if its pores are open and either remains in state `0' (first species of fungal automata) or its previous state (second species of fungal automata) if its pores are closed. The Woronin bodies closing the pores are also governed by ECA rules. We analyse a structure of the composition space of cell-state transition and pore-state transitions rules, complexity of fungal automata with just few Woronin bodies, and exemplify several important local events in the automaton dynamics

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

When--and how--can a cellular automaton be rewritten as a lattice gas?

Both cellular automata (CA) and lattice-gas automata (LG) provide finite algorithmic presentations for certain classes of infinite dynamical systems studied by symbolic dynamics; it is customary to use the term `cellular automaton' or `lattice gas' for the dynamic system itself as well as for its presentation. The two kinds of presentation share many traits but also display profound differences on issues ranging from decidability to modeling convenience and physical implementability. Following a conjecture by Toffoli and Margolus, it had been proved by Kari (and by Durand--Lose for more than two dimensions) that any invertible CA can be rewritten as an LG (with a possibly much more complex ``unit cell''). But until now it was not known whether this is possible in general for noninvertible CA--which comprise ``almost all'' CA and represent the bulk of examples in theory and applications. Even circumstantial evidence--whether in favor or against--was lacking. Here, for noninvertible CA, (a) we prove that an LG presentation is out of the question for the vanishingly small class of surjective ones. We then turn our attention to all the rest--noninvertible and nonsurjective--which comprise all the typical ones, including Conway's `Game of Life'. For these (b) we prove by explicit construction that all the one-dimensional ones are representable as LG, and (c) we present and motivate the conjecture that this result extends to any number of dimensions. The tradeoff between dissipation rate and structural complexity implied by the above results have compelling implications for the thermodynamics of computation at a microscopic scale.Comment: 16 page