categories of quantum automata and quantum computers

Categories of Quantum Automata,
N- Łukasiewicz 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 Schrödinger representation, with both initial and boundary conditions in space-time. A new theorem is proven which states that the category of quantum automata 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 ([4]) 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- Łukasiewicz algebras has a subcategory of centered n- Łukasiewicz algebras (ref. [2]) which can be employed to design and construct subcategories of quantum automata based on n-Łukasiewicz diagrams of existing VLSI. Furthermore, as shown in ref.([2]) the category of centered n-Łukasiewicz 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 ([4,5]) by either standard or quantum automata.

Bibliography

1
Baianu, I.1971.``Organismic Supercategories and Qualitative Dynamics of Systems." Bull. Math.Biophysics., 33, 339-353.

2
Georgescu, G. and C. Vraciu 1970. ``On the Characterization of Łukasiewicz Algebras." J. Algebra, 16 (4), 486-495.

3
Baianu, I.C. 1977. ``A Logical Model of Genetic Activities in Łukasiewicz Algebras: The Non-linear Theory." Bulletin of Mathematical Biology, 39:249-258 (1977).

4
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.

5
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 Łukasiewicz Topos. " , Axiomathes,vol.17,(in press): 46 pp.



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