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quantum automata and computation
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A quantum automaton can be simply described as an extension of an automaton with quantum states instead of the sequentially determined states, inputs and outputs of a sequential, or state machine. The precise mathematical definitions of quantum automaton, variable automaton, and quantum computation
were first introduced formally in refs. [1] and [2] in relation to relational models in Quantum Relational Biology (loc.cit).
Quantum computation and quantum machines (or nanobots) were much publicized in the early 1980's by Richard Feynman (Nobel Laureate in Physics: QED),and, subsequently, a very large number of papers- too many to cite all of them here- were published on this topic by a rapidly growing number of quantum theoreticians and some applied mathematicians.
Definition 1.1 A simple definition of quantum automaton is obtained by considering instead of the transition function of a classical sequential machine, the (quantum) transitions in a finite quantum system with definite probabilities determined by quantum dynamics. The quantum state space of a quantum automaton
is thus defined as a quantum groupoid over a bundle of Hilbert spaces, or over rigged Hilbert spaces. Formally, whereas a sequential machine, or state machine with state space S, input set I and output set O, is defined as a quintuple: $(S, I, O, \delta : S \times S \rightarrow S, \lambda: S \times I\rightarrow O)$ , a quantum automaton is defined by a triple $(\emph{H}, \Delta: \emph{H} \rightarrow \emph{H}, \mu)$ , where H is either a Hilbert space or a rigged Hilbert space of quantum states and operators
acting on H, and $\mu$ is a measure related to the quantum logic, LM, and (quantum) transition probabilities of this quantum system.
Remark. Other definitions of quantum automata and quantum computations have also been reported that are closely related to recent experimental attempts at constructing quantum computing devices.
Two examples of such definitions are briefly considered next.
Definition 1.3 : quantum automata were defined in refs.[ 1] and [ 2] 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 was 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 established 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 ([3]) 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 $n$ - ukasiewicz algebras has a subcategory of centered $n$ -  ukasiewicz algebras [16] (which can be employed to design and construct subcategories of quantum automata based on $n$ - ukasiewicz diagrams of existing VLSI. Furthermore, as shown in ref.([16] the category of centered $n$ - ukasiewicz algebras and the category of Boolean algebras are naturally equivalent. Variable machines with a varying transition function were previously discussed informally by Norbert Wiener as a possible model for complex biological systems although how this might be achieved in Biocybernetics has not been specifcally, or mathematically presented by Wiener. A `no-go' conjecture was also proposed which states that Generalized (M,R)-Systems complexity prevents their complete computability by either standard or quantum automata. The concepts of quantum automata and quantum computation were initially studied and are also currently further
investigated in the contexts of quantum genetics, genetic networks with nonlinear dynamics, and bioinformatics. In a previous publication (ICB71a)- after introducing the formal concept of quantum automaton-the possible implications of this concept for correctly modeling genetic and metabolic activities in living cells and organisms were also considered. This was followed by a formal report on quantum and abstract, symbolic computation based on the theory of categories, functors and natural transformations [2]. The notions of topological semigroup, quantum automaton,or quantum computer, were then suggested with a view to their potential applications to the analogous simulation of biological systems, and especially genetic activities and nonlinear dynamics in genetic networks. Further, detailed studies of nonlinear dynamics in genetic networks were carried out in categories of $n$ -valued,  ukasiewicz Logic Algebras that showed significant dissimilarities [] from the widespread Bolean models of human neural networks that may have begun with the early publication of [18]. Molecular models in terms of categories, functors and natural transformations were then formulated for uni-molecular chemical transformations, multi-molecular chemical and biochemical transformations []. Previous applications of computer modeling, classical automata theory, and relational biology to molecular biology, oncogenesis and medicine were
extensively reviewed and several important conclusions were reached regarding both the potential and limitations of the computation-assisted modeling of biological systems, and especially complex organisms such as Homo sapiens sapiens [3]. Novel approaches to solving the realization problems of Relational Biology models in Complex System Biology are introduced in terms of natural transformations between functors of such molecular categories. Several applications of such natural transformations of functors were then presented to protein biosynthesis, embryogenesis and nuclear transplant experiments. Other possible
realizations in Molecular Biology and Relational Biology of Organisms were then suggested in terms of quantum automata models of Quantum Genetics and Interactomics. Future developments of this novel approach are likely to also include: Fuzzy Relations in Biology and Epigenomics, Relational Biology modeling of Complex Immunological and Hormonal regulatory systems, $n$ -categories and generalized $LM$ -Topoi of  ukasiewicz Logic Algebras and intuitionistic logic (Heyting) algebras for modeling nonlinear dynamics and cognitive processes in complex neural networks that are present in the human brain, as well as stochastic modeling of genetic networks in  ukasiewicz Logic
Algebras (LLA).
Molecular models were previously defined in terms of categories, functors and natural transformations were formulated for unimolecular chemical transformations, multi-molecular chemical and biochemical transformations [13]. Dynamic similarities or analogies between categories of classical, quantum or complex systems and their transformations were then naturally represented in terms of adjoint functors and the corresponding natural equivalences.
Remark. Previous applications of computer modeling, classical automata theory, and relational biology to molecular biology, neural networks, oncogenesis and medicine were extensively reviewed in a previous monograph and several important conclusions were reached regarding both the potential and the severe limitations of the algorithm driven, recursive computation-assisted modeling of complex biological systems [3].
- 1
- Baianu, I.C.: 1971a, Categories, Functors and Quantum Algebraic Computations, in P. Suppes (ed.), Proceed. Fourth Intl. Congress Logic-Mathematics-Philosophy of Science, September 1-4, 1971, the University of Bucharest.
- 2
- Baianu, I.C.: 1971b, Organismic Supercategories and Qualitative Dynamics of Systems. Bulletin of Mathematical Biophysics, 33 (3): 339-354.
- 3
- 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.
- 4
- I.C. Baianu.:
 ukasiewicz-Topos Models of Neural Networks, Cell Genome and Interactome Nonlinear Dynamics). CERN Preprint EXT-2004-059. Health Physics and Radiation Effects (June 29, 2004).
- 5
- I. C. Baianu, J. F. Glazebrook, R. Brown and G. Georgescu.: Complex Nonlinear Biodynamics in Categories, Higher dimensional Algebra and
 ukasiewicz-Moisil Topos: Transformation of Neural, Genetic and Neoplastic Networks, Axiomathes, 16: 65-122(2006).
- 6
- Baianu, I.C.: 1970, Organismic Supercategories: II. On Multistable Systems. Bulletin of Mathematical Biophysics, 32: 539-561.
- 7
- Baianu, I.C. and D. Scripcariu: 1973, On Adjoint Dynamical Systems. The Bulletin of Mathematical Biophysics, 35(4), 475-486.
- 8
- Baianu, I.C.: 1973, Some Algebraic Properties of (M,R) - Systems. Bulletin of Mathematical Biophysics 35, 213-217.
- 9
- Baianu, I.C. and M. Marinescu: 1974, A Functorial Construction of (M,R)- Systems. Revue Roumaine de Mathematiques Pures et Appliquees 19: 388-391.
- 10
- Baianu, I.C.: 1977, A Logical Model of Genetic Activities in
 ukasiewicz Algebras: The Non-linear Theory. Bulletin of Mathematical Biophysics, 39: 249-258.
- 12
- Baianu, I.C.: 1980, Natural Transformations of Organismic Structures. Bulletin of Mathematical Biophysics 42: 431-446
- 12
- Baianu, I. C.: 1987a, Computer Models and Automata Theory in Biology and Medicine., in M. Witten (ed.), Mathematical Models in Medicine, vol. 7., Pergamon Press, New York, 1513 ÃÃÃÃÃÃÃÃÃÃÃÃÃû-1577; CERN Preprint No. EXT-2004-072 .
- 13
- Baianu, I.C.: 2004, Quantum Nano-Automata (QNA): Microphysical Measurements with Microphysical QNA Instruments, CERN Preprint EXT-2004-125.
- 14
- Baianu, I. C., Glazebrook, J. F. and G. Georgescu: 2004, Categories of Quantum Automata and N-Valued
 ukasiewicz Algebras in Relation to Dynamic Bionetworks, (M,R)-Systems and Their Higher Dimensional Algebra, Abstract and Preprint of Report
- 15
- Baianu I. C., Brown R., Georgescu G. and J. F. Glazebrook: 2006, Complex Nonlinear Biodynamics in Categories, Higher Dimensional Algebra and
 ukasiewicz-Moisil Topos: Transformations of Neuronal, Genetic and Neoplastic networks, Axiomathes 16 Nos. 1-2, 65-122.
- 16
- Georgescu, G. and C. Vraciu 1970. On the Characterization of
 ukasiewicz Algebras., J Algebra, 16 (4), 486-495.
- 17
- BMB1: Mathematical Biology reports
- 18
- McCullough, E. and M. Pitts.1945. Bull. Math. Biophys. 7, 112-145.
- 19
- MBR2: Eprint at cogprints.org/3674/ Mathematical Biology reports2.
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See Also: automata, quantum nano-automata, quantum groupoids, groupoid C*-dynamical system, category of automata, category of quantum automata, topic entry on foundations of mathematics, algebras, quantum space-times, ETAS interpretation, quantum category, topic entry on applied mathematics, Richard Feynman, examples of abelian categories, groupoid and group representations related to quantum symmetries, mathematical foundations of quantum field theories, topic on applied mathematical physics and physical mathematics, supercomputers
| Other names: |
nano-automata |
| Also defines: |
quantum automaton, quantum computers |
| Keywords: |
quantum computers, quantum groupoids, quantum groups, quantum applications, quantum relational biology |
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Cross-references: recursive, algorithm, dynamic similarities, developments, nuclear, complex system, conclusions, computer, transformations, chemical transformations, logic algebras, applications, potential, semigroup, natural transformations, functors, theory, symbolic computation, implications, genetic networks, complete, conjecture, complex, naturally equivalent, Boolean algebras, diagrams, design, algebras, Lie algebroids, characterization, terms, cells, functions, represent, open, category, faithful representation, subcategory, colimits, limits, boundary conditions, equations, quantum state spaces, automata, groupoid, automorphism, transformation, operators, unitary representations, locally compact, triplet, quantum algebraic topology object, key, invariants, extended quantum symmetries, representations, quantum operator algebras, solutions, reflect, definitions, theorem, dimensions, blocks, transition probabilities, quantum logic, measure, and operators, state space, Hilbert spaces, quantum groupoid, quantum state space, quantum system, finite, sequential machine, transition function, simple, number, QED, Richard Feynman, machines, relational biology, relation, variable, mathematical definitions, state machine, states, quantum states, automaton, extension
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This is version 167 of quantum automata and computation, born on 2008-07-01, modified 2009-02-02.
Object id is 10734, canonical name is QuantumAutomataAndQuantumComputation2.
Accessed 2198 times total.
Classification:
| AMS MSC: | 03D05 (Mathematical logic and foundations :: Computability and recursion theory :: Automata and formal grammars in connection with logical questions) | | | 03D15 (Mathematical logic and foundations :: Computability and recursion theory :: Complexity of computation) | | | 55U35 (Algebraic topology :: Applied homological algebra and category theory :: Abstract and axiomatic homotopy theory) | | | 37B10 (Dynamical systems and ergodic theory :: Topological dynamics :: Symbolic dynamics) |
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