Interpreting Xor Intuitionistic Fuzzy Connectives from

Quantum Fuzzy Computing

Anderson Avila, Renata Reiser, Maurício Pilla and Adenauer Yamin

PPGC - CDTEC, UFPEL, Pelotas, Brazil

Keywords:

Intuitionistic Fuzzy, Quantum Computing, Xor Operator.

Abstract:

Computer systems based on intuitionistic fuzzy logic are capable of generating a reliable output even when

handling inaccurate input data by applying a rule based system, even with rules that are generated with im-

precision. The main contribution of this paper is to show that quantum computing can be used to extend the

class of intuitionistic fuzzy sets with respect to representing intuitionistic fuzzy Xor operators. This paper

describes a multi-dimensional quantum register using aggregations operators such as t-(co)norms based on

quantum gates allowing the modeling and interpretation of intuitionistic fuzzy Xor operations.

1 INTRODUCTION

Fuzzy logic (FL) and its extensions as the Atanassov’s

fuzzy intuitionistic logic (A-IFL)(Atanassov and Gar-

gov, 1998; Atanassov, 2017) together with quantum

computing (QC) are relevant research areas consoli-

dating the analysis and the search for new solutions

for difﬁcult problems faster than the classical logi-

cal approach by extending results from conventional

computing.

Similarities between these areas in the represen-

tation and modelling of uncertainty have been ex-

plored (Pykacz, 1993; Kreinovich et al., 2009; Man-

nucci, 2006). The uncertainty of human being’s rea-

soning can be modelled in A-IFL as a mathemati-

cal model inheriting the indeterminacy from mem-

bership and non-membership degrees which are not

necessarily complementary. It provides techniques

helping physicists and mathematicians to transform

their uncertainty ideas into new computational pro-

grams (Kosheleva et al., 2015).

The uncertainty of the real world is concerned

with fundamental concepts of QC by making use of

properties of quantum mechanics as the superposition

which suggest an improvement in the efﬁciency re-

garding complex tasks. In addition, simulations using

classical computers improve the development and val-

idation of basic quantum algorithms, anticipating the

knowledge related to their behaviours when executed

in a quantum computer.

In spite of quantum computers being restricted to a

few research centers and laboratories, the studies from

quantum information and quantum computation are a

reality nowadays.

In this context, new methods dealing with fuzzy

approaches to quantum computational logics have

been proposed (Chiara et al., 2018). Moreover, it

contributes to increase the interest in quantum algo-

rithm applications representing fuzzy and intuitionis-

tic fuzzy systems.

1.1 Synergy between FL and QC

Potentialities from quantum parallelism and superpo-

sition of quantum states are explored by new method-

ologies. See, e.g. providing a model for humanoid be-

haviours based on FL (Raghuvanshi and Perkowski,

2010) and using a density operator and logical con-

nectives deﬁned by quantum gates (Bertini and Lep-

orini, 2017). In other areas, QC is used to im-

prove computations as Econometric Modeling (Sri-

boonchitta et al., 2019) and to integrate concepts

from quantum genetic algorithm and fuzzy neural net-

work (Li et al., 2010) as a model employed to control

an inverted pendulum system.

From a logical perspective, many-valued fuzzy

approach is a consolidated research area (Hájek,

1998) and recent efforts integrating the quantum com-

putational logic are considered (Chiara et al., 2018).

See, e.g. (Freytes et al., 2010) making use of quantum

states to represent information processing in QC.

In (Rigatos and Tzafestas, 2002), a parallel control

fuzzy algorithm is proposed showing that both QC

and fuzzy learning algorithms are relevant research

288

Avila, A., Reiser, R., Pilla, M. and Yamin, A.

Interpreting Xor Intuitionistic Fuzzy Connectives from Quantum Fuzzy Computing.

DOI: 10.5220/0008169702880295

In Proceedings of the 11th International Joint Conference on Computational Intelligence (IJCCI 2019), pages 288-295

ISBN: 978-989-758-384-1

areas which can proﬁt from each other.

In this work, the modelling and interpretation of

A-IFL via QC provides the description of intuitionis-

tic fuzzy connectives by using quantum registers and

quantum transformations from the traditional model

of quantum circuits.

The information regarding each intuitionistic

fuzzy set is represented by pairs of quantum registers,

guaranteeing the inherent unitarity of quantum states

and quantum transformations well as the ﬂexibility

of the complementarity relation of membership and

non-membership functions characterizing intuitionis-

tic fuzzy sets.

Some results of such research area are mainly re-

lated to interpretation of fuzzy connectives via QC, as

negation, conjunction, disjunction, implications and

xor operators (Avila et al., 2015) and to intuitionistic

fuzzy connectives (Reiser et al., 2016).

This work extends that approach by the interpreta-

tion of two classes of intuitionistic fuzzy Xor connec-

tives via QC, the ⊗

and ⊕

intuitionistic fuzzy Xor

operators, which can be expressed as composition of

conjunctions (S

), disjunctions (T

) and the comple-

mentary operators (N

). But different from other rep-

resentable intuitionistic fuzzy connectives, an intu-

itionistic fuzzy Xor connective cannot be expressed

as an N-dual pair of Xor and EXor fuzzy connectives.

So, quantum states and quantum operators provide

interpretation for intuitinistic fuzzy values and con-

nectives, respectively. In particular, Xor and EXor

can be applied to explore regular applications based

on symmetric functions by making use of reversible

logic (He et al., 2017) and can be also extended to

QC (Perkowski et al., 2001).

1.2 Paper Outline

The remainder of this paper is organized as follows.

Section 2 presents the foundations on A-IFL. Sec-

tion 3 brings the main concepts of QC. In Section 4,

the study and modeling of intuitionistic fuzzy Xors

using QC is described. An interpretation of classi-

cal A-IFS (Atanassov, 2016; Mannucci, 2006) from

quantum states is also considered, presenting the op-

erations on A-IFS modelled from quantum transfor-

mations, relating the Atanassov’s intuitionistic fuzzy

approach(Atanassov, 1986) to two classes of Xor op-

erators and also considering representable Xor con-

nectives obtained by composition of standard nega-

tion together with the Product and Algebraic Sum.

Finally, conclusions and further work are discussed

in Section 5.

2 FUZZY LOGIC APPROACH

The A-IFL is a type-2 fuzzy logic conceived as a gen-

eralization of FL overcoming the limitations related to

fuzzy sets for dealing with problems where the rules

applied to the system could not be deﬁned with preci-

sion, mainly related to non-membership degree which

cannot be deﬁned as a complement of its membership

degree.

An element x∈X belongs to the subset A such that

0 ≤µ

(x) + ν

≤1, meaning that a non-membership

degree ν

(x) is not necessary the complement of its

membership degree µ

(x). Thus, A is given as:

A = {(x, µ

(x), ν

(x)) : x ∈ X ,0 ≤ µ

(x) + ν

≤ 1} (1)

Taking µ

(x) = x

, ν

(x) = x

∈ [0, 1] for an element

x ∈ χ, the set of all Atanassov’s intuitionistic fuzzy

values is given as

U = {(x

, x

) : x

+ x

≤ 1}. The

least and greatest element on

U are given as

0 = (0, 1)

and

1 = (1, 0), respetively.

Considering the research of A-IFL, it makes pos-

sible to extend the usual logic connectives, as follows:

1. Conjunction, usually modelled by a triangular

norm (t-norm) operator (Klement et al., 2013),

which is an additive aggregation used in the

framework of intersection in A-IFL;

2. Disjunction which is frequently modelled by a tri-

angular conorm (Klement et al., 2013) (t-conorm)

representing the t-norm dual operation;

3. Complement as a negation, a non-increasing

function, deﬁned in the Atanassov’s seminal

work (Atanassov, 2006), reverting the extremes of

the unit interval [0, 1].

4. Xor operators, considering in this work two rep-

resentable classes ⊕

and ⊗

(Bedregal et al.,

2013), both deﬁned by composition of the above

connectives verifying the symmetry, associativity,

neutral element and boundary conditions.

Now, the connectives used to make the correlation

between QC and A-IFL will be described in terms

of representability based on fuzzy negations and

aggregations.The standard intuitionistic fuzzy nega-

tion (Atanassov, 2006) is expressed as follows:

( ˜x) = (x

, x

), ∀˜x = (x

, x

) ∈

U. (2)

Meanwhile, for all ˜x = (x

, x

), ˜y = (y

, y

) ∈

U, the

intersection and union can be deﬁned, respectively, in

terms of a t-norm T and a t-conorm S, given as:

( ˜x, ˜y) = T

((x

, x

), (y

, y

)) = (T (x

, y

), S(x

, y

));

( ˜x, ˜y) = S

((x

, x

), (y

, y

)) = (S(x

, y

), T (x

, y

)).

We consider the Product t-norm T

and Algebraic

Sum S

, respectively described as follows:

( ˜x, ˜y)=(T

, y

), S

, y

))=(x

, x

+ y

+ x

);

( ˜x, ˜y)=(S

, y

), T

, y

))=(x

+ y

+ x

, x

)).

Interpreting Xor Intuitionistic Fuzzy Connectives from Quantum Fuzzy Computing

289

The classical xor expressions considered for this

work are described as follows:

A ⊕B ≡ (¬A ∧B) ∨(A ∧¬B);

A ⊗B ≡ (A ∨B) ∧(¬A ∨¬B).

Deﬁnition 2.1. The Atanassov’s intuitionistic fuzzy

Xor operator is a function E

→

U verifying:

E1: E

(

0) = E

(

1) =

0 and E

(

1) = E

(

0) =

E2: E

( ˜x, ˜y) = E

( ˜y, ˜x);

E3: ˜y

≤ ˜y

⇒ E

(

0, ˜y

) ≤ E

(

0, ˜y

);

E4: ˜y

≤ ˜y

⇒ E

(

1, ˜y

) ≥ E

(

1, ˜y

Proposition 2.1. The functions ⊕

, ⊗

→

given as follows:

⊕

( ˜x, ˜y) = S

( ˜x), ˜y), T

( ˜x, N

( ˜y))) (3)

⊗

( ˜x, ˜y) = T

( ˜x, ˜y), S

( ˜x), N

( ˜y))) (4)

are intuitionistic fuzzy Xor operators.

Proof. Consider ⊕

operator. It holds that:

1: The following boundary conditions are

veriﬁed in the endpoints of unit interval:

⊕

(

0)=S

(

0),

0), T

(

0, N

(

0)))=S

(

0)=

⊕

(

1)=S

(

1),

1), T

(

1, N

(

1)))=S

(

0)=

⊕

(

1)=S

(

0),

1), T

(

0, N

(

1)))=S

(

0)=

⊕

(

0)=S

(

1),

0), T

(

1, N

(

0)))=S

(

1)=

2 : ∀˜x, ˜y ∈

U the following is veriﬁed

( ˜x, ˜y) = S

( ˜x), ˜y), T

( ˜x, N

( ˜y)))

= S

( ˜y, N

( ˜x)), T

( ˜y), ˜x)) = E

( ˜y, ˜x);

3 : If ˜y

≤ ˜y

then the following is veriﬁed

⊕

(

0, ˜y

) = S

(

0), ˜y

), T

(

0, N

( ˜y

)))

≤ S

(

0), ˜y

), T

(

0, N

( ˜y

))) = E

(

0, ˜y

);

4 : If ˜y

≤ ˜y

then the following is veriﬁed

⊕

(

1, ˜y

) = S

(

1), ˜y

), T

(

1, N

( ˜y

)))

≥ S

(

1), ˜y

), T

(

1, N

( ˜y

))) = E

(

1, ˜y

);

Therefore, Proposition 2.1 is veriﬁed.

Based on those expressions, one can use the oper-

ators T

, S

, N

to construct the Atanassov’s intuition-

istic fuzzy xor (⊕

, ⊗

) respectively given as follows:

⊕

( ˜x, ˜y) =

( ˜x), ˜y), T

( ˜x, N

( ˜y)))

((x

, x

), (y

, y

)), T

((x

, x

), (y

, y

))))

((T (x

, y

), S(x

, y

)), (T (x

, y

), S(x

, y

))))

=(S(T (x

, y

), T (x

, y

)), T (S(x

, y

), S(x

, y

))) (5)

⊗

( ˜x, ˜y) =

( ˜x, ˜y), S

( ˜x), N

( ˜y)))

((x

, x

), (y

, y

)), S

((x

, x

), (y

, y

)))

((S(x

, y

), T (x

, y

)), (S(x

, y

), T (x

, y

)))

=(T (S(x

, y

), S(x

, y

)), S(T (x

, y

), T (x

, y

))) (6)

By considering T

and S

in Eqs. (5) and (6), the

xor (⊕

, ⊗

) operators can be respectively expressed

as follows:

⊕

( ˜x, ˜y) = ((x

+ x

−x

= (x

+ x

+ y

−x

−

−x

+ x

)). (7)

⊗

( ˜x, ˜y) = ((x

+ x

+ y

−x

= −x

−x

+ x

= (x

+ x

−x

)). (8)

According to (Mannucci, 2006), fuzzy sets can be

obtained by quantum superposition of classical fuzzy

states associated with a quantum register.

3 QUANTUM COMPUTING

In QC, the qubit is the basic information unit, being

the simplest quantum system, deﬁned by a unitary and

bi-dimensional state vector.

Qubits are generally described, in Dirac’s nota-

tion (Nielsen and Chuang, 2003), by the following

expression

|ψi = α|0i+β|1i

when the coefﬁcients α and β are complex numbers

for the amplitudes of the corresponding states in the

computational basis (state space), respecting the con-

dition |α|

+|β|

= 1, which guarantees the unitary of

the state vectors of the quantum system, represented

by (α, β)

(Kaye et al., 2007).

The state space of a quantum system with mul-

tiple qubits is obtained by the tensor product of the

space states of its subsystems. Considering a quan-

tum system with two qubits, |ψi = α|0i+ β|1i and

|ϕi = γ|0i + δ|1i, the state space comprehends the

tensor product given by

|Πi= |ψi⊗|ϕi = α ·γ|00i+ α ·δ|01i+β ·γ|10i+ β ·δ|11i.

The state transition of a quantum systems is per-

formed by controlled and unitary transformations as-

sociated with orthogonal matrices of order 2

, with

N being the number of qubits within the system, pre-

serving norms, and thus, probability amplitudes (Imre

and Balázs, 2005). For instance, the NOT opera-

tor (Pauli-X transformation) and its application over

1-dimensional and 2-dimensional quantum systems

are presented in the following.

X|ψi =



0 1

1 0









; (9)

⊗2

|Πi=







0 0 0 1

0 0 1 0

0 1 0 0

1 0 0 0













α ·γ

α ·δ

β ·γ

β ·δ













α ·γ

α ·δ

β ·δ

β ·γ







(10)

FCTA 2019 - 11th International Conference on Fuzzy Computation Theory and Applications

290

Furthermore, the action of a Toffoli QT is also

shown in next Eq. (11), describing a controlled op-

eration for a 3-dimensional quantum system

T |χi = T (ψ ⊗ϕ ⊗σ).

In this case, the NOT operator is applied to the

third qubit |σi when the current states of the ﬁrst two

qubits |ψi and |ϕi are both |1i:

Similarly to QTs of multiple qubits which were

obtained by the tensor product performed over unitary

transformations, Eq.(11) presents the matrix structure

of such QT, when |X i is the initial state:

T|X i=







1 0 0 0 0 0 0 0

0 1 0 0 0 0 0 0

0 0 1 0 0 0 0 0

0 0 0 1 0 0 0 0

0 0 0 0 1 0 0 0

0 0 0 0 0 1 0 0

0 0 0 0 0 0 0 1

0 0 0 0 0 0 1 0































(11)

In order to obtain information from a quantum

system, it is necessary to apply measurement opera-

tors, deﬁned by a set of linear operators M

, called

projections. The index m refers to the possible mea-

surement results. If the state of a 1-dimensional quan-

tum system is |ψi immediately before the measure-

ment, the probability of an outcome occurrence is

given by p(|ψi) =

|ψi

hψ|M

†

|ψi

. When measuring

a qubit |ψi with α, β 6= 0, the probability of observ-

ing |0i and |1i are, respectively, given by the follow-

ing expressions:

p(0) = hφ|M

†

|φi = hφ|M

|φi = |α|

;

p(1) = hφ|M

†

|φi = hφ|M

|φi = |β|

After the measuring process, the quantum state

|ψi has |α|

as the probability to be in the state |0i

and |β|

as the probability to be in the state |1i.

In multidimensional systems, the operators M

and p

(m) denote the m-projection and correspond-

ing probability measure, both performed on the n-

qubit.

4 INTERPRETING XOR

OPERATORS BASED ON QC

The description of intuitionistic fuzzy sets from the

QC viewpoint extends the work in (Mannucci, 2006)

by modeling an element ˜x = (x

, x

) by a pair of one-

dimensional qubit quantum states (|x

i, |x

i) where:

i =

1 −x

|0i+

√

|1i; (12)

i =

1 −x

|0i+

√

|1i. (13)

By modeling fuzzy operators in QC(Avila et al.,

2015), the Product t-norm T

and Algebriac Sum t-

conorm can be represented through the Toffoli gate

(T) and the standard negation through the Pauli-X

gate (N).

So the ﬁrst step to generate the quantum represen-

tation for both Xor operators ⊕

and ⊗

is to ap-

ply De Morgan’s law related to t-(co)norms T (S) and

fuzzy negation N as present in Eqs. (5) and (6), result-

ing on the following expressions for their membership

and non-membership degrees when considering ( ˜x, ˜y)

as input:

⊕

= N(T (N(T (x

, y

)), N(T (x

, y

)))) (14)

⊕

= T (N(T (N(x

), N(y

))), N(T (N(x

), N(y

)))) (15)

⊗

= T (N(T (N(x

), N(y

))), N(T (N(x

),N(y

)))) (16)

⊗

= N(T (N(T (x

, y

)), N(T (x

, y

)))) (17)

Taking ˜x = (x

, x

), ˜y = (y

, y

), the initial quan-

tum stated |φiis the 10-dimensional quantum register:

|φi = |x

i⊗|x

i⊗|y

i⊗|0i⊗|0i⊗|0i⊗|0i.

Then, the quantum representation of the Xor op-

erator ⊕

can be obtained by translating Eqs. (14)

and (15), respectively resulting in the following QT

compositions:

⊕

◦N

5,6,9

◦T

5,6

◦N

5,6

◦T

1,4

◦T

2,3

(18)

⊕

◦T

7,8

◦N

2,3,8

◦T

2,3

◦N

2,3

◦N

1,4,7

◦T

1,4

◦N

1,4

(19)

Analogously, considering the initial quantum state

|φi, the quantum representation of the Xor operator

⊗

can be obtained by translating Eqs. (16) and (17),

respectively resulting in QT compositions given as

follows:

⊗

◦T

5,6

◦N

2,4,6

◦T

2,4

◦N

2,4

◦N

1,3,5

◦T

1,3

◦N

1,3

(20)

⊗

◦N

10,7,8

◦T

7,8

◦N

7,8

◦T

1,3

◦T

2,4

(21)

For both quantum representations of membership

functions, it is used 10 qubits: 2 pairs for the in-

puts ( ˜x, ˜y), 4 ancillaries qubits to store intermediate

results and 1 pair for the ﬁnal result. The member-

ship degree obtained is stored on qubit 9 and the non-

membership degree on qubit 10.

Interpreting Xor Intuitionistic Fuzzy Connectives from Quantum Fuzzy Computing

291

|0i

T1 T2

T5 T6

T10

T11 T12

Figure 1: Quantum circuit modelling ⊕

( ˜x, ˜y).

|0i

T1 T2

T5 T6

T10

T11 T12

Figure 2: Quantum circuit modelling ⊗

( ˜x, ˜y).

4.1 Modelling ⊕

Quantum Operator

See in Fig. 1 the quantum circuit for the ⊕

Xor oper-

ator, resulting from the composition of Eq.(18) (from

T1 to T5) and Eq. (19) (from T6 to T12) which is an-

ticipating the measure operations.

Columns in Table 1 show the non-void amplitude

evolution for the most relevant points of this quantum

circuit, with T0 denoting the initial quantum state and

T12 is the ﬁnal quantum state resulting on ⊕

Xor

operator obtained from the application of all compo-

sition quantum operators. And, in such columns, the

changed qubits are highlighted.

After performing the circuit in Fig. 1, the measure

operator M

is applied, that is, on the 9

qubit and

related to |1i, it has probability

⊕

(1) = x

+ x

−x

corresponding to the membership degree of ⊕

Xor

operator, obtained by the µ

⊕

expression. Therefore,

⊕

(1) = µ

⊕

( ˜x, ˜y)

Analogous, the resulting measure operator M

has

the probability expressed as follows

⊕

(1)(0) = 1 −x

−x

+ x

corresponding to the complement of the membership

degree of ⊕

( ˜x, ˜y), meaning that

⊕

(1)(0) = 1 −µ

⊕

( ˜x, ˜y).

Thus, the 9

qubit is given by the following ex-

pression:

|µ

⊕

i =

1 −µ

⊕

( ˜x, ˜y)|0i+

⊕

( ˜x, ˜y)|1i.

In addition, by applying the measure operator

, that is, on the 10

qubit and related to |1i, the

result quantum state has probability

⊕

(1) = x

+ x

+ y

−x

+ x

corresponding to the non-membership degree of

⊕

( ˜x, ˜y). Then, it means that

⊕

(1) = ν

⊕

( ˜x, ˜y).

FCTA 2019 - 11th International Conference on Fuzzy Computation Theory and Applications

292

Table 1: Evolution of superposition quantum registers in modelling quantum circuit: ⊕

( ˜x, ˜y).

non-void amplitudes T 0 T 1 T 2 T 5 T 8 T 11 T 12

(1 −x

)(1 −x

)(1 −y

) 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000

(1 −x

)(1 −x

)(1 −y

0001000000 0001000000 0001000000 0001000000 0001001000 0001001000 0001001000

(1 −x

)(1 −x

(1 −y

) 0010000000 0010000000 0010000000 0010000000 0010000000 0010000100 0010000100

(1 −x

)(1 −x

0011000000 0011000000 0011000000 0011000000 0011001000 0011001100 0011001101

(1 −x

(1 −y

)(1 −y

) 0100000000 0100000000 0100000000 0100000000 0100000000 0100000100 0100000100

(1 −x

(1 −y

0101000000 0101000000 0101000000 0101000000 0101001000 0101001100 0101001101

(1 −x

(1 −y

) 0110000000 0110100000 0110100000 0110100010 0110100010 0110100110 0110100110

(1 −x

0111000000 0111100000 0111100000 0111100010 0111101010 0111101110 0111101111

(1 −x

)(1 −y

) 1000000000 1000000000 1000000000 1000000000 1000001000 1000001000 1000001000

(1 −x

)(1 −y

1001000000 1001000000 1001010000 1001010010 1001011010 1001011010 1001011010

(1 −x

(1 −y

) 1010000000 1010000000 1010000000 1010000000 1010001000 1010001100 1010001101

(1 −x

1011000000 1011000000 1011010000 1011010010 1011011010 1011011110 1011011111

(1 −y

)(1 −y

) 1100000000 1100000000 1100000000 1100000000 1100001000 1100001100 1100001101

(1 −y

1101000000 1101000000 1101010000 1101010010 1101011010 1101011110 1101011111

(1 −y

) 1110000000 1110100000 1110100000 1110100010 1110101010 1110101110 1110101111

1111000000 1111100000 1111110000 1111110010 1111111010 1111111110 1111111111

In analogous way, the resulting measure operator

has the probability

⊕

(0) = 1 −x

−x

−y

+ x

−x

corresponding to the complement of the non-

membership degree of ⊕

( ˜x, ˜y), that is,

⊕

(0) = 1 −ν

⊕

( ˜x, ˜y).

Thus, the 10

qubit is given as follows:

|ν

⊕

i =

1 −ν

⊕

( ˜x, ˜y)|0i+

⊕

( ˜x, ˜y)|1i.

Concluding, the interpretation of intuitionistic

fuzzy values related to the ⊕

Xor connective is pro-

vided by the pair (|µ

⊕

i, |ν

⊕

i) of quantum registers.

Remark 4.1. As a relevance, the entanglement of

such qubits makes the fuzzy quantum circuits differ-

ent from circuits modeling other logical approaches.

For instance, by taking the results from ⊕

Xor

operator one can easily observe that when the mea-

surement applied to the 10

qubit is related to |1i,

then it returns 7

and 8

qubits also related to |1i,

meaning that these three qubits collapse to value 1.

This is a phenomenon that does not occur in stan-

dard fuzzy set theory: the result of measurement (ob-

servation) affects the state of other arguments.

So, these values can be used in next calculations

performed by other functions but involving such qubit

systems, independently of the reuse of circuit inputs,

which are also restored from 1

to 4

qubits.

4.2 Modelling ⊗

Quantum Operator

Fig. 2 shows the ⊗

quantum circuit related to com-

position of Eq.(20) (from T1 to T7) and Eq.(21) (from

T8 to T12) without the measure operations.

Analogously, based on the entanglement of such

qubits, one can easily observe that when the measure-

ment 10

qubit is related to |0i simultaneously, the

and 6

qubits are also related to |0i, meaning that

these three qubits collapse to value 0.

Moreover, see columns in Table 2 presenting the

evolution of the non-void amplitudes for the most rel-

evant points of this circuit, with T0 being the initial

quantum state and T12 is the ﬁnal quantum state, ie,

the quantum state resulting after the application of all

the quantum operators related to the ⊗

( ˜x, ˜y).

After executing this circuit, applying the measure

operator M

, that is, on the 9

qubit and related to

|1i, it has the following distribution of probability:

⊗

(1) = x

+ x

+ y

−x

−

−x

+ x

corresponding to the membership degree of ⊗

( ˜x, ˜y),

then we obtain that

⊗

(1) = µ

⊗

( ˜x, ˜y)

. (22)

Analogously, the measure M

has the following

probability

⊗

(0) = 1 −x

−x

−y

+ x

−x

which means that the complement of the membership

degree of ⊗

Xor operator is given as follows

⊗

(0) = 1 −µ

⊗

( ˜x, ˜y). (23)

Thus, By Eqs. (22) and (23) we have that the 9

qubit can be given as follows:

|µ

⊗

i =

1 −µ

⊗

( ˜x, ˜y)|0i+

⊗

( ˜x, ˜y)|1i.

Interpreting Xor Intuitionistic Fuzzy Connectives from Quantum Fuzzy Computing

293

Table 2: Evolution of superposition quantum registers in modelling quantum circuit: ⊗

( ˜x, ˜y).

non-void amplitudes T 0 T 3 T 6 T 7 T 8 T 11 T 12

(1−x

)(1−x

)(1−y

) 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000

(1 −x

)(1 −x

)(1 −y

) 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000

(1 −x

)(1 −x

)(1 −y

0001000000 0001000000 0001010000 0001010000 0001010000 0001010000 0001010000

(1 −x

)(1 −x

(1 −y

) 0010000000 0010100000 0010100000 0010100000 0010100000 0010100000 0010100000

(1 −x

)(1 −x

0011000000 0011100000 0011110000 0011110010 0011110010 0011110010 0011110010

(1 −x

(1 −y

)(1 −y

) 0100000000 0100000000 0100010000 0100010000 0100010000 0100010000 0100010000

(1 −x

(1 −y

0101000000 0101000000 0101010000 0101010000 0101011000 0101011000 0101011001

(1 −x

(1 −y

)

0110000000 0110100000 0110110000 0110110010 0110110010 0110110010 0110110010

(1 −x

0111000000 0111100000 0111110000 0111110010 0111111010 0111111010 0111111011

(1 −x

)(1 −y

) 1000000000 1000100000 1000100000 1000100000 1000100000 1000100000 1000100000

(1 −x

)(1 −y

1001000000 1001100000 1001110000 1001110010 1001110010 1001110010 1001110010

(1 −x

(1 −y

) 1010000000 1010100000 1010100000 1010100000 1010100000 1010100100 1010100101

(1 −x

1011000000 1011100000 1011110000 1011110010 1011110010 1011110110 1011110111

(1 −y

)(1 −y

) 1100000000 1100100000 1100110000 1100110010 1100110010 1100110010 1100110010

(1 −y

1101000000 1101100000 1101110000 1101110010 1101111010 1101111010 1101111011

(1 −y

) 1110000000 1110100000 1110110000 1110110010 1110110010 1110110110 1110110111

1111000000 1111100000 1111110000 1111110010 1111111010 1111111110 1111111111

Moreover, applying the measure operator M

that is, measurement on the 10

qubit and related to

|1i, the resulting probability is given as follows:

⊗

(1) = x

+ x

−x

which corresponds to the non-membership degree of

⊗

( ˜x, ˜y), therefore, we obtain that

⊗

(1) = ν

⊗

( ˜x, ˜y). (24)

Analogous, the measure M

has the probability

⊗

(0) = 1 −x

−x

+ x

corresponding to the complement of the non-

membership degree of ⊗

( ˜x, ˜y), that is,

⊗

(0) = 1 −ν

⊗

( ˜x, ˜y). (25)

And ﬁnally, by Eqs. (24) and (25) we have that the

qubit is given by:

|ν

⊗

i =

1 −ν

⊗

( ˜x, ˜y)|0i+

⊗

( ˜x, ˜y)|1i.

Concluding, the interpretation of intuitionistic

fuzzy values related to the ⊗

Xor connective is pro-

vided by the pair (|µ

⊗

i, |ν

⊗

i) of quantum registers.

5 CONCLUSIONS

This paper describes the interpretation of two classes

of Xor operations on A-IFS through concepts of QC.

It was modelled using a quantum register using oper-

ations over fuzzy sets described by QTs.

Therefore, the presented approach to interpreta-

tion of intuitionistic fuzzy valued from quantum reg-

isters and quantum states shows another basic con-

struction in the speciﬁcation of fuzzy expert systems

from QC, in order to obtain new information tech-

nologies based on intuitionistic fuzzy approach.

Computer systems based on A-IFL and performed

over quantum computers may be able to generate

an output considering the manipulation of inaccurate

data and also dealing with imprecision in the model of

rule-based system, by taking advantage of properties

as quantum parallelism.

Further work aims at consolidation of this speciﬁ-

cation including not only other fuzzy connectives but

also constructors (e.i. automorphisms and reductions)

and the corresponding extension of (de)fuzzyﬁcation

methodology from formal structures provided by QC.

ACKNOWLEDGEMENTS

This work is partially supported by the Brazil-

ian grants: 309533/2013-9 (CNPq), 448766/2014-

0 (MCTI/CNPQ), PROCAD/CAPES/Brasil Finance

Code 001, and PqG-FAPERGS Edital 02/2017

(17/2551-0001207-0).

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