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.
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