Toward a Quantum Fuzzy Approach for

Emotion Modeling in Parent-Child Interactivity

Cec

ılia Botelho

1 a

, Larissa Schonhofen

1 b

, Helida Santos

2 c

, Giancarlo Lucca

3 d

Adenauer Correa Yamin

1 e

and Renata Hax Sander Reiser

1 f

Federal University of Pelotas, LUPS, Pelotas, RS, Brazil

Federal University of Rio Grande, C3, Rio Grande, RS, Brazil

Catholic University of Pelotas, PGEEC, Pelotas, RS, Brazil

Keywords:

Quantum Computing, Fuzzy Systems, Emotion Modeling, Parent-Child Interaction, Qiskit.

Abstract:

This study presents an integrated framework combining Quantum Fuzzy computing concepts with emotion

modeling and simulations of intelligent agents. It explores the distinctions between Quantum Fuzzy and

Classical Computing, focusing on parent-child relationships. Simulations performed on the Qiskit platform

highlight signiﬁcant differences in the results produced by these two approaches. The research emphasizes

how membership degrees(MD) are represented in the quantum circuit model by interpreting fuzzy operations

through unitary quantum transformations. Established fuzzy connectives, such as the exclusive OR, serve as

an algebraic basis for constructing quantum operators and circuit representations. The algorithms demonstrate

substantial potential for extension, allowing for modeling interactions among multiple agents using multi-

dimensional quantum registers. Simulations within Qiskit offer a solid foundation for implementing these

algorithms on real quantum platforms, paving the way for further exploration in this interdisciplinary ﬁeld.

1 INTRODUCTION

Quantum Computing (QC) introduces a revolution for

solving classical complex problems, offering an expo-

nentially superior processing capability compared to

classical computers. Using qubits, the fundamental

units of information in quantum systems, this tech-

nology enables the simultaneous execution of multi-

ple entangled and parallel operations. Based on the

principles of Quantum Mechanics, such as entangle-

ment and superposition, quantum algorithms promise

to solve computational challenges more efﬁciently,

transforming sectors such as ﬁnance, logistics, and ar-

tiﬁcial intelligence.

Fuzzy Logic (FL) handles concepts beyond binary

values. By dealing with incomplete and imprecise

information, FL enables the manipulation of multi-

https://orcid.org/0000-0002-2167-7139

https://orcid.org/0009-0003-8243-686X

https://orcid.org/0000-0003-2994-2862

https://orcid.org/0000-0002-3776-0260

https://orcid.org/0000-0002-7333-244X

https://orcid.org/0000-0001-9934-3115

valued fuzzy sets, reﬂecting the complexity of natu-

ral language. Flexible algorithms provide a gradual

representation of knowledge, facilitating ﬂexibility in

decision-making processes, especially in expert sys-

tems where ambiguity and uncertainty are frequent.

These characteristics make fuzzy logic particularly

effective for modeling human emotional interactions,

as it captures the inherent ambiguity and ﬂuidity of

emotions. Unlike binary models, which overly re-

duce emotions into ﬁxed states, fuzzy logic represents

them as a continuum, allowing gradual transitions and

intertwined states such as apprehension, fear, or ter-

ror. This allows for a more reﬁned approach to dy-

namic and context-dependent relationships. Conse-

quently, systems based on fuzzy rules promote the in-

terpretability of computational outcomes.

The quantum technology market is expanding,

with a projected compound annual growth rate of

25% between 2024 and 2034 (IDTechEx Research,

2023). This growth is driven by advancements in

three main areas: computing, sensors, and quantum

communications. The development of hardware for

quantum computing is expanding in research centers

and data centers, while quantum sensors are ﬁnding

Botelho, C., Schonhofen, L., Santos, H., Lucca, G., Yamin, A. C. and Reiser, R. H. S.

Toward a Quantum Fuzzy Approach for Emotion Modeling in Parent-Child Interactivity.

DOI: 10.5220/0013323700003890

Paper published under CC license (CC BY-NC-ND 4.0)

In Proceedings of the 17th International Conference on Agents and Artiﬁcial Intelligence (ICAART 2025) - Volume 3, pages 1297-1303

ISBN: 978-989-758-737-5; ISSN: 2184-433X

1297

applications in sectors such as precision navigation

and medicine. In this context, the interrelationship

between quantum technologies, such as lasers, scan-

ners, and sensors, and quantum-fuzzy applications be-

comes evident.

Following recent scientiﬁc literature, we can ex-

plore the interaction between QC and FL. One ex-

ample is the fuzzy connectives modeling via the ex-

tension of quantum operators (de Avila et al., 2019).

Quantum implementations of fuzzy connectives us-

ing multi-qubit gates have also been investigated (Yo-

geesh et al., 2023), along with studies discussing the

use of quantum movements in representing emotions

in humanoid robots modeling (Deng et al., 2021). Ad-

ditionally, the concept of entropy has been applied

in the analysis of inference systems based on the

quantum-neuro-fuzzy perspective, demonstrating the

relevance of this approach for complex data analysis

related to emotion modeling (Ferreira and Almeida,

2020).

This work applies a Quantum Fuzzy strategy to

interpret family interactions, focusing on modeling

emotions through membership degrees represented by

the U quantum gate. To implement this approach, we

developed quantum circuits that utilize the principles

of superposition and entanglement in a quantum sim-

ulation environment. The simulations formally repre-

sent the membership values, reﬂecting the intensity of

the behaviors between the parent-child agents.

The paper is organized as follows: Section II ex-

plores the fundamental concepts of FL and how it re-

lates to quantum computing. In Section III, we dis-

cuss the modeling and simulation of emotions in fam-

ily dynamics using fuzzy logic applied to quantum

circuits. Section IV describes a case study that models

emotions in a family context. Lastly, we present the

ﬁnal considerations and proposals for future research

in Section V.

2 PRELIMINARIES

2.1 Basic Concepts of Fuzzy Logic

FL is a mathematical extension of traditional Boolean

logic, providing a logical foundation for dealing with

imprecise or uncertain data. Fuzzy set theory gen-

eralizes the classical one, by smooth transitions be-

tween associated classes (Zadeh, 1965). Further-

more, their multivalued generalizations, later formal-

ized in (Zadeh, 1975), enabled new applications in

many ﬁelds.

Let U ̸=

0 be the universal set. Classical set the-

ory is based on the characteristic function f

: U →

{0,1}, where f

(x) = 1 if x ∈ A, and f

(x) = 0 if

x /∈ A, with U being the universal set. This function

associates each element x ∈U ̸=

0 with a value in the

discrete set {0,1}.

A fuzzy set A in U is characterized by the mem-

bership function f

: U →[0,1] where, for each x ∈U,

(x) indicates the MD of each element x in the fuzzy

set A.

A fuzzy set A in U can also be described as a set

of ordered pairs, where each element x ∈ U is asso-

ciated with its respective MD f

(x) ∈ [0,1], that is,

A = {(x, f

(x)) | x ∈ U}. Extending this context, a

multivalued fuzzy set can be deﬁned by n-tuples in

the multivalued logic approach.

Let A and B be fuzzy sets in U ̸=

0, represented

by the membership functions f

, f

: U → [0,1], re-

spectively. Taking f

∪

, f

∩

: U → [0, 1], the union and

intersection between A and B are, respectively, given

as:

A∪B={(x, f

∪

(x)) |x ∈U}, f

∪

(x) =max{f

(x), f

(x)};

A∩B={(x, f

∩

(x)) |x ∈U}, f

∩

(x) =min{f

(x), f

(x)}.

The operators max,min : [0,1]

→ [0,1] represent

triangular norms and conorms and can be replaced by

other functions of the corresponding classes, as seen

in (Klement and Navara, 1999).

Moreover, according to (Bustince et al., 2003), let

′

: U → [0,1]. The fuzzy set A

′

expresses the fuzzy

complement of A in U considering the standard nega-

tion N

: [0,1] → [0,1] given by N

(x) = 1 −x, and is

deﬁned by:

′

= {(x, f

′

(x)) |x ∈U }, and f

′

(x) = 1− f

(x).

A function E : [0,1]

→ [0, 1] is called exclusive

OR (or XOR) if, for all x,y ∈ [0, 1], it satisﬁes the

properties:

E1: E(0, 0) = E(1,1) = 0 and E(1,0) = E(0,1) = 1

(boundary conditions);

E2: E(x, y) = E(y,x) (symmetry);

E3: x ≤y ⇒E(0,x) ≤ E(0,y) (0-partial isotonicity);

E4: x ≤y ⇒E(1,x) ≥ E(1,y) (1-partial antitonicity).

Example 1. Let E

: [0,1]

→ [0,1] be the Xor class,

(x,y) = x + y −2xy, (1)

extending the binary classical operation expressed as

A ⊗B = (A ∪B) −(A ∩B).

An aggregation function A : [0,1]

→ [0,1] veri-

ﬁes, for all x,y,x

′

∈ [0,1], the following properties:

A1 A(0, 0) = 0 and A(1,1) = 1;

A2 x ≤ x

′

and y ≤ y

′

⇒ A(x, y) < A(x

′

) (strict iso-

tonicity).

ICAART 2025 - 17th International Conference on Agents and Artiﬁcial Intelligence

1298

Additional properties can also be demanded:

A3 A(x, y) = A(y,x) (symmetry);

A4 A(x, x) = x (idempotence);

A5 A(0, 1) =

;

A6 A(λx, λy) = λA(x,y), for all λ ∈[0, 1] (homogene-

ity);

A7 A(λ + x,λ + y) = λ + A(x, y) for all λ ∈[0, 1] (lin-

earity).

Example 2. The arithmetic mean, given by:

A(x,y) =

(x + y), (2)

veriﬁes seven properties, from A1 to A7.

2.2 Basic Concepts of Quantum

Computing

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

deﬁned by a two-dimensional unit state vector ψ =

(α,β)

, usually described in Dirac’s notation (Nielsen

and Chuang, 2000) by the expression: |ψ⟩ = α|0⟩+

β|1⟩, where the coefﬁcients α and β are complex

numbers corresponding to the amplitudes of their re-

spective states, satisfying the normalization condition

|α|

+ |β|

= 1. So, it ensures the system’s state vec-

tor, represented by (α, β)

, is unitary. The amplitudes

conﬁgure a state of quantum superposition, giving

rise to the phenomenon of quantum parallelism.

The state space of a multidimensional quantum

system is obtained by the tensor product of the state

spaces of its component systems. Considering a quan-

tum system of two qubits, |ψ⟩ = α

|0⟩+ β

|1⟩ and

|ϕ⟩ = α

|0⟩+ β

|1⟩, the related state space is com-

posed by the tensor product:

|ψ⟩ ⊗ |ϕ⟩ = α

|00⟩ + α

|01⟩ + β

|10⟩ +

|11⟩.

A state change in a quantum system is performed

via a unitary quantum transformation (QT), repre-

sented by orthogonal square matrices of order 2

where N is the number of qubits in the transformation.

Taking θ ∈ [0,

], λ,φ ∈ [0, 2π], an one-dimensional

QT is represented by:

F =



cos

−e

iλ

sin

iφ

sin

i(φ+λ)

cos



. (3)

In particular, when θ =

, λ = π, and φ = 0, then

F = H, known as the Hadamard gate. The application

of the unitary gate H ⊗H on a classical state |01⟩ gen-

erates a superposition state mathematically described

by:

H⊗H|01⟩=

√



1 1

1 −1



⊗

√



1 1

1 −1



|01⟩=







−1







And when θ = λ = φ = 0, then F = Id represents

the Identity. Furthermore, when θ = π, λ = π, and

φ = 0, then F = X represents the Not gate, referred to

as the Pauli X gate. Additionally, let j =

√

−1 be the

imaginary unit. The QT associated with the quantum

gate V qubit (

√

X) is given by the matrix expression:

V =



1 + j −1 + j

−1 + j 1 + j



The evident exponential growth in the spatial and

temporal complexity of quantum algorithms justiﬁes

the use of simulators to assist in the interpretation and

perform computed algorithms.

The amplitudes of multidimensional quantum

states are governed by the normalization condition,

which is not always achieved through the tensor prod-

uct of the corresponding states of the qubits (the basic

states of the computational basis). In this case, we

have an entangled state (Nielsen and Chuang, 2000).

For a characterization of two-dimensional entan-

gled states, we consider the classical states |00⟩, |01⟩,

|10⟩, and |11⟩ as basic vectors of a two-dimensional

quantum state. The entangled states are the lin-

ear combinations |s

′

⟩ = α

|00⟩+ β

|11⟩ and |s

′′

⟩ =

|01⟩+ β

|10⟩, with α

,β

,α

,β

being normalized

complex amplitudes and α

+ β

= α

+ β

= 1.

Example 3. The composition of one entangled state

with another generates a new entanglement. See, e.g.,

the three-dimensional quantum state |s

⟩, given as:

⟩ = |s

′

⟩⊗

√

(|0⟩+ |1⟩)

So, the entangled qubits are intertwined in such

a way that their individual properties cannot be de-

scribed independently. When an entangled qubit is

subjected to a measurement and its state is deter-

mined, the state of the other entangled qubit is in-

stantly affected, regardless of the distance between

them, known as the “spooky action at a distance”.

The measurement operation on the current state of

a quantum system is deﬁned by a set of linear projec-

tions M

, acting on the quantum states (Nielsen and

Chuang, 2000). Let the state be given by |ψ⟩. After

the measurement, the output probability is given by:

p(|ψ⟩) =

|ψ⟩

⟨ψ|M

†

|ψ⟩

The measurement operations satisfy the complete-

ness relation given as:

∑

†

= I. In one-

dimensional systems, we have:



1 0

0 0



= M

†

; and M



0 0

0 1



= M

†

Toward a Quantum Fuzzy Approach for Emotion Modeling in Parent-Child Interactivity

1299

For a qubit |ψ⟩, with α,β ̸= 0, we observe proba-

bilities to measure |0⟩ and |1⟩, resulting in:

• p(|0⟩) = ⟨φ|M

†

|φ⟩ = ⟨φ|M

|φ⟩ = |α|

• p(|1⟩) = ⟨φ|M

†

|φ⟩ = ⟨φ|M

|φ⟩ = |β|

Therefore, after measuring the |ψ⟩ state, we have

|α|

as the probability of being in the classical state

|0⟩; and |β|

as the probability of being in the other

state, |1⟩.

The Parent-Child algorithm is based on the QC

Model, considering sequential composition. In this

quantum context, the signiﬁcance of superposition

and entanglement in the representation and the par-

ents’ dynamics of emotions will be discussed. Ad-

ditionally, we will address the implementation of its

quantum circuits using the Qiskit framework, which

is integrated with the Python programming language,

simulating and executing quantum circuits.

Next, we will describe the case study.

3 EMOTION MODELING VIA

QUANTUM FUZZY APPROACH

This section discusses the theoretical foundation from

previous sections for simulating fuzzy systems that

represent the modeling of emotions of effective agents

using quantum computing. The Parent-Child case

study exempliﬁes this methodology. The principles

of quantum computing, such as superposition and en-

tanglement, are considered to model and analyze such

complex emotions, considering interpretations of un-

certainty through the application of fuzzy logic.

Let U be a universe with cardinality ∥U∥ = n de-

ﬁned in the set of the ﬁrst natural numbers in N ,

N = {1, 2,. ..,n}. For each element x

, we can as-

sociate its MD f

) and non-membership degree

1 − f

) to a one-dimensional quantum register ob-

tained from the following superposition state:

)

⟩ = |X

⟩ = [

)|1⟩+

1 − f

)|0⟩].

(4)

Thus, applying the tensor product, a n-

dimensional quantum state (n-qubits) represents

all the elements x

. Let U ̸=

0,|U | = n and let A

be a fuzzy set deﬁned by the membership function

: U → [0,1]. For each x

∈ U, a n-dimensional

fuzzy state is given by the expression:

)

⟩ =

1≤i≤n

[

)|1⟩+

1 − f

)|0⟩].

Highlighting the differences in relation to non-

classical correlations, we can explore the emotional

correlations, applying quantum states and operators.

This research explores the emotion intensities,

expanding on the ideas seen in (Raghuvanshi and

Perkowski, 2010). The angle of the qubit is illus-

trated as meridians in the Bloch sphere (Nielsen and

Chuang, 2000) and the intensity of the emotion is

shown as a point between the north and south poles,

|0⟩and |1⟩. We consider fuzzy interpretations to illus-

trate different types and intensities of emotions, such

as the joy variable, which varies from serenity (low)

to ecstasy (high).

Example 4. In Eq. (3), if λ = π, φ = 0 and

θ = 2arc tan

)

1 − f

)

we have the QT F is given as given by



1 − f

)

1 − f

)



, (5)

verifying the following properties:

• F

†

= F

−1

= F

;

• F

|0⟩ =

1 − f

)|0⟩+

)|1⟩;

• F

|1⟩ =

)|0⟩−

1 − f

)|1⟩;

• f

) = 1 ⇒ F

= X;

• f

) =

⇒ F

= H.

The Parent-Child (PC) algorithm is graphically

represented by a quantum circuit considering the se-

quential composition of unitary transformations per-

forming superpositions and entanglements to repre-

sent the dynamics of emotional modeling. Addition-

ally, we analyze the implementation of its quantum

circuits using the Qiskit platform, exploring its fea-

tures in Python library for designing, simulating, and

executing quantum circuits.

4 CASE STUDY:

PC-INTERACTIVITY

The PC-problem models the mood change of a child

based on the level of interactivity of their parents. If

both caregivers are interactive, the child will also be

happy. In other cases, the child will be in a “half

happy” and “half unhappy” state, interpreted as a su-

perposition state

√

(|0⟩+ |1⟩).

The Parent-Child algorithm in (Raghuvanshi and

Perkowski, 2010) is based on the QC model, where

the emotion intensity is modeled as projections on the

Bloch Sphere (Nielsen and Chuang, 2000), a stereo-

graphic representation of qubits, given by a point be-

tween the north and south poles, representing |0⟩ and

ICAART 2025 - 17th International Conference on Agents and Artiﬁcial Intelligence

1300

|1⟩, respectively. Thus, the type of emotion is mod-

eled by the phase angle of the related qubit, geomet-

rically represented by meridians on the Bloch Sphere.

The most positive emotional activity is at |1⟩, and at

|0⟩, the least positive.

In this work, a general interpretation of the Parent-

Child algorithm based on fuzzy aggregations extends

this interpretation to multiple agents, modeling more

complex interactions within the family structure and

considering, e.g., a stepfather and a stepmother.

Figure 1: Fuzzy Modeling Circuit of the Parent-Child Inter-

activity.

Fig. 1 describes the C1 circuit and presents the

fuzzy approach based on f

and f

membership

functions, respectively modeled by F

and F

quan-

tum gates, given as matrices obtained from Eq. (5)

that consider controlled gates based on the “Square

Root of the Not” gate V =

√

The sequential composition via controlled opera-

tors includes the following description:

• CV

, which executes the operator

√

X (V) on the

qubit (target) when the 2

qubit is in state |1⟩;

and

• CV

, which executes the operator

√

X (V) on the

qubit when the 2

qubit is in state |1⟩.

Now, the emotional modeling based on a fuzzy ag-

gregation function is formalized in the next proposi-

tion.

Proposition 1. The fuzzy arithmetic mean provides a

behavioral interpretation for the C1 circuit in Fig. (1).

Proof. Consider the fuzzy arithmetic mean opera-

tor, as given by Eq. (2). When f

) = x

and

) = x

, we have F

|0⟩ =

√

1 −x

|0⟩+

√

|1⟩

and F

|0⟩ =

√

1 −x

|0⟩+

√

|1⟩. Additionally, tak-

ing x

+ x

̸= 0 and S

= (F

⊗ F

⊗ Id)(|X

⟩ ⊗

⟩⊗|0⟩) as given by Eq. (4), the resulting entan-

gled state S

, obtained from the temporal evolution, is

summarized as follows:

=(M

⊗C

)(S

)

j−1

2(x

)

(1−x

|011⟩+

j−1

2(x

)

(1−x

|101⟩+

−

√

2(x

)

|111⟩ and p

+ x

(6)

Thus, based on the algebraic expressions extracted

from the qFuzzyAnalyser Library (Buss et al., 2024)

and, related to the above measurement, the results

promote the quantum-fuzzy interpretation via the

arithmetic mean, as given by Eq. (2). Thus, inde-

pendently from the membership functions attributions

and F

, it is interpreted by the quantum gate

given by the M

⊗C

) composition.

Therefore, the probability of the child’s behavior

changing from |0⟩ to |1⟩ is given by the arithmetic

mean performed by the intensity of the behaviors be-

tween the parents-child agents.

4.1 Classical PC-Interactivity

Table 1 presents an analysis based on classical inputs,

ﬁxing the 3

qubit as |0⟩. In addition, S

= S

since

= F

= Id. Note that the ﬁrst two qubits, repre-

senting the parents’ mood, do not change during the

evolution from S

to S

. Furthermore, measuring the

qubit (related to the child’s mood) the following

holds:

• In the ﬁrst row, MD f

) = f

) = 0 return the

same initial classical states, with probability p = 0

for the 3

qubit in |1⟩. In these cases, the circuit

interpretation guarantees that, like the parents, the

child’s mood remains in |0⟩;

• In the last row, interpreting the parents’ happy at-

titude as f

)= f

)=1 results in a change in

the child’s emotional behavior, from unhappy to

happy;

• In the other rows, modeling only one of the par-

ents as happy, the measure of the 3

qubit always

returns a state in superposition, with probability

p = 0.5 to evaluate whether the child maintains

the same mood or experiences a mood change.

Table 1: Temporal Evolution for the Classical Parent-

Child Interactivity.

|000⟩ |000⟩ |000⟩ p=0, S

=|001⟩

|010⟩

j+1

|010⟩+

j−1

|011⟩

j+1

|010⟩+

j−1

|011⟩ p=

, S

=|011⟩

|100⟩ |100⟩

j−1

|100⟩+

j+1

|101⟩ p=

, S

=|101⟩

|110⟩ |111⟩ |110⟩ p=1, S

=|111⟩

So, in the emotion modeling of parent-child in-

teraction as described in Fig. (1), the parent’s happi-

ness inﬂuences the child to become (or remain) happy.

However, this inﬂuence on mood change is not ev-

ident when both parents are unhappy. Whenever at

least one of them is happy, there is a 50% chance of

changing the child’s mood, either to a happy mood

(proactive attitude) or to an unhappy one (passive at-

titude).

Toward a Quantum Fuzzy Approach for Emotion Modeling in Parent-Child Interactivity

1301

Table 2: Temporal Evolution of the Fuzzy Model of the Parent-Child Interactivity.

|000⟩

|000⟩+

√

|010⟩)

(|000⟩+

√

( j+1)|010⟩+

√

( j−1)|011⟩)

(|000⟩+

√

( j−1)|010⟩+

√

( j+1)|011⟩) p=

, S

=|011⟩

|010⟩

√

|000⟩+

|001⟩

√

(|000⟩−

j−1

|010⟩+

j−1

|011⟩)

√

(|000⟩−

j−1

|010⟩+

j−1

|011⟩) p=

, S

=|011⟩

|100⟩

|100⟩+

√

|110⟩)

|100⟩+

√

( j+1)|110⟩+

√

( j−1)|111⟩

j+1

|100⟩+

j−1

|101⟩+

√

|111⟩ p=

, S

j−1

|101 +

√

|111⟩

|110⟩

√

|100⟩+

|110⟩

√

|100⟩+

j+1

|110⟩+

j−1

|111⟩)

√

( j+1)|100⟩+

√

( j−1)|101⟩+

|111⟩) p=

, S

√

(

√

3( j−1)

|101⟩+

√

|111⟩)

4.2 Fuzzy PC-Interactivity: Algebraic

Discussion

In this case study, we consider quantum-fuzzy inputs

for one of the parents interpreted by the 2

qubit. The

other agents, related to the qubits 1 and 3, remain in

the classical state |0⟩. Therefore, we consider F

Id and



√

3 −1



(θ = 2

rad = 120

◦

Table 2 summarizes the results of the Parent-Child

interactivity. In this case study, the resulting prob-

ability of a single measurement on the 3

qubit in

|1⟩, according to the algebraic expression presented

in Eq. (6), results in the following interpretations:

• In the 1

row, when the input variable received

MD f

) = 0 and f

) = 0.75, the output vari-

able received a membership degree f

(y) =

0.375;

• In the 2

row, when the input variable received

MD f

) = 0 and f

) = 0.25, the output

variable received the lowest membership degree:

(y) = 0.125;

• In the 3

row, MD f

) = 1, and f

) = 0.75

returning the highest MD f

(y) = 0.875;

• And ﬁnally, in the last row, the input variable re-

lated to MD f

) = 1 and f

) = 0.25, implies

a MD f

(y) = 0.625 for the output variable.

The probability distributions seen earlier can also

be obtained through simulations on the Qiskit plat-

form, as shown in the histograms in Fig. 2. Observe

that the best option for changing the child’s mood is

related to the highest degree of the parents in the fuzzy

set, interpreting happiness as the linguistic variable.

4.3 Fuzzy PC-Interactivity: Qiskit

Simulations

The histograms in Fig. 2 illustrate the simulation via

quantum-fuzzy inputs related to the execution of the

quantum circuit. These histograms report results from

Table 2, analyzing the temporal evolution of the fam-

ily dynamics for the four initial states, and the 3

qubit as |0⟩.

The X-axis of the histogram depicts the possible

states of the three qubits after executing the circuit

and measurement. The Y-axis indicates the frequency

of each state, based on a total of 1000 circuit execu-

tions. The frequencies of the |110⟩ and |010⟩ states

are particularly important for our analysis, as they re-

ﬂect distinct scenarios within the family dynamic in

question.

The third qubit starts in the |0⟩ state in the pre-

sented scenarios. The ﬁrst row of Table 2 corresponds

to the ﬁrst histogram. After executing the circuit, the

child represented by the third qubit becomes happy

387 times and remains in the initial mood 613 times.

In the second histogram, the child becomes happy 123

times and remains sad 877 times. In the third his-

togram, the child changes the mood 873 times and

stays in the initial state 127 times. Finally, the child’s

mood changes 625 times while keeping it 375 times.

These simulations suggest that happy parents can in-

ﬂuence the child to become or remain satisﬁed.

Figure 2: Histogram: Fuzzy Approach to the Parent-Child

Interactivity.

5 FINAL CONSIDERATIONS

Based on algebraic expressions integrating fuzzy con-

nectives, including the fuzzy exclusive “or” and fuzzy

arithmetic means, we focused on the parent-child

dilemma as a simple case to investigate the applica-

tion of a generalized quantum gate in the quantum-

fuzzy context. This allows more precise in the model-

ing three agents’ emotions, in a 3-dimensional quan-

tum system.

ICAART 2025 - 17th International Conference on Agents and Artiﬁcial Intelligence

1302

The simulation conducted on the Qiskit platform

revealed distinct patterns in the probability distribu-

tions, providing perceptions of the emotional dynam-

ics of family interactions. Additionally, the imple-

mentation of fuzzy operators, considering the model

of Quantum Circuits, emphasizes the importance of

superposition and entanglement in the emotional rep-

resentation.

The quantum approach reﬁnes the modeling of

interactions between multiple agents. The analysis

of the histograms obtained during the simulation al-

lowed valuable insights into the applicability of quan-

tum computing to tackle complex real-world prob-

lems, especially in the modeling of emotions. Fu-

ture works may explore the fundamentals of quantum-

fuzzy theory in modeling emotions for human-like be-

havior in future intelligent robots. Moreover, the re-

sults can be extended for new research based on data

fusion for artiﬁcial intelligent systems (Tiwari et al.,

2024).

Expanding the dimensional model, the natural lan-

guage related to emotional and social contexts can

also essentially improve their practical applicability

on real systems. Besides, applying quantum neural

networks (QNN) and transferring the simulations per-

formed in Qiskit to real quantum hardware (as the

IBM quantum platform) is the next research step, val-

idate our Quantum Fuzzy models in real-world sce-

narios. Thus, when more agents are involved, like

restructured families (stepfather/stepmother and half

brothers/systems) and the Parent-Children Interactive

model will involve more than three agents, justify the

use of quantum simulators. So, further work based

on emerging technologies and combining the poten-

tials of QC and FL, can model emotions collaborates

in relevant scenarios as affective computing, social

robotics, and neurorobotics research areas.

ACKNOWLEDGEMENTS

The authors thank the funding agencies:

CAPES, CNPq (309160/2019-7; 311429/2020-

3, 150160/2023-2), PqG/FAPERGS (21/2551-

0002057-1), FAPERGS/CNPq (23/2551-

0000126-8), and PRONEX (16/2551-0000488-9).

\section*{ACKNOWLEDGEMENTS}

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