PlanetPhysics/Categories of Quantum Automata and Quantum Computers

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

Quantum automata were defined (in ref. ) 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  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. ) 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. 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 by either standard or quantum automata.