4,666 research outputs found
Post-surjectivity and balancedness of cellular automata over groups
We discuss cellular automata over arbitrary finitely generated groups. We
call a cellular automaton post-surjective if for any pair of asymptotic
configurations, every pre-image of one is asymptotic to a pre-image of the
other. The well known dual concept is pre-injectivity: a cellular automaton is
pre-injective if distinct asymptotic configurations have distinct images. We
prove that pre-injective, post-surjective cellular automata are reversible.
Moreover, on sofic groups, post-surjectivity alone implies reversibility. We
also prove that reversible cellular automata over arbitrary groups are
balanced, that is, they preserve the uniform measure on the configuration
space.Comment: 16 pages, 3 figures, LaTeX "dmtcs-episciences" document class. Final
version for Discrete Mathematics and Theoretical Computer Science. Prepared
according to the editor's request
On algebraic cellular automata
We investigate some general properties of algebraic cellular automata, i.e.,
cellular automata over groups whose alphabets are affine algebraic sets and
which are locally defined by regular maps. When the ground field is assumed to
be uncountable and algebraically closed, we prove that such cellular automata
always have a closed image with respect to the prodiscrete topology on the
space of configurations and that they are reversible as soon as they are
bijective
Quantum Walks and Reversible Cellular Automata
We investigate a connection between a property of the distribution and a
conserved quantity for the reversible cellular automaton derived from a
discrete-time quantum walk in one dimension. As a corollary, we give a detailed
information of the quantum walk.Comment: 15 pages, minor corrections, some references adde
On the structure of Clifford quantum cellular automata
We study reversible quantum cellular automata with the restriction that these
are also Clifford operations. This means that tensor products of Pauli
operators (or discrete Weyl operators) are mapped to tensor products of Pauli
operators. Therefore Clifford quantum cellular automata are induced by
symplectic cellular automata in phase space. We characterize these symplectic
cellular automata and find that all possible local rules must be, up to some
global shift, reflection invariant with respect to the origin. In the one
dimensional case we also find that every uniquely determined and
translationally invariant stabilizer state can be prepared from a product state
by a single Clifford cellular automaton timestep, thereby characterizing these
class of stabilizer states, and we show that all 1D Clifford quantum cellular
automata are generated by a few elementary operations. We also show that the
correspondence between translationally invariant stabilizer states and
translationally invariant Clifford operations holds for periodic boundary
conditions.Comment: 28 pages, 2 figures, LaTe
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