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ISSN 1064-2307, Journal of Computer and Systems Sciences International, 2007, Vol. 46, No. 6, pp. 908961. Pleiades Publishing, Ltd., 2007.Original Russian Text L.V. Litvintseva, I.S. Ulyanov, S.V. Ulyanov, S.S. Ulyanov, 2007, published in Izvestiya Akademii Nauk. Teoriya i Sistemy Upravleniya, 2007, No. 6,pp. 71126.

INTRODUCTION

In complex and essentially nonlinear dynamic mod-els of control objects (CO) with weakly formalizedstructure and random parameters, it is quite difficult todetermine an optimal structure of an automatic controlsystem (ACS), in which, e.g., a conventional propor-tionalintegraldifferentiating (PID) controller isemployed at the lower (executive) level. Especially, thisdifficulty reveals itself in problems of designing thestructures of automatic control systems of this type inthe presence of random noise different in its nature andunder uncertainty information about the control goals.

In the design methodology of intelligent control sys-tems (ICS) based on soft computing technology, thestructure of a fuzzy controller (FC) is considered as oneof the variants of designing conventional automaticcontrol systems. Beginning from the moment of origi-nation (1974), the models of fuzzy controllers havedemonstrated improved ability of controlling dynamiccontrol objects that have a weakly formalized structureor function in the conditions of uncertainty of the infor-mation source. As the experience and simulation resultshave shown that, in the listed control situations fre-

quently, a conventional (based on the principles of aglobal negative feedback) PID controller cannot man-age the control problem posed. The use of the soft com-puting technology (based on genetic algorithms (GA)and fuzzy neural networks (FNN)) has extended thefield of efficient application of fuzzy controllers byusing new functions in the form of learning and adapta-tion. However, it is very difficult to design a globallyappropriate and robust structure of the intelligent con-trol system. This limitation is especially typical ofunpredicted control situations, when the control objectsoperates in sharply changing conditions (a failure ofsensors or noise in the sensor system, the presence of a

time delay in control signals or measurement, a sharpchange in the structure of the control objects or itsparameters, etc.).

In a number of practical cases, conditions of thistype can be predicted, but it is difficult to realize arobust control in unpredicted situations based on thedesigned (for a fixed situation) knowledge base (KB) ofa fuzzy controller (even of the whole set of predictedrandom situations). It seems to us that one of the exist-ing solutions is to form a finite number of knowledge

ARTIFICIALINTELLIGENCE

Quantum Fuzzy Inference for Knowledge Base Designin Robust Intelligent Controllers

L. V. Litvintseva, I. S. Ulyanov, S. V. Ulyanov, and S. S. Ulyanov

OOO MCG QUANTUM, Moscow, Russia

Received January 19, 2006; in final form April 23, 2007

Abstract

The analysis of simulation results obtained using soft computing technologies has allowed one toestablish the following fact important for developing technologies for designing robust intelligent control sys-tems. Designed (in the general form for random conditions) robust fuzzy controllers for dynamic control objectsbased on the knowledge base optimizer (stage 1 of the technology) with the use of soft computing can operateefficiently only for fixed (or weakly varying) descriptions of the external environment. This is caused by possi-ble loss of the robustness property under a sharp change of the functioning conditions of control objects: theinternal structure of control objects, control actions (reference signal), the presence of a time delay in the mea-surement and control channels, under variation of conditions of functioning in the external environment, andthe introduction of other weakly formalized factors in the control strategy. In this paper, a description of the

strategy of designing robust structures of an intelligent control system based on the technologies of quantumand soft computing is given. The developed strategy allows one to improve the robustness level of fuzzy con-trollers under the specified unpredicted or weakly formalized factors for the sake of forming and using newtypes of processes of self-organization of the robust knowledge base with the help of the methodology of quan-tum computing. Necessary facts from quantum computing theory, quantum algorithms, and quantum informa-tion are presented. A particular solution of a given problem is obtained by introducing a generalization of strat-egies in models of fuzzy inference on a finite set of fuzzy controllers designed in advance in the form of new

quantum fuzzy inference.

The fundamental structure of quantum fuzzy inference and its software toolkit in theprocesses of designing the knowledge base of robust fuzzy controllers in on-line, as well as a system for simu-lating robust structures of fuzzy controllers, are described. The efficiency of applying quantum fuzzy inferenceis illustrated by a particular example of simulation of robust control processes by an essentially nonlineardynamic control object with randomly varying structure.

DOI: 10.1134/S1064230707060081


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QUANTUM FUZZY INFERENCE FOR KNOWLEDGE BASE DESIGN 909

bases of the fuzzy controller for the set of fixed controlsituations.

The question arises about how to determine which ofknowledge bases can be used at a particular timeinstant.

In this case, the choice of a generalized strategyto provide the opportunity to switch the flow of controlsignals input from different knowledge bases of thefuzzy controller and to adapt their output signal (if nec-

essary) to the current conditions of functioning of theknowledge bases of a control object is especiallyimportant. A simple variant of solving this problem isto use the method of weighted coefficients and aggre-gate the output signals from each independent fuzzycontroller. Regrettably, this method has limited capabil-ities (as simulation results have shown) since, fre-quently, the distribution of weighting factors has to bedetermined in on-line dynamics (see in what followsand [1, 2]), and the search procedure has combinatorialnature.

In this paper it is shown that a solution of the prob-lems of this type can be found based on introducing the

self-organization principle in the course of designingknowledge bases of a fuzzy controller, which is imple-mented and has a software toolkit using the developedmodel of quantum fuzzy inference with application ofthe methodology of quantum soft computing and sys-tems engineering (System of Systems Engineering)using the synergetic self-organization principle [35].

In particular, the implementation of the process ofself-organization of robust knowledge bases in thisapproach is performed using generalization of strate-gies of fuzzy logical inference in the form of quantumfuzzy inference. The structure of quantum fuzzy infer-ence and a system for simulating robust knowledgebases for fuzzy controller, which illustrate the effi-ciency of application of quantum fuzzy inference, aredescribed. The model of the developed quantum fuzzyinference is regarded as a new type of the search quan-tum algorithm on the generalized space of knowledgebases of a fuzzy controller, and a generalized robustcontrol signal is designed as the output result.

The model of quantum fuzzy inference proposed inthis paper harness particular individual knowledgebases of a fuzzy controller, each of which is obtainedwith the help of an knowledge base optimizer (KBO)for the corresponding conditions of functioning of acontrol object and fixed control situations in a randomexternal environment. The process of designing partic-

ular individual knowledge bases of a fuzzy controllerwith the help of the knowledge base optimizer for givencontrol situations is performed in accordance with thedesign technology and is considered in detail in [1, 6](see stage 2 in Fig. 3 in [7]).

In particular in [7], based on a comparison of simu-lation results obtained in [1, 6], it was shown that forsufficiently wide range of variation of parameters char-acterizing a given

control situation, the knowledge baseoptimizer yields an essential gain (compared with other

software tools) in order to achieve the required robust-ness level of knowledge bases. Other industrial tools forforming knowledge bases, such as the FNN ANFIS(built-in module in the Matlab simulation system) orAFM (an ST Microelectronics development [6]), etc.have an increased sensitivity to the variation of param-eters (characterizing a given control situation) com-pared with the KBO tools and result in the loss of con-

trol robustness as was strictly shown in [7]. As a result,in fixed control situations, fuzzy controllers withknowledge bases (designed with the help of the Knowl-edge base optimizer) have improved robustness, andthe corresponding control laws contain less redundantinformation

and thus are used as an input signal forquantum fuzzy inference (in accordance with the devel-oped technology of designing robust knowledge bases(Fig. 3 in [7]). It is worth noting that the presence ofredundant information in control laws is physical real-ity, which takes place because of the use in the pro-cesses of optimization of knowledge bases of a randomchoice in the form of genetic algorithms, as well as itfollows from the laws of information theory on the

necessity of presence of redundancy in unreliable datatransmission channels with noise. This is the inevitablecost of the possibility to obtain a solution of the prob-lem of optimal control of an essentially nonlinear con-trol plant in the conditions of uncertainty in the sourceinformation and the multi-criteria nature of the optimi-zation conditions.

In the case of an unpredicted control situation, theadditional information redundancy in the control lawsof the fuzzy controllers arises as the total result of theinadequate reaction of the CO (control plant) (in theform of a new control error) and of the logically incor-rect interpretation of the initialization of the corre-

sponding production rules in the Knowledge bases usedby the fuzzy controllers (trained only on given controlsituations).

The model of quantum fuzzy inference is a new typeof the quantum search algorithm on the generalizedspace of structured data and, based on the methods ofquantum computing theory [810], it allows one tosolve efficiently control problems that could not besolved earlier at the classical level. The developedapproach is first applied in the theory and practice ofintelligent control systems. Therefore the present paperis a generalization and development of the resultsobtained in [1, 2, 6, 7].

Remark 1.

From the point of view of the complex-ity theory of computations and the quantum algorithm[810], the developed quantum algorithm (QA) isreferred to the class of polynomial with bounded error,the BPP class (bounded-error probabilistic polynomialtime), and its quantum generalization is referred to theBQP class. Therefore, it is efficient by the definition.This means that, structurally, the quantum algorithm inquantum fuzzy inference has a polynomial complexity;i.e., the randomized algorithm has polynomial, rather


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than exponential (as in classical search algorithms)dependence on the input signals, and a bounded proba-bility (at the level 3/4) of the measurement accuracy ofthe result of computations is sufficient for (making) anefficient decision making.

In the proposed model of the quantum algorithm forquantum fuzzy inference the following actions are real-ized [35]: (1) the results of fuzzy inference are pro-

cessed for each independent FC fuzzy controller;(2) based on the methods of quantum information the-ory, valuable quantum information hidden in indepen-dent (individual) knowledge bases is extracted; and(3) in on-line, the generalized output robust control sig-nal is designed in all sets of knowledge bases of thefuzzy controller.

In this case, the output signal of quantum fuzzyinference in on-line is an optimal signal of control ofthe variation of the gains of the PID controller, whichinvolves the necessary (best) qualitative characteristicsof the output control signals of each of the fuzzy con-trollers, thus implementing the self-organization princi-

ple. Therefore, the domain of efficient functioning ofthe structure of the intelligent control system can beessentially extended by including robustness

, which isa very important characteristic of control quality. Therobustness of the control signal is the ground for main-taining the reliability and accuracy of control underuncertainty conditions of information or a weakly for-malized description of functioning conditions and/orcontrol goals [7].

In this paper, we describe the fundamental structureof quantum fuzzy inference and its software toolkit inthe processes of designing knowledge bases of robustself-organized fuzzy controllers in on-line. In particu-

lar, the functional organization of the system for simu-lating robust knowledge bases for fuzzy controllers thatallows one to improve the efficiency of applying quan-tum fuzzy inference is presented. Necessary facts fromquantum computing theory and quantum informationtheory used for development and substantiation of thestructure of the quantum algorithm in quantum fuzzyinference are also presented.

1. STATEMENT OF THE PROBLEM

One of the main problems of the modern technologyof designing fuzzy controllers is the design and appli-cation of robust knowledge bases in the structures ofintelligent control system [1] in order to increase theirability to be self-learned (trained), self-organized, andself-developed. In connection with this fact, we pose inthis paper the following problem: develop a model ofquantum fuzzy inference for designing robust knowl-edge bases in intelligent controllers that ensure theachievement of a guaranteed control in unpredicted(abnormal) control situations. In the end, the solution ofthis problem results in increasing the robust level of thestructure of the intelligent control system.

The problem of designing the structure and knowl-edge bases of the most intelligent fuzzy controllers fora given control situation were considered in previouspublications of the authors (see a detailed description in[1, 6, 7], and are used in this paper as source data. Inparticular, in [7, Figs. 1 and 2], the interrelationbetween the measures of quality control and the typesof intelligent computing tools was presented in detail.

The relations between stability, controllability, androbustness were also investigated based on the thermo-dynamic approach [7, Fig. 1, level 1, and Fig. 3]. Thecorresponding quantitative measures and regularitieswere included in the software toolkit (support) of theknowledge base optimizer.

In the modern control theory, various aspects oflearning and adaptation processes of fuzzy controllerswere investigated. Many of the learning schemes werebased on the error back propagation algorithm and theirmodifications [1]. The adaptation processes are basedon iterative models of random algorithms. These ideaswork well in designing control processes in the condi-

tions of absence of weakly formalized noise of theexternal environment or unknown noise in the sensorsystem, etc. In more complex unpredicted control situ-ations, the learning and adaptation methods using errorback propagation methods or iterative random algo-rithms did not guaranteed the achievement of therequired robustness and accuracy level of control pro-cesses. The efficient solution to this problem with thehelp of the knowledge base optimizer for particularcontrol situations was developed in [1, 2, 7].

It was shown in [3] that to achieve self-organization[7, Fig. 1, level 3] in the structure of intelligent controlsystems, it is necessary to use quantum fuzzy inference.Figure 1 presents the general block diagram of quantum

fuzzy inference. The principles of operation of quan-tum fuzzy inference and its particular blocks are con-sidered in detail in Section 5 below.

The table presents the structure of the intelligentcontrol system including the model of quantum fuzzyinference and describes its advantages and disadvan-tages. The model of quantum fuzzy inference is basedon the physical laws of quantum computing theory [810], namely, unitary, and invertible quantum operatorsare used. In the general form, the quantum algorithmconsists of three basic unitary operations: superposi-tion, quantum correlation (quantum oracle and entan-gled operators) and interference. The fourth operator,

the operator of measurement of the results of quantumcomputing is irreversible (classical).

Quantum computing based on the listed types ofoperators is referred to a new type of computationalintelligence [10]. In what follows (to provide a morecomplete understanding of the principles of operationof the quantum algorithm in quantum fuzzy inference),we give a brief description of the listed quantum oper-ators, their interrelations and properties. The necessaryfacts from quantum information theory and the theory


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of quantum correlation processes are presented. Thisinformation allows one to understand more completelyand deeply the solution of the following problem that isdifficult and fundamentally important for the theoryand control systems: determine the role and influence ofquantum effects on the improvement of the robust levelof the designed intelligent control processes

for thesake of extracting additional quantum information hid-

den (and accessible only partially) in the correlationclassical states of control laws and designed only basedon classical methods of soft computing. Additionalinformation and a detailed presentation of these prob-lems with a mathematical proof of the required propo-sitions can be found in [10].

2. QUANTUM COMPUTING: EXAMPLESAND PROPERTIES OF THE BASIC OPERATORS

As an example, we consider the conventional math-ematical formalism of describing models of the basicquantum operators from the point of view of the secondquantum problem of describing the quantum algorithm

(see Subsection 5. 2). This formalism can be expressedin the language of quantum states or transformations,but we are also interested in the possibility of an ade-quate description of quantum states and effects in thelanguage of logical inference: the application of theconventional formalism, its power, and expressivepower as a quantum system of fuzzy logical inference

[10, 11].

Example 1. Quantum bit as a quantum state.

A con-ventional bit may be in one of the two states, 0 or 1.Thus, its physical state can be represented as b

= a

1

0 +

a

2

1

, which has one of the following forms: either a

1

= 1and a

2

= 0, then b

= 0, or a

1

= 0 and a

2

= 1, then b

= 1.

On the contrary, the state of a quantum bit |

is givenby a vector in a two-dimensional complex space. Here,the vector has two components, and its projections onthe bases of the vector space are complex numbers. Thequantum bit is represented (in the Dirac notation in theform of a ket vector) as | = |0 + |1

or in the vector

notation |

= , |

= (bra vector). If|

= |0

,

then |0

= . The amplitudes

and

are complex

numbers such that the condition

* +

*

= 1 holds,

where * is the operation of complex conjugation; and

|0

and |1

form a pair orthonormal basis vectors calleda state of the computational basis.

If

and

take zerovalues, then

defines a classical, pure state. Otherwise,it is said that

is in the state of superposition of two

classical

basis states.

Geometrically, the quantum bit is in a continuousstate between |0

and |1

until its state is measured. Thenotion of amplitude of the probabilities of the quantumstate is a combination of the concepts of state and

T

1

0.

phase. In the case when the system consists of twoquantum bits, it is described as tensor product. Forexample, in the Dirac notation, a two-quantum systemis given as

The number of possible states of the combined sys-tem increases exponentially when a quantum bit is

added. This fact results in the problem of estimating thequantum correlation, which occurs for quantum bits ina compound system (see example 4).

In quantum mechanics, a quantum state |

isexpressed by the operator of density of a state

. Thedensity matrix

of a quantum system has the following

properties: =

(Hermitian matrix);

> 0

(positivematrix); and Tr

= 1 (normalized matrix). If the state ofthe quantum system is known exactly, then the systemis described by the density operator in the matrix form

= ||

and is in apure

state. Otherwise, the systemis in a mixed

state. In this case, we have a mix of differentpure states described by the density operator

=

n the ensemble {

p

i

|

i

}

. The matrices that

satisfy the listed conditions generate a convex set.Therefore, they can be written in the form

where |

k

are unit vectors of the Hilbert space andp

k

> 0,

= 1

. The coefficientp

k

for a given k

can be inter-

preted as the probability of the event that the quantumsystem is in a pure state |

k

.

However, this physical interpretation depends on therepresentation

=

which is not uniquedifferent states may have thesame density matrix. In particular, vectors may be takenorthonormal and interpreted as eigenvectors of the den-sity matrix

with eigenvaluesp

k

, and the form

=

is called the spectral representation of

. Pure states are

one-dimensional projectors ||

and are extremalpoints of the convex set of density matrices. Thus,based on density matrices of pure states, all the otherdensity matrices corresponding to mixed states as aconvex combination in the form

=

are reproduced, but they do not admit themselves adescription in the form of a nontrivial convex combina-

12| 00| 01| 10| 11| .+ + +=

pi i| i |i

pk k| k |,k

=

pkk

pk k| k |,k

pkk| k |k

pk k| k |,k


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Thestructureandqualitative

characteristicsofQFI-basedself-orga

nizingICS

Structureofcontrolsystem

Levelsofcontrolquality

Advantagesanddisadvantages

L

imitcapabilities

Stability

Controllability

Controlprecision

Adaptation

Learning

Self-organization

Elementsofself-develop-

ment

A

new

levelofcontrolqualitywas

introduced(self-organizationwith

elementsofself-development).

Guaranteesthecontrolquality

invariousspacesofsearchfor

solutionsofquantumfuzzy

inferenceinon-line.

The

processofdesigningaunified

KBis

performedautomaticallyby

theop

eratorsofsuperpositionand

quantuminterferencewiththehelp

ofaw

isecontrollerbasedonthe

principleofminimumo

finformation

entrop

yandmaximumo

fquantum

correlation(maximumo

fquantum

amplitudeoftheprobabilityofa

quantums

tate)

Guaranteesonlynecessary

conditionsforoptimization

oftheprocessofdesigninga

robust

KB.

InQFI,thereisno

possibilityofoptimalcontrol

ofquantumoperators.

Theprocessofglobal

optimizationdependsonthe

choice

ofafitnessfunction

andisperformedbya

combinatorialmethod.

Req

uiresalargeamountof

compu

tationaltime(high

temporalcomplexityofcom-

putatio

ns)


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tion of other matrices. There is a simple criterion fordetermining the type of state of a quantum system: ifthe trace of the density matrix Tr(2) = 1, then the quan-tum system is in a pure state; ifTr(2) < 1, then a mixedstate takes place. The definition and calculation of thetrace of the density operator is given in what follows.

From the point of view of information theory, thequantum bit contains the same amount of informationas the classical bit, despite an infinite set of virtualstates of the quantum bit. The quantum bit can bedescribed by an infinite number of superpositions ofclassical states, but because of the irreversible characterof the measurement process, it is possible to extractonly a simple classical bit of information from a singlestate among the possible ones. Note that the other vir-tual states are destroyed, and information is lost. Theground for this statement (a quantum bit contains nogreater information than the classical one) is the factthat information is extracted as a result of a physicalprocess. For the sake of measurement of a quantum bit,its state is changed and, as a result, it passes to one of

the possible basis states. Each quantum bit exists in atwo-dimensional space, its measurement is associatedwith the corresponding basis and expresses the resultonly in one of the two states; i.e., one of the basis vec-tors is associated with a given measurement device.Thus, as in the classical case, in a measurement of aquantum bit, there are only two possible results. Sincethe measurement measures a state of the quantum bit, itis impossible to register states in two different basessimultaneously. In simulating a classical dynamic sys-tem, its state can be measured at the first stage in onebasis, then, at the second stage, it can be measured inanother basis. In a true quantum system, this is impos-sible since the wave function, describing the state of the

quantum bit, is destroyed in measurement. Moreover,quantum states in a true quantum system cannot becloned; i.e., there are objective physical limitations inview of which we cannot conduct the measurement intwo ways, using, e.g., copying a quantum bit and itsregistration in different bases [12]. In contrast to thequantum bit, the state of a classical bit can be copiedand we are able to measure in different computationalbases. In addition, the unknown quantum bit cannot besplit into complementing parts [13]; i.e., the informa-tion that is contained in an unknown state of a quantumbit is indivisible.

Thus, in quantum mechanics, operations that are

impossible in the classical mechanics are admissible,and vice versa, in classical mechanics, there are opera-tors for solving problems that are inadmissible in quan-tum mechanics.

Remark 2. Computational basis {|+, |}. Todescribe and measure a quantum bit, we used above thecomputational basis {|0, |1}. However, this choice ofthe computational basis is not unique. It is possible touse various sets of vectors as orthonormal bases. Forexample, it is admissible to represent basis vectors in

the form of states {|+, |} defined as

, respectively. Using this

representation of basis vectors, we can pass to the con-ventional basis

Example 2. Formation of the superposition stateusing Hadamard operator (WalshHadamard). Theexistence of the state of superposition and effect ofmeasurement of a quantum state (see example 6) phys-ically means that there is information hidden from theobserver, which is contained in a closed quantum sys-tem (before the moment of its excitation from an exter-nal perturbation) in the form of observation of a quan-tum state. The system remains to be closed up to aninteraction with the external environment (i.e., up to the

action of system observation. The following question isthe most important: How can we use efficiently infor-mation hidden in superposition (see in what followsexamples in Section 3 and the results of simulation inSection 8)? In the conventional formalism of quantumcomputations, quantum operators are described in anequivalent matrix form. The multiplication of thematrix of an operator by a state vector means the actionof the operation on the investigated system.

For example, the action of the Hadamard matrix (H)on the system | = |0 can be represented as

Similarly,

i.e., the Hadamard transform generates a state of thequantum bit in the form of superposition of two classi-cal states. The formation of the superposition withequivalent amplitudes of probabilities is an importantstep for many quantum algorithms. ApplyingHn in thecorresponding basis states |x n, x {0, 1}n, we

1

2------- 0| 1| +( ) 1

2------- 0| 1| ( ),

0| 12------- +| | +( ) 1| 12

------- +| | ( ).= =

H| H0| 12

------- 1 1

1 1

1

0

12

------- 1

1= = =

=1

2------- 1

0

0

1+

1

2------- 0| 1| +( ).=

H| H1| 12

------- 1 1

1 1

0

1

1

2------- 1

1= = =

=1

2------- 1

0

0

1+

1

2------- 0| 1| ( ),=


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obtain as a result the equivalent form of the Hadamardtransform

wherexz =x1z1 + +xnzn forx = 0 andx = 1. Thus, thesuperposition with equivalent probability amplitudes

for each basis state is obtained by the application

of the operatorHn to the state |0.The value of the superposition state for the theory of

computational processes becomes more understand-able if we interpret the resulting superposition state asa set of2n classical trajectories (paths) of computationswith equivalent weights, based on which the quantumcomputer physically conducts computations in parallel.In this sense, superposition plays the role of the firststage in order to organize quantum parallelism.

Example 3. Quantum parallelism and models of

computations with a quantum oracle. The consideredeffect is one of the most important in quantum comput-ing and is used (together with superposition) in manymodels of the quantum algorithms. It is especiallywidely applied in various models of black box orquantum oracle in constructing quantum algorithmsof different classes [810, 1419], e.g., for calculatingfunctions of the following form: g : {0, 1}n {0, 1}m.Since the mappingx (x,g,(x)),x {0, 1}n is invert-ible, there is a unitary transformation Ug, which is sim-ulated efficiently by classical computations (x, g(x)), sothat |x,y |x,yg(y) for somey {0, 1}m. Note thatadditional quantum bits necessary for implementing

invertible schematic transformations are not consideredhere. The transformation Ufthat describes the black box(as a particular case Ug) is a unitary transformation in

the form of a Boolean functionf: {0, 1}n {0, 1}. If|y is the initial state |0, then after application of thetransformation Uf, the output of the transformationf(x)is |x,f(x).

The physical meaning of quantum parallelism is inthe presence of the effect of parallelism of computa-tions after using the transformation Uffor the superpo-sition state representing different values x. For exam-ple, applying Ufto the state

we have as a result

i.e., the superposition of all possible values of the com-puted function.

H x| 12

------- 1( )xzz| ,z 0 1,==

1

2n

---------

x y,| 0,| = , | Hn

0| 12

n--------- z|

z 0 1,{ }n= = ,

Uf 0,| 1

2n

--------- z f z( ),| ,z 0 1,{ }n

=

Thus, the estimation of values of the functionf(x) ofonly one iteration step is sufficient to compute in paral-lel the values off(x) from all possible input argumentsx. This effect is equivalent to the application of proper-ties of the black box (single use of the internal quantumscheme). However, in reality, only one value of thefunction f(x) is accessible in the measurement of theresult of computingf(x) in the superposition of all pos-

sible states, since, because of the effect of destructionof states in the superposition, only one state randomlymeasured is available. The discussion of the choice of amodel of a quantum oracle for quantum fuzzy comput-ing and its substantiation are given in Section 6 in whatfollows.

Remark 3.On computing function values with thehelp of the phase. The application of the operator Ufto acontrolled quantum bit in the superposition state | : =

(|0 |1) yields the following result:

Omitting |, we can define a new transformation of thetype Vf : |x (1)f(x)|x. Therefore, Vf allows one tocompute values of the function f(x) in terms of thephase. In other words, the computation is possiblethrough the transformation of values offfrom the basisstates to amplitudes relative to given basic states. Now,applying Vf to the vector |, we obtain the followingstate:

(2.1)

This corresponds to the application to the controlledquantum bit of the operation

Example 4. Simulation of quantum correlation(entanglement) with the help of CNOT-similar opera-tors. A correct estimation of the power of quantumcomputing is possible only with the help of finding cor-relations between the values of variables in the quan-tum algorithm at different time instants. As an example,we consider a system of two quantum bits A andB. Inaccordance with the law of tensor product of vectorspaces, the dimension of the space of the com-pound system is determined as the product ofdimensions of the spaces and =; i.e.,as {|00, |01, |10, |11} . The computational basis of thestates for the compound system is expressed interms of the basis states of the systemsA andB {|0, |1}by tensor product |x1,x2 = |x1|x2(x 1,x2) {0, 1}2.Note that in the compound system of quantum bits,

1

2-------

Ufx ,| 1

2

------- x f x( ),| x 1 f x( ),| ( )=

= 1( )f x( )x| 12

------- 0| 1| ( ) 1( )f x( )x ,| .=

'| Vf | 1

2n

--------- 1( )f x( )z| z 0 1,{ }n

.

= =

UfH n 1+( )

0| n 1| ( )[ ].


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there are states that cannot be expressed in terms of ten-sor product of particular components of quantum bits.This property is called entanglementor nonseparabilityof quantum states.

Assume that |AB consists of two states. If there aretwo states |A in and |B in B such that |AB =|A|B, then the state is called unentangled. Other-wise, it is an entanglementor is unseparable [8, 10, 20].

As examples, we can present the system of two quan-tum bits known as the Bell state of the EPR-state (inhonor of the pioneers of these examples Bell, Einstein,Podolsky, and Rosen)

When these states are measured as subsystems in acompound system of entangled states, for a knownresult about a state of the subsystem, we can determineexactly the state of the other subsystem (without con-ducting measurements on it). Therefore, there existsanother variant in the interpretation of these states. Ifwe consider one of the states as entangled, then thismeans that the state cannot be factorized in the state ofthe product of the subsystems of two states. Thus, if anoperator is applied to one of the components of theentangled state, then the result of the action is not fac-torized over the other components, but is calculated forthese components directly based on one of the mea-sured components.

Acting by the Hadamard operatorHin the Bell state|+ on the first component, we obtain as a result

or in the matrix form

i.e., we have a new entangled state, which is used in thefeedback of the quantum genetic search algorithm forenriching the quantum correlation of the entangledstate (see Fig. 1). Successively acting on the secondcomponent of the obtained state by the Hadamard oper-atorH2, we arrive at

| 12

------- 00| 11| ( ); | 12

------- 01| 10| ( ).==

H11

2------- 00| 11| +( ) 1

2------- H1 0| ( ) 0| H1 1| ( ) 1| +{ }=

=1

2--- 00| 10| 01| 11| + +( ),

H11

2------- 00| 11| +( )

=1

2-------

1 0 1 0

0 1 0 1

1 0 1 0

0 1 0 1

1

2-------

1

0

0

1

1

2---

1

1

1

1

,=

H2H11

2------- 00| 11| +( ) 1

2------- 00| 11| +( ).=

Regrettably, the complexity of implementation of oper-ations by quantum gates in this approach grows withthe number of entangled states connected with the con-trolled bit.

Remark 4.Efficient simulation of the quantum algo-rithm on conventional computers. Entangled states inquantum computing are regarded as an additional phys-ical resource that allows one to increase essentially the

design power compared with the conventional modelsof computations. The number of parameters necessaryfor describing non-entangled (pure) states in the givenHilbert space n (represented as the tensor product ofquantum bits) increases only linearly when the numbern of quantum bits grows. However, to describe the gen-eral form of the state (non-entangled or entangled), anexponential number (2n) vector coefficients arerequired. Therefore, the problem of the physicalresource of quantum computing cannot easily besolved.

This problem was addressed in detail from the gen-eral positions of the theory of quantum computing in

[21]. In particular, it was shown that for the quantumalgorithm (operating with pure states), to improve theefficiency compared with the classical analogues whenthe dimension the input quantum bits increases, anunlimited number of entangled states are required.Moreover, the quantum algorithm can be simulated bythe conventional tools (classical algorithms) efficientlyonly under the presence of small amount of quantumcorrelation and a fixed level of tolerance of computa-tional operations in the quantum algorithm. Indepen-dently, in [22] it was shown how to simulate the quan-tum algorithm with a relatively small quantum correla-tion by classical algorithms efficiently. Thecomputational cost increases linearly with the number

of input quantum bits, and it grows exponentially withthe increase of the required amount of quantum corre-lation. The independent generalization of this approachwas presented in [23] and the corresponding softwarehardware toolkit for efficient simulation of the quantumalgorithm on conventional computers was developed.This approach is harnessed in this paper for simulationin on-line of robust knowledge bases for intelligentfuzzy controllers.

The presented arguments and results confirm thepreferable role of quantum correlation as a drivingforce of quantum computing (on pure states of the evo-lution of quantum dynamics).

Example 5. Simulation of quantum interference withthe help of the Hadamard transform and the quantumFourier transform (QFT). To increase the probability ofmeasurement and extraction of the desired (marked)solution, the main unified idea in the processes ofdesigning various models of the quantum algorithm isto use the phenomenon of constructive/destructiveinterference as a tool of extracting results of efficientcomputations of the quantum algorithm. To increase theprobability of extracting a successful solution, the


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constructive interference is applied, and to reducepoor solutions the destructive interference is used.The constructive (destructive) effect can be illustratedclearly by the example of application the Hadamardtransform to the states{

|0

,

|1

}, and

It is obvious thatH|0 = |+ and

H|1 = |; i.e., the state of superposition of classicalstates is reproduced in the form of quantum bits. Notethat the application of the Hadamard transform to thestates |0 and |1 generates states with the same proba-bility distribution. Since the state |+ is the superposi-tion of both classical states |0 and |1, under repeatedapplication of the Hadamard transform to |+, the clas-sical model of logical inference (Kolmogorov model)presumes the same probability of the resulting classicalstate (the principle of probability conservation). How-ever, because of operation in quantum computing withthe concept of amplitude of probabilities [24, 25], theapplication of the Hadamard transform to the state |+

yields the following result:H|+ = (|+ + |) = |0.

Thus, the interference effect between the probabili-ties of both classical states becomes apparent. On theone hand, the interference (in view of its physical char-acter) reinforced the probability amplitude of a classi-

| 12

------- 0| 1| ( )=

.

1

2-------

cal state |0 (constructive interference) and weakenedessentially (up to zero) the probability amplitude of theother classical state |1 (destructive interference). Act-ing on the superposition of possible solutions, interfer-ence implements the process of forming the final phaseof quantum computing and is (as well as quantum cor-relation) a physical resource for reinforcing quantumcomputing, as well as for solving various problems ofdesigning models of quantum algorithm. For example,applying the transform Hn to the state |' in form(2.1), we obtain as a result the quantum state

which provides a background for designing a quantumgate, e.g., in solving the DeutschJozsa problem [10].

In the Shor model of quantum algorithm, in factor-izing a product into prime numbers, the quantum Fou-rier transform provides interference: the operatorQFTnIn acts on each basis vector belonging to a lin-ear combination of the initial vector |. This meansthat any vector in this combination reproduces thesuperposition of base vectors. The complex weightingcoefficients of the basis vectors are equal in modulus(i.e., the amplitudes of probabilities are equal), but havedifferent phases. Each basis vector is a weighted sum ofprobability amplitudes obtained from different

1

2n

----- 1( )xz f x( )+ z| ,x 0 1,{ }n

z 0 1,{ }n

Fig. 1. The structure of the generalized quantum block.


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sequences of basis vectors. This sum can increase ordecrease the resulting probability amplitude.

Since this effect is similar to the effect of interfer-ence of classical waves, it is said that the operator

plays the role of the interference operator. From the math-ematical point of view, when the operator QFTnIn actson a state, all columns of the resulting matrix areemployed in computing, and the interference is per-formed between the weighting coefficients from differ-ent sequences of basis vectors [26]. Consider the spe-cific features of a very important (and frequentlyaddressed in papers on the description of the founda-tions of quantum mechanics) quantum operatordescribing the irreversible process of extracting theresult of quantum computing.

Example 6. Measurements in different computa-tional bases. The postulate of quantum measurement

was introduced by von Neumann. Only projective mea-surements were considered, in which standard quan-tum observable quantitiesA have a spectral representa-tion in terms of orthogonal projective operators. Thepostulate states that, at the time of measurement ofA,the state vector of the quantum system is reduced to aneigenvector of the observable quantityA correspondingto the measurement result.

Thus, the basis of the main approach to designingmodels of measurements in quantum computing is pro-vided by the postulate of von Neumann projections: theresult of the action of an observation of a state overan observable quantityA as a result of measurement is

one of the eigenvalues of A; i.e., a measurementdestroys the state and renormalize it. For a finite-dimensional Hilbert space, this means mathematicallythe following: let be a Hilbert space of dimension nof a quantum system S. Assume that

is the spectral representation of the observable quantityA, where i are eigenvalues and i is the eigenvectorcorresponding to i; is an orthonormal basis of

the space and Pi is the projection on the proper spaceof the operatorA (spanned on the eigenvectors) with theeigenvalues i. A possible value of the projective mea-surement corresponds to an eigenvalue i of the observ-able quantityA. The observable quantityA of the state

| = is reproduced as the value i with the

probability , where , , are such that

QFTn[ ]ij1

2n

--------- 2J i 1( ) j 1( )2

n-------------------------------

exp=

A i i| i |i 1=

n

iPii 1=

n

= =

i{ }i 1=n

i| i 1=n

cijj 1=

k

2

ci1 cik

= = = = = i, and the system state

after the measurement is defined as

where = Thus, before measurement, for

the quantum system in the state |, a possible measure-ment result is defined asp(m) =|Pm|, and after mea-surement, the system is renormalized in the state |' =

The completeness of equations is established

by the fact from probability theory

(2.2)

The postulate on projective measurements has beendeveloped in different directions.

Postulate of a generalized quantum measurement.The model of a generalized quantum measurement (a

closed quantum system S in a finite-dimensional statespace) is described by a set {Mm} of measurement oper-

ators on the Hilbert space of the quantum system S,where the subscript m specifies the possible result of themeasurement process. Measurement operators satisfy

the completeness condition: =I. If the state

of the system S is | before measurement, then theprobability to receive the output value m is determined

as p(m) = . In this expression and in

what follows, the symbol T means the conjugationoperation for a unitary operator. After measurement,the system S is renormalized as

From this measurement model, models of projectivemeasurements and positive operator-valued (POV)measurement measures follow as a particular case. Forexample ifMm satisfies two additional constraints,Mm =

and = , then we obtain the model

of projective measurements presented above. If the

conditionE = holds, we have positive operator-

valued measurement measures.

In quantum information theory, there are strict rulesand laws describing processes of information extrac-tion from an unknown quantum state. The result of pro-jective measurements of quantum bits has to be formu-lated in classical terms. More precisely, any projectivemeasurement of a quantum bit yields only one classicalinformation bit. Therefore, despite the existence of aninfinite set of possible quantum states of a quantum bit,these states are indistinguishable. There are no mea-surement processes in the framework of the von Neu-mann model with the help of which we can extract

i1 i2 ik 1 ikij' ij| ij |,j 1=

kij'

ij

cij2

---------------------.

Pm |

p m( )-----------------.

1 p m( )m

Pm .m

= =

MmTMmm

MmTMm .

'| Mm | p m( )

-----------------.=

MmT

MmMm ' mm 'Mm,

MmTMm


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LITVINTSEVA et al.

more information than a single expected informationbit from the quantum bit. The identification of the stateof the quantum bit is not complete; i.e., for an unknownstate of the quantum bit |, it is impossible to deter-mine its true state using projective measurements. Themeasurement of the state of the quantum bit | = a|0 +b|1 corresponding to the observation {1, 2} (where1(2) is the subspace spanned on the state |0 , (|1) or,

in other words, in accordance with the standard compu-tational basis {|0 , (|1}) has the bit 0 (1) with the prob-ability |a|2 (|b2) as the output result, and the state | =a|0 + b|1 collapses to the state |0 (|1). The whole otherinformation that belongs to the superposition is irre-versibly lost. Therefore for an observer, the quantum bitis represented as a random variable with a certain prob-ability distribution. However, the quantum bit | = a|0 +b|1 can also be measured relative to other computa-tional bases in an infinite number of ways. For example,the dual computational basis is used frequently

Then

and the measurement | in this basis yields 0 (or 1)with the probability

Remark 5. Quantization of classical operators indifferent computational bases (on the correspondencebetween the quantum and classical operators). Con-sider as an example the problem of quantization of agiven classical operator. Assume that a single-bit nega-tion operation (NOT gate) is the classical operator,which converts the bit (a) into its complement (2), (1 a),(a = 0, 1). It is not difficult to show that, as the quantumcomponent of this operation, it is sufficient to choose

the unitary Pauli matrix of type x = However,

if we take the negation operation for the quantum bit inthe form of the matrix a= |aa|, where |a =

, then the relation a 1 a holds exactly.

On the other hand, the Pauli matrix z = also

= +| 0'| 12

------- 0| 1| +( ) | , 1'| = 12

------- 0| 1| ( )= ,

0| 12

------- 0'| 1'| +( ) 1| , 12

------- 0'| 1'| ( ).= =

| 12

------- a b+[ ] 0'| a b[ ] 1'| +( )=

1

2--- a b+

2or

1

2--- a b

2

.

0 11 0

.

1 a

a

,x

1 0

0 1

implements the negation operation NOT a 1 aunder the condition that the computational basis is cho-sen in another way, i.e., in the form a = |a' 'a|, where

|a' = . This simple example shows the

dependence of the procedure of establishing the quan-tum-logical correspondence on the choice of states inthe computational basis. Moreover, even if a computa-tional basis is chosen, there is a set of variants for fur-ther description of the operation. For example, if an

operation a is chosen, then the operator =

also implements the bit conversion a

1 a for given values of the angles (, ). The standardnegation operation for = = 0 is only a variant, but notthe uniquely possible one. This fact is explained by thephysical nature of quantum states, which are describedby rays rather than vectors in the Hilbert space. Thepresented arguments take place even in establishing thereverse correspondence between quantum and classicaloperators; i.e., different classical operators can corre-spond to a quantum operator. For example, both the

identity operation a a and the negation operation

a 1 a correspond to the quantum operator z.

Consider the example of application of measure-ment models in quantum computing. Assume that acompound quantum system of two quantum bits is

given in the form of the state vector in the complex state4 in the computational basis

(2.3)

Note that (2.3) is a generalized entangled state. If thefirst bit is measured in the state |, then there exist thefollowing two possible results: the first bit is equal tozero (m = 0) or it equals one (m = 1). In the first case,the corresponding operator of generalized measure-ment is determined as M0 = |0000| + |0101|. In thesecond case, M1 =|1010| + |1111|. The probabilityof the event that the first bit in the state | is zero is cal-culated in the formp(0) = = |a0|2 + |a1|2.After measurement, the state is determined as

z

1

2-------

1

1( )a

.

x

0 ei

ei

0 x

z

z

| a0 00| a1 01| a2 10| a3 11| .+ + +=

M0M0

'| M0 |

p 0( )---------------

M0 |

M0TM0

-------------------------------------a0 00| a1 01| +

a02

a12

+

----------------------------------- .= = =


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LITVINTSEVA et al.

quantum mechanics consists of two constituents, thefirst of which operates with the concept of quantumstate of the investigated quantum system, and the sec-ond one is oriented to quantum dynamics (evolution ofthe quantum system). The presence of two approachesis connected with the particlewave duality of thedescription of dynamic objects of quantum mechanics.Note certain specific features and properties of quan-

tum evolution and the interrelation of measurementsprocesses with the duality of the representation ofdynamic objects in quantum mechanics. Consider thewave representation of quantum mechanics. Assumethat the state of a quantum system is described in the

following form: | = , where =1, and{|i} determines the set of orthogonal vectors. Ifi 2,then we say that the dynamics of the given quantumsystem have a wave character. Otherwise i = 1, thequantum system describes the dynamic behavior of asingle particle, and we deal with the corpuscular behav-ior of the considered system.

The trace of a matrix and processes of quantummeasurements.,. Let us consider the trace of a matrixA,

Aii: Tr(A) . The matrix trace has the cyclicproperty and is linear. Suppose that the state of twophysical systems A and B is described by the densityoperator . The reduced density operator for the sys-tem A can be written in the following form: A TrB(AB), where the partial trace TrB is the mapping ofoperators known as the partial trace over the system

Thus, the partial trace of an operator is a tool for quan-titative description of the observed subsystems belong-ing to a compound system.

To distinguish quantum states, quantum measure-ments are required. A projective measurement over asubsystem is similar to the operation of taking the par-tial trace. Assume that we have a state GHZ of the fol-lowing form:

(an entangled three-particle state of three quantum bitsBC). Then the partial trace over one quantum bit, e.g.,over the system, can be written as

where = |0B0C0B0C|, = |1B1C1B1C|; i.e., aftertaking the partial trace over system , the subsystem

ai i| i aii

Aiii

A TrB AB( ) TrB

ABa1| a1 | b1| b1 |=( )=

a1| a1 |( )Tr b1| b1 |( ).

GHZ| 12

------- 0A0B0C| 1A1B1C| +( )=

TrA GHZ| GHZ |( )1

2--- 00

BC( ) 12--- 11

BC( ),+=

00BC 11

BC

has the probabilityp = and is either in the state

or in . According to the postulate of quantum

measurements, if a projective measurement in the basis{|0, |01} was carried out over the system, then theresult is the state |0 (corresponding to ) or the state

|01 (corresponding to ) with the probabilityp =

Consider the quantum state |w of three quantum

bits of the form |w = (|1A0B0C +|0A1B0C +

|0A0B1C). Let us take the trace over the quantum bitA

where = (|0B1C + |1B0C) determines the Bell

state. Thus after the projective measurement over thesystem in the basis {|0, |1} the system with the

probabilityp = can be in the state |0 (corresponding

to ) and with the probabilityp = , in the state |1

(corresponding to ).

Assume that Q is compound system (of two quan-tum bits Q1 and Q0), which is in the Bell state

The density operator Q has the representation

(2.4)

Let us calculate the partial trace based on the state ofthe second quantum bit; i.e., the system Q1. As a result

1

2---,

00BC 11

BC

00BC

11

BC 1

2---.

1

3-------

TrA w| w |( )1

3---00

BC 2

3--- BC

+| BC+ |,+=

BC+| 1

2-------

1

3---

00BC 2

3---

BC+|

Q| 0100| 1110|

2--------------------------------- .=

Q0100| 1110|

2---------------------------------

0100 | 1110 |

2---------------------------------

=

=1

2--- 0100| 0100 | 0100| 1110 |(

1110| 0100 | 1110| 1110 |+1

2---

1 0 0 1

0 0 0 0

0 0 0 0

1 0 0 1

.=


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the reduced density operator 0 has the followingform:

Note that it was taken into account that the states areorthogonal, 01|01 = 1, 01|11 = 0, etc. Note that in thisexample the reduced state 0 is mixed since

This result characterizes the specific feature of thequantum system that is principally absent in the classi-cal system [8]. Indeed, the state of a compound systemis pure, Tr[(Q)2] = 1, and it is established exactly (max-imum information). However, the first quantum bit is ina mixed state. This means that information about thestate of this quantum bit is not maximal, and the systemitself has more chaotic behavior than the compoundsystem. A similar property, which strange for classicalsystems, when complete information about the com-pound system is available, and for the subsystems onlya part of information is known, characterizes the abilityof quantum systems to be self-organized by using thesuper-correlation property (quantum correlation) con-tained in entangled states. It is the presence of quantumcorrelation that explains the possibility to organize apure state in two mixed compound states.

The Schredinger equation describes the evolution ofthe state of a quantum system. The generalized quan-tum dynamic processes are formalized by quantumoperators. The complementarity principle presumes theuse of particlewave duality in describing a quantumobject. To extract information on the behavior of thequantum object, it is necessary to conduct quantummeasurements. Unitary operations are connected withthe wave representation of a quantum object. However,non-unitary operators violate the wave pattern of sys-tem description and result in a corpuscular representa-tion. In the quantum case, completeness and correct-ness of relations require that the condition Tr() = 1 issatisfied, which is valid for quantum measurements bycondition (2.2). The trace operation means that we candetect a particle in the state space with definiteness,

Q0

0 Tr1 0( )=

=1

2--- Tr1 0100| 0100 |( ) Tr1 0100| 1110 |( )[

Tr1 1110| 0100 |( ) Tr1 1110| 1110 |( )+ ]

=1

2--- 00| 00 | 01 01 | 00| 10 | 11 01 | 10| 00 | 01 11 | [

+ 10| 10 | 11 11 | ]1

2--- 00| 00 | 10| 10 |+[ ]=

=1

2--- 1 0

0 1 1

2---I.=

Tr 0( )2[ ] Tr I

2---

2 1

2--- 1.


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LITVINTSEVA et al.

a2 = b2. This physical measure is maximal when thestate is separable (i.e., for either a = 0 or b = 0).

On the other hand, it is known that the partial traceover one of the states in the form of the von Neumannentropy (see the definition in what follows)

where the reduced density operator

is the measure of entanglement of the state (quantumcorrelation). The quantum correlation takes the maxi-mum valueE= 1 for a2 = b2, and it takes the minimumvalueE= 0 if either a = 0 or b = 0. Thus, the greater isthe quantum correlation of the state, the lower is theinterference visibility, and vice versa.

Another popular correlation measure, called nega-tivity N, is defined as a negative value of the doubledvalue of the least eigenvalue of the transposed density

matrix. Therefore,N= 2|ab|. In this case, the comple-mentarity measure is expressed as N2 + 2 = 1. Then,the interrelation between the entangled state and inter-ference follows from the constraint in the form of uni-tarity of quantum operators a2 + b2 = 1.

Thus in models of quantum algorithm, the measuresof quantum correlation and interference are not inde-pendent, and the efficiency of finding a successful solu-tion with the help of the quantum algorithm depends ontheir interrelation.

Example 8.Design of quantum algorithmic gatesand quantum programming. The ground of the methodof designing quantum algorithmic schemes and similar

methods for forming new types of quantum algorithmsis provided by the system for designing quantum algo-rithmic gates (QAGC) described in [8, 10, 23]. As in thegeneral structure of the quantum algorithm, the struc-ture of the system for designing QAG (quantum algo-rithmic gates) is based on the formalization of thedescription of three main quantum operators (superpo-sition, quantum correlation (of entangled states), andinterference) and measurement in the form of elemen-tary evolutionary unitary operators. In accordance withthe quantum scheme of the QA (quantum algorithm),these operators are combined by tensor and direct prod-ucts in a unified evolutionary quantum unitary operator[10, 23]. Structurally, the quantum algorithmic gatesacts on the initial canonical basis vector and forms acomplex linear combination of the constituent classicalvectors (called superposition) in the form of basis vec-tors as the output result of the action of the superposi-tion operator. The superposition contains informationabout the solution of the investigated problem as one ofthe components. After forming the superposition, in thequantum algorithmic gates, the operators of quantumcorrelation, interference, and measurement are appliedin order to extract information about the desired solu-

E S re d( ) a2

a2

log b2

b2

log ,=

re d Tr2 '| ' | Tr2 | |= =

a2

01| 01 | b2

11| 11 |.+

tion. In quantum mechanics, the measurement processhas the irreversible character and is a non-deterministicoperation, which results in measurement of only one ofthe basis vectors in the formed superposition. The prob-ability of each basis vector to be a measurement resultin the composition of the superposition for a givencomputational basis depends on the complex coeffi-cient (probability amplitude). The process of termina-

tion of the iterative action of the quantum algorithmicgates is performed by a program method based on theprinciple of minimal information entropy of an intelli-gent quantum state, containing valuable informationabout the desired solution [10, 26]. quantum algorith-mic gates can be implemented with the help of a soft-warehardware toolkit of evolutionary quantum com-puting.

In quantum programming, there exists a proof ofcompleteness of the description of a quantum algorith-mic gate by the corresponding programming languagesof improved semantic expressiveness [10]. By analogywith the existence of equivalence in the theory of com-puting, which is based on classical algorithms, weknow a hypothesis on equivalence between the repre-sentation of expressions of quantum operations at thesyntactic level with conservation of the completenessof their description for the sake of inclusion in quantumprogramming languages of semantic expressiveness ofquantum operators (at the functional level of thedescription of actions of quantum operators). One ofthe natural steps in this direction is the development ofprinciples of logical inference and test of the truth ofpropositions in quantum programming languages foravoiding contradictions in the obtained consequence oflogical inference. We refer to these language, e.g., theQML (Quantum Programming Language) [10, 2734].

Consider how a consistent description of the defini-tion of action of the Hadamard operator is implementedin QML in the following form:

Let us estimate the completeness and truth of thisexpression, which is equivalent to the test of truth of thefact that the successive action of Hadamard operatorsH(Hx) leads to the result equivalent to x at the func-tional level. For this purpose we use the followingmodel of logical inference in QML [29]:

H(Hx) = if (ifxthen (false+(1)true)else(false + true))

then (false + (1)true)

else (false + true) by commuting conversion for if

= ifx

then if (false + (1)true)


else (false + true)

else if (false + true)


else (false + true)

Hx ifx then false + 1( )true( )else false true+( ).=


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by if

= ifx

then (false false + true + true)

else (false +false + true true)

by simplification and normalisation

= ifxthen true elsefalse

by -rule for if=x

Elements of the theory of testing completeness andtruth of the semantics of the functional description ofthe quantum algorithm in the QML functional quantumprogramming language were described in [10, 2734].

3. QUANTUM INFORMATION PROCESSESAND LAWS OF QUANTUM

INFORMATION THEORY

The Shannon information entropy is defined as

The von Neumann entropy has the following form:

In the particular case, when the matrix is diagonal, theShannon and von Neumann entropies are equal. How-ever, the laws and consequences of quantum informa-tion theory have a number of fundamental distinctionsunder the quantum generalization of Shannon classicalinformation theory [8, 10, 3537]. Let us investigatebriefly some of these specific features employed in themodels of quantum fuzzy inference by examples.

Example 9. Consider the specific features of the

description and information analysis of Bell entangledstates

Since the Bell state with density operator (2.4) is pure,Q represents a pure ensemble. Therefore, there is nouncertainty in the quantum state; i.e., the von Neumannentropy is zero SN(Q) = 0. The reduced density opera-tor 0 for the quantum bit |00 is the partial trace over thesystem Q, i.e.,

Therefore, the quantum uncertainty in the state |00 isdetermined by the von Neumann entropy as SN(Q) = 1.Thus, the information analysis of uncertainty in thecompound quantum system allows one to explainclearly the presence of unusual (non-classical) proper-ties: if we ignore a part of information about the state ofa subsystem, this results in an increase of quantumuncertainty. As a result, quantum uncertainty in a part

H p( ) pi pilog .

i

=

SN ( ) Tr log( ).=

Q| 0100| 1110|

2---------------------------------= .

0 Tr1 Q( )

1

2--- 0| 0 | 1| 1 |+( )

1

2---

1 0

0 1

.= = =

(subsystem) Q0 is greater than in the complete (com-pound) quantum system Q. Classical systems do nothave this effect in view of the properties of the Shannoninformation entropy.

Example 10. Consider the Bell state | = (|00 +

|11) in the Hilbert space AB =AB, whereA=B = 2. The density matrices AB = ||, A, andA|B determined as

The matrix of conditional density is A|B = AB (IAB)1(in this case the cross density matrix AB and the mar-ginal density matrixIAB commutes). It follows fromthe definition of the von Neumann entropy that SN(A) =SN(B). Then, we have

since S(A|B) = 1.Therefore, in contrast to the Shannon classical infor-

mation theory, the von Neumann quantum conditionalentropy can take negative values when entangled statesare considered. This fact is connected directly with thequantum non-separability of entangled states, and theyare interpreted as super-correlated states. Thus, the neg-ativeness of the conditional entropy indicates the pres-ence of entangled states in a compound quantum sys-tem and determines the lower bound of their correlation[8, 10, 38, 39].

The validity of this fact was also established in theShor and Grover quantum search algorithms [10, 26]and is harnessed in solving the problem of efficient ter-

mination of the quantum algorithm. We also note thatnot all base classical relations and inequalities havequantum analogues. For example, in the classical case,we have

while, in the quantum case, the upper bound is given bythe inequality

1

2-------

A

1

2--- 0

01

2---

, A B

1 0 0 1

0 0 0 0

0 0 0 0

1 0 0 1

,= =

AB

1

2--- 0 0

1

2---

0 0 0 0

0 0 0 0

1

2--- 0 0

1

2---

.=

S AB( ) S B( ) C A B( )+ 1 1 0,= = =

I x : y( ) min H x( ) H y( ),[ ] .

S X: Y( ) 2min S X( ) S Y( ),[ ] .


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Quantum information theory has strictly substanti-ated rules of how to extract information from anunknown quantum state [8, 10, 3537]. An optimalquantum process of extraction of valuable informationfrom individual knowledge bases designed for fixedcontrol situations on the basis of soft computing isbased on four facts of quantum information theory pre-sented below. In particular, it was proved that there

exist an efficient quantum data compression, couplingof classical and quantum parts of information in thequantum state, and total correlation in the quantumstate is a mix of classical and quantum correlations, andthere is a hidden (observed) classical correlation in thequantum state. In what follows, we consider briefly thephysical meaning of the listed facts and their role in theprocesses of designing optimal processes and controlsignals based of quantum fuzzy inference.

Fact 1.Efficient quantum data compression. In theclassical information theory, Shannon showed howmuch (under given accuracy) we can compress limit-edly a message comprising N independent symbols

(xa), where each symbol enters the message with theprior probability pa, using the concept of informationentropy. The Shannon information entropy H(pa) isdefined as

The following proposition was proved: a block of codesof length NH bits is sufficient for coding all typical(most frequently occurred) sequences without accountof the methods of coding non-typical sequences of themessage. Note that the probability of error coding

(information loss) does not exceed a given threshold .In quantum information theory, symbols are densitymatrices. Two variants are possible. In the first variant,the density matrices correspond to an ensemble of purestates |a. In the second variant, the ensemble is formedby density matrices a with the probabilitypa. Considerthe ensemble of states for the second variant. In thiscase, the matrix of density of messages comprisingNsymbols is described as (N) = , where =

The von Neumann entropy of messages has the sim-ple relation with the ensemble entropy, S =Tr(ln).

The following inequality that relates the Shannon infor-mation entropy and the von Neumann entropy isknown: S((N) =NS(); i.e., the value of the Shannoninformation entropy exceeds the value of the von Neu-mann entropy. This means that the application of quan-tum information theory allows one to perform deepercompression of classical information [8, 10].

Fact 2.Coupling (separation) of information in thequantum state in the form of classical and quantumcomponents. Consider the model of a generalized mea-

H pa( ) pa p2 alog .a

=

pa a| a |.a

surement (see Section 2) in the state for which

the density matrix has the following definition:

Then, the final state of the subsystemB is

The entropy of the reduced state is The

amount of classical information received in the mea-surement i is expressed with the probability pi as theShannon information entropy H(p). If the quantum

states belong to orthogonal subspaces, then the

entropy of the final state (after a measurement) is the

sum of the reduced quantum entropy and

the classical information, i.e.,

Thus, the amount of information contained in thequantum state can be separated (coupled in the form)into the quantum and classical components [40]. There-fore, in simulating robust structures of intelligent con-trol systems, the classical information is simulated byusing the knowledge base optimizer, and its deficitcanbe defined as [8, 10]

Therefore, we can extract additional amount ofvaluable quantum information from individual knowl-edge bases for subsequent use in designing intelligentcontrol of improved level. Note that quantum proce-dures for compression and reduction of redundantinformation contained in classical control signals areapplied (using the corresponding models of quantumcorrelation in the quantum algorithm of quantum fuzzyinference).

Fact 3.Amount of total, classical, and quantum cor-

relation. Entangled states or, in the general form, quan-tum correlation are typical physical resources of quan-tum computing. However, not all types of correlationhave pure quantum nature. In other words, total corre-lations are mixes of classical and quantum correla-tions [41].

For optimal design (efficiently simulated on conven-tional computers) of a given class of quantum algo-rithms, it is important to know the type (or form) of thenecessary classical correlation. For example, if it is pos-

AiAiT,

Bi AiBAi

T

Tr AiBAiT( )

----------------------------- .=

AiBAiT

i piB

i

i .=

piS Bi( )

i .

Bi

piS Bi( )

i

S pi Bi( )

i

H p( ) piS Bi( ).

i

+=Classical

Quantum

I S piBi

i

piS Bi

( )i H p( ).= =

Total Quantum Classical


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sible to determine the classical component of the corre-lation, then, using optimal positive operator-valued)measurement measures (see the details in Section 2), itis admissible to extract the maximum amount of infor-mation in the classical form contained in the quantumstate with the minimum increasing of entropy [42, 43].The amount of total correlation can be split into theclassical and quantum components. This measure is

equivalent to the measure of maximum classical/quan-tum mutual information I(A : B), retaining the directphysical interpretation of the interrelations between thecorresponding measures [44].

Fact 4.Hidden (observable) classical correlation inthe quantum state. In quantum information theory, thefollowing unexpected fact was established. The condi-tion of proportional increase of the amount of informa-tion

determined by local measurementsMAMB in the stateAB can be violated under some extremal constraints onthe initial mixed state . For example, the initial amountof information in the form of a single classical informa-tion bit sent fromA toB can be enlarged at the receiptstage by a definite amount in the quantitative measureICl() [45]. This fact is explained from the position ofthe observation phenomenon of the classical correla-tion in the quantum state . Since the amount of infor-mationICl() is proportionally increased at the classicallevel, the phenomenon of correlation observation is apure quantum effect arising due to the fact that quantum

non-orthogonal states are indistinguishable. Therefore,there exist quantum two-particle states that contain alarge amount of classical correlation non-observable atthe classical level because of the amount of classicalinformation in the transmission channel that is non-pro-portionally small for its observation (limited data trans-mission ability).

There are (2n + 1) quantum bits, using which a sin-gle-bit message increases twice the optimal amount ofclassical mutual information as a result of measure-ments between the subsystems. In general, for n/2 bitssent, the specified amount of information increases to n

bits. It is impossible to obtain the specified effect at theclassical level because of the laws of classical physics.In this case, the following fact is remarkable: the statesthat support the specified effect are not necessarilyentangled, and the corresponding classical dataexchange channel can be implemented by the Had-amard transform. The presented facts yield the infor-mation resource of the background of quantum fuzzyinference employed in simulation of robust knowledgebases for intelligent fuzzy controllers.

ICl ( ) I A : B( ),M

A M

B

max=

4. TOTAL AND HIDDEN (OBSERVABLE)CORRELATION IN QUANTUM STATES

There is a belief that the expected computationalpower of quantum computing issues from the presenceof quantum resource. Entangled states, or quantum cor-relation in the general case are bright examples of this.However, as was mentioned in Section 3, not all corre-lation forms have purely quantum nature; i.e., the total

correlation is a mix of classical and quantum correla-tions. The knowledge of how and where the classicalcorrelation is used in the quantum correlation is animportant point. For example, if it is possible to deter-mine and select the classical correlation component,then using optimal measurement, we can extract certainadditional information amount in the classical formhidden in the quantum state with a minimal increasingof entropy.

Physically, the listed correlation types are character-ized by the amount of work (noise) that is necessary todo for eliminating (destroying) the correlation. For thetotal correlation, the amount of work for complete

destroying is required. For the quantum correlation, theamount of work to the destruction into separable statesis sufficient. However, even in the case of classical cor-relation, the maximum correlation is destroyed aftereliminating the quantum correlation. The total amountof correlation, measured by the minimum production ofrandomization and equivalent to the requirement oftotal destruction of all forms of correlation in the stateAB, is equivalent to the quantum amount of mutualinformation [46, 47].

4.1. Classical and Quantum Correlations

The classical mutual information contained in thequantum state AB (before its measurement) can be esti-mated in a natural way as the maximum mutual infor-mation which can be extracted by local measurementsMA MB in the state AB

Here, I(A : B) is the classical mutual informationdefined in the form

whereHis the information entropy andpAB,pA, andpBare the density functions of the probability distribution

of the mutual and individual results obtained by localmeasurementsMA MB in the state [45].

Example 11.Mutual information and classical cor-relation. To understand the role of classical correlationand its interrelations with the concept of mutual infor-mation, we define quantum mutual information for thetwo-particle state AB of the quantum system in theStratonovich form

ICl ( ) IM

A M

B

max A : B( ).=

I A : B( ) H pA( ) H pB( ) H pAB( ),+

I A : B( ) S A( ) S B( ) S AB( ).+=


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Consider a compound systemAB in the state AB, whichcan be in the state A and B with the probabilitiesp and(1 p), respectively. For this case of the compound sys-tem AB, mutual information can be calculated in thefollowing form [42]:

(4.1)

IfAB is a separable state, then its relative entropy in theentangled state is zero.

The physical interpretation of the value ICl() ismany-valued [45, 48]: ICl() manifests itself as maxi-mum classical correlation extracted by a purely localmeasurement procedure from the state ;ICl() corre-sponds the classical definition when the state is clas-sical, i.e., diagonal in some (locally used) computa-tional basis and corresponds to the classical distribu-tion; if is a pure state, then ICl() specifies thecorrelation determined by the Schmidt basis and equiv-alent to the measure of entangled pure states;ICl() = 0if and only if

It is known that some suitable measures of quantumcorrelation have to satisfy certain axiomatic properties:(1) quantum correlation is non-local and cannotincrease in local measurement procedures (monotonic-ity property); (2) complete proportionality; (3) growthof proportionality; and (4) continuity in .

Physically, property (2) means that the state proto-col constituted of the non-correlated initial state using lquantum bits or 2l classical bits (for data transmissionthrough a quantum communication channel) and apply-ing local operations cannot generate more than 2l cor-relation bits. Property (3) presumes that in transmittinga message ofl quantum bits or 2l quantum bits the cor-relation in the initial state does not increase and doesnot exceed 2l bits.

Properties (1)(4) fulfill completely, in particular,for classical mutual informationI(A :B) when the mes-sage transfer is performed by the classical method. For

example, the properties of the complete proportionalityand growth of proportionalityI(A :B) for the classicalcase follow from the fact that

so that whenA sends a classical systemA' toB, we have

I A : B( ) 2H 12--- 1 p

21 p( )2++[ ]

=

H1

2--- 1 1 3p

23p++[ ]

.

AB A B.=

max H pA( ) H pB( ),( ) H pAB( ) H pA( ) H pB( )+ ,

ICl ( ) I A BA';( ) I AA ', B( ) H pA '( ).+=

Then the property of complete proportionality followsfrom the property of proportionality growth. The sameis true for quantum mutual information

It was unexpected that the property of proportional-

ity growth is violated forICl() in an extremal way in thecase of mixed initial states simple classical bit sentfromA toB can result in an increase ofICl() to a certainbig value. We consider this phenomenon as an opportu-nity of observation of classical correlation in the quan-tum state . If the property of growth of proportionalityICl() takes place at the classical level, then the phenom-enon of observed classical correlation is a pure quan-tum effect. This result immediately follows from indis-tinguishability of non-orthogonal quantum states.

Example 12. Assume that an initial state , the typeof message transfer, and the corresponding amount of

data transferred are given. The increase of correlationcan be characterized by the following functions:

The operator is an operation on a two-particlestate, which consists of local operations and containsno more than 2l classical or l quantum bits in a mes-sage. This is reflected in the corresponding superscripts(l) or [l]. Denote by and ' the states before and afterthe operations with message exchange, ' = (). Theamount of correlation hidden (unobservable)

in the state with l quantum bits in one-way messageexchange can be bounded by the following condition[45]:

For small values ICl(), the amount of hidden (unob-servable) correlation in the two-way exchange isbounded from above

IQ A : B( ) S A( ) S B( ) S AB( ).+=

ICll( )

ICl l( ) ( )( )

l( )max=

one-way classical message( );

ICll[ ]

ICl l[ ] ( )( )

l[ ]max=

two-way classical message( ).

IClL( ) ( ),

IClL( )

( ) ICl ( ) l 2l

1( )ICl ( ).+

ICll[ ] ( ) ICl ( ) 2l O d

2ICl ( ) ICl ( )log( ).+


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4.2. Hidden (Observable) Classical correlationin a Quantum State

Let us discuss the situation in which someamount of correlation is not accessible to observa-tion in one-way message exchange. The initial stateis determined by the subsystemsA andB on the cor-responding subspaces of dimensions 2dand din the

form

(4.2)

where the operators U0 = I and U1 change the initial

computational basis by the united one as |i|U1|k| =

i, k. Then, the playerB takes randomly a state

|k of d states in two possible randomized bases(depending on the case when either t = 0 or t = 1in (4.2)

At the same time, the observerA has complete infor-mation about the quantum state of the observer B.Receiving the required amount of information in the

form = + 1,A sends a value ttoB, which,

in turn, applies the operator Ui to its state and measure

the value kin the computational basis. As a result,A and

B have the measurement kand t, which yields =

+ 1 correlation bits.The state evolutes according to the following

scenario [45]. Let d= 2n. Then,A chooses randomlyk of length n bits and sends to B either a message

about the state |k or Hn|k depending on a randomvalue of the bit either t= 0 or t= 1. Here, H is theHadamard transform; A sends t to B and, later,observes the created correlation. It was experimen-tally established that the application of the Had-amard transform and measurement of the state of thequantum bits is sufficient in order to implement the

procedure of preparing the state and then extractthe classical correlation hidden in the state '. The

initial correlation is a small quantity =

After the complete measurement MA in the

one-way message exchange, the final value of the

amount of information in the quantum state is deter-

mined asICl(') = = ; i.e., the amount of

available information increases.

Remark 6. Note that the complete measurementMA in the basis {|k|t} (Section 2) is optimal forthe systemA. The output value of the measurementresult gives exactly information about what purestate of the ensemble is chosen. Therefore, there isan opportunity to apply the classical local processof processing (of the measurement result) forobtaining information about the distribution ofresults for other measurements. For the system A,the choice of the optimal measurement allows thesystemB to extract from ICl() amount of informa-tionIAcc about the ensemble of uniformly distributed

states {|k, (U1 =H)|k}k=0, , d1.

4.3. Available Informationabout the Ensemble of Mixed States

In the general case, the available information aboutthe ensemble of mixed states = {pi, i} is defined as

the maximum mutual information between the mea-sured state with the index iand the result of its measure-ment. The amount of available information IAcc() canbe characterized as the maximum value of informationextracted from the quantum state with the help of posi-tive operator-valued measurements (section 2) with ele-ments of only rank 1 [8].

Assume thatM= {j|jj|} means a positive opera-tor-valued measurement with elements of rank 1, whereeach state |j is normalized and j > 0. Then,IAcc() canbe calculated as

(4.3)

12d------ k| k t t |( )A Utk| k |Ut

T( )B,t 0=

1

k 0=

d 1

=

1

a-------

ICll( ) ( ) dlog

ICll( ) ( )

dlog

ICll( ) ( )

1

2--- d.log

ICll( ) ( ) dlog

IAcc ( ) pi pilogi

pij j i j pi j i j j i j ----------------------------logi

i

+M

max ,=

Classical component Quantum component


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springer quantum

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