Cellular automata computation

Ii89] Li, W., Complex patterns generated by next nearest neighbor cellular automata, Computers and Graphics 13 (1989). 

Following photoinduced electron injection (a) from the Ru or Os, the hole produced (b) resides on the distal Os so that the electron-hole recombination is five orders of magnitude slower than electron injection. Binuclear mixed-valence complexes have been proposed as components in quantum dot cellular automata computational schemes. ... 

Peng-Zhi Pan, Fei Yan, Xia-Ting Feng. 2012. Modeling the cracking process of rocks from continuity to discontinuity using a cellular automaton. Computers ... 

Fig. 30. Computer simulation by the cellular automaton technique of spatial pattern evolution during CO oxidation on Pt(100). (From Ref. 32.)...

Partial differential equations represent one approach (a computationally intensive one) to simulating reaction-diffusion phenomena. An alternative, more approximate, but often less expensive and more intuitive technique employs cellular automata. A cellular automaton consists of an array of cells and a set of rules by which the state of a cell changes at each discrete time step. The state of the cell can, for example, represent the numbers of particles or concentrations of species in that cell, and the rules, which depend on the current state of the cell and its neighbors, can be chosen to mimic diffusion and chemical reaction. 

Bertrand et al. (2007) proposed a model adopting cellular automaton, where the system is divided in millions of cubic cells. Five different states are allowed for each cell polymer (P), solvent (S), porosity (E), solid dmg (D), and solubilized drug (SD). The nonerosion probability of a specific cell in the matrix Fy is then computed ... 

Before showing how this can be done, we must first define more precisely what a cellular automaton (CA) is. A CA model for a system is constructed as follows The dynamical evolution is imagined to take place on a set of cells. Each cell is defined by a finite set of values of a dynamical variable = sk-k = l,...,n. The values of the s in a cell change at discrete time instants according to a rule that depends on the value of s in the cell as well as that of its neighbors. Even very simple rules, 3i, can produce complicated dynamics. In fact, CA rules that are Turing machines, that is, are capable of universal computation, can be devised. 

Twamley, J. Quantum-cellular-automaton quantum computing with endohedal fullerenes. Phys. Rev. A 67, 052318 (2003)... 

Von Neumann was the first to design a cellular automaton capable of universal computation. Since then many others became interested in the problem and either improved on his original construction or have developed alternative constructions. An early example was Alvy Ray Smith (1971), who constructed a series of progressively simpler cellular automata capable of universal computation. The first in the series was a two-dimensional cellular automaton Mg that simulates a given Turing machine M in real time —that is, there is a one-to-one correspondence between the time steps of Mg and the time steps of M. 

In Sects. 5.6 - 5.8 an introduction to Feynman s idea and its generalizations is given. We explain how to construct a locally connected serial quantum computer and a quantum cellular automaton. 

Notice that, if making a measurement, the cellular automaton will most probably be in a state with sites that have different computational ages. Some of the sites will contain part of a final result, others might still have to catch up, others might be ahead already. However, it is possible to give a quantum description of an automaton, where sites that contain a final result will all have the same age [52]. 

Obviously, it is again possible to use Peres notation Instead of the creation and annihilation operators, projection operators can be employed. However, for the cellular automaton, this notation turns out to be more complicated. Thus, we will adopt throughout the original Feynman notation for the cellular automaton and also for the Feynman computer when the two are compared. 

Not every computation can be performed by the computers with local Hamiltonians described so far. Neither the Feynman computer nor the one-dimensional cellular automaton is universal in the sense of complexity theory. In [21], Feynman s ideas are used to describe a two-dimensional quantum cellular automaton, where the classical version of the automaton is universal. As for the one-dimensional automaton, it is most likely to measme a state where part of the automaton has done more computational steps than another. It is not possible to obtain global synchrony as in the classical case since there is no global synchronous updating which can be performed by local means [32]. Thus, it is not clear whether one should apply the term universal to those automata. 

In other cases, where different parts of the computer may be updated simultaneously, the same problem as for the cellular automaton occurs Upon measurement, parts of the computer may have evolved further than others. This is not a problem for the machine that will now be described, since until the final result is stored, only one local region is updated by applying U. 

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