0.1 Quantum automaton and quantum computation
Definition 0.1. A quantum automaton can be simply defined 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).
Note: 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 0.2. One obtains a simple definition of quantum automaton 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,δ : S × S → S,λ : S × I → O), a quantum automaton is defined by
a triple (H, Δ : H → H,μ), where H is either a Hilbert space or a rigged Hilbert space of
quantum states and operators acting on H, and μ is a measure related to the quantum logic,
LM, and (quantum) transition probabilities of this quantum system.
Remark.
Quantum ‘computation’ becomes possible only when macroscopic blocks of quantum
states can be controlled via quantum preparation and subsequent, classical observation.
Obstructions to ‘building’, or constructing quantum computers are known to exist in
dimensions greater than 2 as a result of the standard K-S theorem. Subsequent definitions of
quantum computers reflect attempts to either avoid or surmount such difficulties often
without seeking solutions through quantum operator algebras and their representations
related to extended quantum symmetries which define fundamental invariants that are
key
Definition 0.3. : Alternatively, as a quantum algebraic topology object, a quantum
automaton is defined by the triplet (G,H−ℜG,Aut(G)), where G is a locally compact quantum
groupoid, H-ℜG are the unitary representations of G on rigged Hilbert spaces ℜG of quantum
states and quantum operators on H, and Aut(G) is the transformation, or automorphism,
groupoid of quantum transitions.
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 0.4. : 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.
Related Results: Quantum Automata Applications to Modeling Complex Systems.
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-Łukasiewicz algebras has a subcategory of centered n- Łukasiewicz algebras [15] (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.([15] the category of
centered n-Łukasiewicz 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, Łukasiewicz
Logic Algebras that showed significant dissimilarities [?] from the widespread Bolean
models of human neural networks that may have begun with the early publication of [17].
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 [?]. 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
Łukasiewicz 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 Łukasiewicz Logic Algebras
(LLA).
0.2 Quantum Automata, Quantum Computation and Quantum Dynamics Represented by
Categories, Functors and Natural Transformations.
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 [12]. 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 [11].
References
[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] I.C. Baianu.: Łukasiewicz-Topos Models of Neural Networks, Cell Genome and
Interactome Nonlinear Dynamics). CERN Preprint EXT-2004-059. Health Physics and
Radiation Effects (June 29, 2004).
[4] I. C. Baianu, J. F. Glazebrook, R. Brown and G. Georgescu.: Complex Nonlinear
Biodynamics in Categories, Higher dimensional Algebra and Łukasiewicz-Moisil
Topos: Transformation of Neural, Genetic and Neoplastic Networks, Axiomathes, 16:
65–122(2006).
[5] Baianu, I.C.: 1970, Organismic Supercategories: II. On Multistable Systems. Bulletin
of Mathematical Biophysics, 32: 539-561.
[6] Baianu, I.C. and D. Scripcariu: 1973, On Adjoint Dynamical Systems. The Bulletin
of Mathematical Biophysics, 35(4), 475–486.
[7] Baianu, I.C.: 1973, Some Algebraic Properties of (M,R) – Systems. Bulletin of
Mathematical Biophysics 35, 213-217.
[8] Baianu, I.C. and M. Marinescu: 1974, A Functorial Construction of (M,R)– Systems.
Revue Roumaine de Mathematiques Pures et Appliquees 19: 388–391.
[9] Baianu, I.C.: 1977, A Logical Model of Genetic Activities in Łukasiewicz Algebras:
The Non-linear Theory. Bulletin of Mathematical Biophysics, 39: 249-258.
[10] Baianu, I.C.: 1980, Natural Transformations of Organismic Structures. Bulletin of
Mathematical Biophysics, 42: 431-446
[11] Baianu, I. C.: 1987, Computer Models and Automata Theory in Biology and
Medicine., in M. Witten (ed.), Mathematical Models in Medicine, vol. 7., Pergamon Press,
New York, 1513–1577; CERN Preprint No. EXT-2004-072 .
[12] Baianu, I.C.: 2004, Quantum Nano–Automata (QNA): Microphysical Measurements
with Microphysical QNA Instruments, CERN Preprint EXT–2004–125.
[13] Baianu, I. C., Glazebrook, J. F. and G. Georgescu: 2004, Categories of Quantum
Automata and N-Valued Łukasiewicz Algebras in Relation to Dynamic Bionetworks,
(M,R)–Systems and Their Higher Dimensional Algebra, Abstract and Preprint of Report
[14] Baianu I. C., Brown R., Georgescu G. and J. F. Glazebrook: 2006,
Complex Nonlinear Biodynamics in Categories, Higher Dimensional Algebra and
Łukasiewicz–Moisil Topos: Transformations of Neuronal, Genetic and Neoplastic
networks, Axiomathes 16 Nos. 1–2, 65–122.
[15] Georgescu, G. and C. Vraciu 1970. On the Characterization of Łukasiewicz Algebras.,
J Algebra, 16 (4), 486-495.
[16] BMB1: Mathematical Biology reports
[17] McCullough, E. and M. Pitts.1945. Bull. Math. Biophys. 7, 112-145.
[18] MBR2: Eprint at cogprints.org/3674/ Mathematical Biology reports2.
[19] A Java quantum circuit simulator: jQuantum; click on the link to download the Java
quantum circuit simulator
[20] A C++ Quantum Library
[21] The Haskell Library for Quantum computations
[22] A Quasi–Quantum Computer.
[23] A Quantum circuit emulator: QCAD
[24] An introduction to Quantum Computers
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