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PlanetPhysics/Categories of Quantum Automata and Quantum Computers

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\section{Categories of Quantum Automata, \\ N-- \L ukasiewicz Algebras and Quantum Computers}

Quantum automata were defined (in ref.[1]) as generalized, probabilistic automata with quantum state spaces. Their next-state functions operate through transitions between quantum states defined by the quantum equations of motions in the Schr\"{o}dinger representation, with both initial and boundary conditions in space-time. A new theorem is proven which states that the \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 (sequential machines) are bicomplete. A second new theorem establishes that the standard automata category is a subcategory of the quantum automata category. The quantum automata category has a faithful representation in the category of Generalized (M,R )--Systems which are open, dynamic bio-networks ([2]) with defined biological relations that represent physiological functions of primordial(s), single cells and the simpler organisms. A new category of quantum computers is also defined in terms of reversible quantum automata with quantum state spaces represented by topological groupoids that admit a local characterization through unique 'quantum' Lie algebroids. On the other hand, the category of n-- \textsl{\L}ukasiewicz algebras has a subcategory of centered n-- \textsl{\L}ukasiewicz algebras (ref. [3]) which can be employed to design and construct subcategories of quantum automata based on n--\textsl{\L}ukasiewicz diagrams of existing VLSI. Furthermore, as shown in ref.([3]) 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 (M,R )--Systems complexity prevents their complete computability ([4]) by either standard or quantum automata.

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[1] [3] [5] [2] [6]

References

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  1. ↑ 1.0 1.1 Baianu, I.1971."Organismic Supercategories and Qualitative Dynamics of Systems." Bull. Math.Biophysics ., 33, 339-353.
  2. ↑ 2.0 2.1 Baianu, I.C. 1987. "Computer Models and Automata Theory in Biology and Medicine" (A Review). In: Mathematical Models in Medicine. ",vol.7., M. Witten, Ed., Pergamon Press: New York, pp.1513-1577.
  3. ↑ 3.0 3.1 3.2 Georgescu, G. and C. Vraciu 1970. "On the Characterization of \L ukasiewicz Algebras." J. Algebra , 16 (4), 486-495.
  4. ↑ Cite error: Invalid <ref> tag; no text was provided for refs named ICB87,BGGB2k7
  5. ↑ Baianu, I.C. 1977. ``A Logical Model of Genetic Activities in \L ukasiewicz Algebras: The Non-linear Theory." Bulletin of Mathematical Biology , 39:249-258 (1977).
  6. ↑ 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. " , Axiomathes ,vol.17,(in press): 46 pp.
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