Talk:PlanetPhysics/Categories of Quantum Automata and Quantum Computers

Original TeX Content from PlanetPhysics Archive

%%% This file is part of PlanetPhysics snapshot of 2011-09-01 %%% Primary Title: categories of quantum automata and quantum computers %%% Primary Category Code: 03.67.-a %%% Filename: CategoriesOfQuantumAutomataAndQuantumComputers.tex %%% Version: 4 %%% Owner: bci1 %%% Author(s): bci1 %%% PlanetPhysics is released under the GNU Free Documentation License. %%% You should have received a file called fdl.txt along with this file. %%% If not, please write to gnu@gnu.org. \documentclass[12pt]{article} \pagestyle{empty} \setlength{\paperwidth}{8.5in} \setlength{\paperheight}{11in}

\setlength{\topmargin}{0.00in} \setlength{\headsep}{0.00in} \setlength{\headheight}{0.00in} \setlength{\evensidemargin}{0.00in} \setlength{\oddsidemargin}{0.00in} \setlength{\textwidth}{6.5in} \setlength{\textheight}{9.00in} \setlength{\voffset}{0.00in} \setlength{\hoffset}{0.00in} \setlength{\marginparwidth}{0.00in} \setlength{\marginparsep}{0.00in} \setlength{\parindent}{0.00in} \setlength{\parskip}{0.15in}

\usepackage{html}

% this is the default PlanetPhysics preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners.

% almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts}

% used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic}

% there are many more packages, add them here as you need them

% define commands here

\usepackage{graphicx} \newtheorem{thm}{Theorem}[section] \newtheorem{cor}[thm]{Corollary} \numberwithin{equation}{section} \newcommand{\norm}[1]{\left\Vert#1\right\Vert} \newcommand{\abs}[1]{\left\vert#1\right\vert} \newcommand{\set}[1]{\left\{#1\right\}} \newcommand{\Real}{\mathbb R} \newcommand{\eps}{\varepsilon} \newcommand{\To}{\longrightarrow} \newcommand{\BX}{\mathbf{B}(X)} \newcommand{\A}{\mathcal{A}}

\begin{document}

\section{Categories of Quantum Automata, \\

N-- \L ukasiewicz Algebras and Quantum Computers}

\htmladdnormallink{Quantum automata}{http://planetphysics.us/encyclopedia/QuantumComputers.html} were defined (in ref.\cite{IB71}) as generalized, probabilistic automata with \htmladdnormallink{quantum state spaces}{http://planetphysics.us/encyclopedia/NonAbelianQuantumAlgebraicTopology3.html}. Their next-state \htmladdnormallink{functions}{http://planetphysics.us/encyclopedia/Bijective.html} operate through transitions between quantum states defined by the quantum equations of \htmladdnormallink{motions}{http://planetphysics.us/encyclopedia/CosmologicalConstant.html} in the Schr\"{o}dinger \htmladdnormallink{representation}{http://planetphysics.us/encyclopedia/CategoricalGroupRepresentation.html}, with both initial and \htmladdnormallink{boundary}{http://planetphysics.us/encyclopedia/GenericityInOpenSystems.html} conditions in \htmladdnormallink{space-time}{http://planetphysics.us/encyclopedia/SR.html}. A new \htmladdnormallink{theorem}{http://planetphysics.us/encyclopedia/Formula.html} is proven which states that the \emph{\htmladdnormallink{category of quantum automata}{http://planetphysics.us/encyclopedia/CategoryOfQuantumAutomata.html} and automata--homomorphisms has both limits and colimits.} Therefore, both categories of quantum automata and classical automata (\htmladdnormallink{sequential machines}{http://planetphysics.us/encyclopedia/AAT.html}) are \emph{bicomplete.} A second new theorem establishes that the standard automata \htmladdnormallink{category}{http://planetphysics.us/encyclopedia/Cod.html} is a subcategory of the quantum automata category. The quantum automata category has a faithful representation in the category of Generalized (\textbf{M,R})--Systems which are open, \htmladdnormallink{dynamic}{http://planetphysics.us/encyclopedia/NewtonianMechanics.html} bio-networks (\cite{ICB87}) with defined biological \htmladdnormallink{relations}{http://planetphysics.us/encyclopedia/Bijective.html} that represent physiological functions of primordial(s), single cells and the simpler organisms. A new \emph{category of quantum computers} is also defined in terms of \emph{reversible} quantum automata with quantum state spaces represented by \htmladdnormallink{topological groupoids}{http://planetphysics.us/encyclopedia/GroupoidHomomorphism2.html} that admit a local characterization through unique 'quantum' \emph{\htmladdnormallink{Lie algebroids}{http://planetphysics.us/encyclopedia/LieAlgebroids.html}}. On the other hand, the category of n-- \textsl{\L}ukasiewicz algebras has a subcategory of \emph{centered} n-- \textsl{\L}ukasiewicz algebras (ref. \cite{GGV70}) which can be employed to design and construct subcategories of quantum automata based on n--\textsl{\L}ukasiewicz \htmladdnormallink{diagrams}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} of existing VLSI. Furthermore, as shown in ref.(\cite{GGV70}) the category of centered n--\textsl{\L}ukasiewicz algebras and the category of Boolean algebras are naturally equivalent. A `no-go' conjecture is also proposed which states that Generalized (\textbf{M,R})--Systems \htmladdnormallink{complexity}{http://planetphysics.us/encyclopedia/Complexity.html} prevents their complete computability (\cite{ICB87,BGGB2k7}) by either standard or quantum automata.

\begin{thebibliography}{99}

\bibitem{IB71} Baianu, I.1971.``Organismic Supercategories and Qualitative Dynamics of Systems." \emph{Bull. Math.Biophysics}., 33, 339-353.

\bibitem{GGV70} Georgescu, G. and C. Vraciu 1970. ``On the Characterization of \L ukasiewicz Algebras." \emph{J. Algebra}, 16 (4), 486-495.

\bibitem{ICB77} Baianu, I.C. 1977. ``A Logical Model of Genetic Activities in \L ukasiewicz Algebras: The Non-linear Theory." \emph{Bulletin of Mathematical Biology}, 39:249-258 (1977).

\bibitem{ICB87} Baianu, I.C. 1987. ``Computer Models and Automata Theory in Biology and Medicine" (A Review). In: \emph{"Mathematical Models in Medicine.}",vol.7., M. Witten, Ed., Pergamon Press: New York, pp.1513-1577.

\bibitem{BGGB2k7} Baianu, I.C., J. Glazebrook, G. Georgescu and R.Brown. 2007. ``A Novel Approach to Complex Systems Biology based on Categories, Higher Dimensional Algebra and A Generalized \L ukasiewicz Topos. " , \emph{Axiomathes},vol.17,(in press): 46 pp.

\end{thebibliography}

\end{document}