quantum automata and computation

quantum automata and computation

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1

Topic on Quantum Automata and Quantum Computations

1.1

Introduction

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

).

1.2

Historical 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.

1.3

Quantum Automata

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

,

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 1.1

.

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 to realizing quantum computation.

Definition 1.2

.

: alternatively, as a quantum algebraic topology object, a

quantum automaton

is defined by the

triplet

(

𝖦

,

H

-

ℜ

𝖦

,

A

u

t

(

𝖦

)

), where

𝖦

is a locally compact

quantum groupoid

,

H

-

ℜ

𝖦

are the

unitary representations

of

𝖦

on rigged Hilbert spaces

ℜ

𝖦

of quantum states and quantum operators on

H

, and

A

⁢

u

⁢

t

⁢

(

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 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.

1.4

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 (

[

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

-Łukasiewicz

algebras

has a subcategory of

centered

n

- Łukasiewicz algebras

[

16

]

(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.(

[

16

]

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

[

ICB77

]

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

[

ICB2k4a

]

. 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

L

⁢

M

–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).

1.4.1

Quantum Automata,Computation and Dynamics Represented by Categories and Functors

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

]

.

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

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.: 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).

5

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).

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 Łukasiewicz
Algebras: The Non-linear Theory.

Bulletin of Mathematical Biophysics

,

39

: 249-258.

11

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, 1513–1577;

http://doe.cern.ch//archive/electronic/other/ext/ext-2004-072.pdf

CERN Preprint No. EXT-2004-072 .

13

Baianu, I.C.: 2004, Quantum Nano–Automata (QNA): Microphysical Measurements with Microphysical QNA Instruments,

http://documents.cern.ch/cgi-bin/setlink?base=preprint&categ=ext&id=ext-2004-125

CERN Preprint EXT–2004–125.

14

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,

http://www.ag.uiuc.edu/fs401/QAuto.pdf

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 Łukasiewicz–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 Łukasiewicz Algebras.,

J Algebra

, 16 (4), 486-495.

17

BMB1:

http://www.kli.ac.at/theorylab/Keyword/R/RelationalBio.html

Mathematical Biology reports

18

McCullough, E. and M. Pitts.1945.

Bull. Math. Biophys

. 7, 112-145.

19

MBR2: Eprint at

cogprints.org/3674/

http://cogprints.org/3674/

Mathematical Biology reports2.

Title

quantum automata and computation

Canonical

name

QuantumAutomataAndComputation

Date of creation

2013-03-22 18:10:17

Last modified on

2013-03-22 18:10:17

Owner

bci1 (20947)

Last modified by

bci1 (20947)

Numerical id

172

Author

bci1 (20947)

Entry type

Topic

Classification

msc 37B10

Classification

msc 55U35

Classification

msc 03D15

Classification

msc 03D05

Synonym

nano-automata

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Defines

quantum automaton

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quantum computers

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