**1. Introduction**

Chaos theory studies nonlinear dynamic systems. Chaos is the interaction between regularity and probability-based unpredictability [1]. Weather and climate, biological and ecological processes, the economy, social structures, and other natural phenomena all exhibit chaotic regimes. The primary feature of chaos is its ability to generate a wide range of complex patterns. For use as cryptographic secret keys, relevant mathematical models may produce a vast amount of data. Confusion and diffusion are two key features of cryptography, and chaos theory has the unique quality of having a direct connection to both features. Furthermore, the deterministic but unexpected dynamics of chaotic systems may be a powerful tool in the development of a superior cryptosystem [2,3].

The fundamental benefit of chaos is that unauthorized users see chaotic signals as noise [2]. Chaotic-based encryption techniques are utilized for military, mobile, and private data [3]. These applications demand real-time, rapid, secure, and reliable monitoring. Most chaos-based secure communication systems use chaos synchronization [4]. Chaos synchronization is vital for achieving security after information has been transferred [5].

**Citation:** Zainel, Q.M.; Darwish, S.M.; Khorsheed, M.B. Employing Quantum Fruit Fly Optimization Algorithm for Solving Three-Dimensional Chaotic Equations. *Mathematics* **2022**, *10*, 4147. https://doi.org/10.3390/math10214147

Academic Editor: Tao Zhou

Received: 21 September 2022 Accepted: 13 October 2022 Published: 6 November 2022

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

Therefore, many cryptographic algorithms have adopted popular chaotic models that depict chaos by employing mathematical models, such as a logistic map.

Chaos-based secure communication has issues. Due to the limitations of chaos theory and techniques for creating chaos, attackers may sometimes determine the chaotic system employed in encryption through state reconstruction. Second, transmission and sampling delays make chaotic synchronization difficult. Due to the limits of digital computer accuracy, computer chaotic maps are always periodic. Therefore, chaos-based public-key cryptography has collisions [6]. Finally, picking the input parameters limits chaos theory. The techniques used to determine these characteristics rely on the data dynamics and the desired analysis, which is often complicated and inaccurate. Due to a chaotic system's complicated nature, many practical characteristics are unknown and difficult to quantify [7]. Parameter estimation is a major issue.

Two parameter estimation methods exist. One is the synchronization method [3,8], which proposes updating parameter estimation based on chaotic system stability. Its methodologies and sensitivities rely on the considered system; hence, updating may be challenging due to the complexity of the chaotic system. Another method is through metaheuristic algorithms. Metaheuristic algorithms are intelligent optimization algorithms [9,10]. It translates parameter estimation into a multidimensional optimization problem using sample data from the original system. It is easier to implement than synchronization. Metaheuristic algorithms are popular for estimating chaotic system parameters [11,12]. Metaheuristic techniques require starting system settings. In many circumstances, the original values cannot be retrieved, making reconstruction and management of the chaotic system difficult. Most of these approaches are also used to estimate chaotic system parameters. Few apply to complex chaotic systems [13].

The fruit fly optimization algorithm (FOA) is simple and easy to comprehend compared with other sophisticated algorithms. FOA only requires adjusting the population size and maximum generation number. Traditional intelligent algorithms need at least three parameters. The influence of numerous factors on algorithm performance is hard to examine; hence, they are generally determined via several tests. An incorrect parameter will impair algorithm performance and complexity [14]. However, there is still a lot of potential for development of FOA variations to obtain greater performance, particularly for complicated practical issues related to convergence speed or avoiding being trapped into the local optimum.

When it comes to population-based optimization methods, variability in the population and unpredictability in the search process are two factors that often play a pivotal role. By using quantum mechanics instead of Newtonian dynamics, the quantum-behaved particle swarm optimization (QPSO) increases the particles' capacity to escape the local optimum. Classical quantum mechanics is the theoretical underpinnings of quantum theory, which aims to appropriate some of the mysteriousness of quantum behavior processes. Integrating quantum theory into the original FA, the quantum firefly algorithm (QFA) is able to combat the loss of variety [15]. Quantum mechanics may be used to explain how fruit flies navigate the environment in search of food; their actions are characterized by a wave function of uncertainty. A quantum-behaved approach can avoid premature convergence and help escape from the local optimum.

#### *1.1. Problem Statement and Motivation*

Chaotic systems are very sensitive to initial parameter choices. Long-term system behavior prediction is difficult. Synchronization and chaos control in nonlinear systems depend on exact parameter values in chaotic systems; if one of these values is uncertain, the system will not perform as intended. Some parameters are unknown or difficult to quantify due to the complexity of chaotic systems (such as secure communication). If we wish to control or synchronize chaotic systems, we must estimate unknown system parameters. Too many factors may cause the parameter estimation algorithm for 3D chaotic systems to become more complex, which in turn increases the amount of effort required

to calculate the results. This is why most algorithms struggle to find the global optimum. As a result of its effectiveness, FA has been used to tackle a wide range of optimization issues, leading to significant progress in a short period of time. The motivation is to take insights from quantum theory to improve upon the FA for estimating the parameters of a 3D chaotic system.

#### *1.2. Contribution and Methodology*

The work presented in this paper is an extension of the work introduced in Ref. [16], where quantum mechanics was used in the fruit fly optimization algorithm to make it easier for particles to get out of the local optimum, so that the chaotic system parameters could be estimated. In this paper, the QFOA was adopted to solve the parameter estimation problem of the Lorenz chaotic system to achieve the synchronization with the aim of transmitting data correctly. Fitness function based on the mean square error was utilized to find the minimum error between the original and estimated ones in different directions. To achieve high performance in terms of time and accuracy, the suggested model selected only some samples from the received signal to check the synchronization early. QFOA variables were tuned to estimate the unknown chaotic system parameters. Then, these estimated parameters were used later, inside the well-known fourth-order Runge–Kutta algorithm, to build the estimated original signal (a chaotic signal with a known structure) to yield synchronization.

The rest of this paper is organized as follows: Section 2 provides a background and literature review of some studies related to estimating the parameters of the chaotic system; Section 3 presents the proposed methodology based on the analysis of the previous techniques; Section 4 reports a complete evaluation of the proposed methodology, along with the results and the discussion; and the final section contains the conclusion based on the previous sections and future directions for research.

#### **2. Background and Related Work**

This section offers some important background related to the proposed model and includes a literature review on parameters estimation of the chaotic system as one of the most important techniques to achieve chaotic synchronization concerns on wireless communication networks.

#### *2.1. Preliminaries*

#### 2.1.1. Chaos Theory

Chaos theory is an alternative description and explanation of the behavior of nonlinear dynamical systems [17]. In mathematical language, a dynamical system is classified as a chaotic system [18–21] if it has the following properties:

• Sensitive to initial conditions—each point in a system is arbitrarily near other points with drastically different behavior. Qualitatively, two paths with a starting separation *δX*<sup>0</sup> diverge.

$$|\delta X(t)| \approx e^{\lambda t} |\delta X\_0|\tag{1}$$

*λ* is the Lyapunov exponent. One positive Lyapunov exponent indicates chaotic behavior, whereas more than one indicates hyperchaotic behavior.


#### 2.1.2. Lyapunov Exponents

The Lyapunov exponents help investigate chaotic or hyperchaotic dynamical systems. Lyapunov exponents categorize dynamical systems so that we can see their behavior. A dynamical system is chaotic if it has one positive Lyapunov exponent, and hyperchaotic if it has more [22–24]. Consider two locations in space, *X*<sup>0</sup> and *X*<sup>0</sup> + Δ*X*0, which form orbits using an equation or set of equations. Sensitive dependency may only occur in particular parts of a system; hence, this separation depends on the beginning value, Δ*x*(*X*0,*t*). For chaotic points, Δ*x*(*X*0,*t*) acts unpredictably. The mean exponential rate of divergence of two near orbits is defined as [25].

$$\lambda = \lim\_{\substack{t \to \infty \\ |\Delta X\_0| \to 0}} \frac{1}{t} \ln \left| \frac{\Delta x(X\_0, t)}{\Delta X\_0} \right| \tag{2}$$

The Lyapunov exponent, *λ*, is used to differentiate orbits. If *λ* < 0, the orbit attracts a stable fixed point or periodic orbit. The more negative the exponent, the better the stability. If *λ* = 0, the system is steady state. A conservative system has this exponent and are Lyapunov stable. In this case, orbits would stay apart. For *λ* > 0, the orbit is chaotic. Nearby points diverge to any arbitrary separation.

To define sphere trajectories, we require linearized systems or variational equations. → *<sup>x</sup>* <sup>=</sup> <sup>→</sup> *F*( → *x* ), where <sup>→</sup> *<sup>x</sup>* <sup>=</sup> (*x*1, *<sup>x</sup>*2,......, *xn*) and <sup>→</sup> *F* = (*f*1, *f*2,......, *fn*). Any ordinary numerical differential equation solution may create ∅( → *x*0). Formally, partial derivatives explain how these perturbations respond. Consider the Lorenz system [26–28]:

$$\begin{cases}
\dot{\mathbf{x}} = \theta\_1(\mathbf{y} - \mathbf{x}) \\
\dot{\mathbf{y}} = \theta\_2 \mathbf{x} - \mathbf{y} - \mathbf{x}z \\
\dot{z} = -\theta\_3 z + \mathbf{x}y
\end{cases} \tag{3}$$

*θ*1, *θ*2, and *θ*<sup>3</sup> are Lorenz parameters. To set up the linearized system for the above equations, the right-hand Jacobian is needed.

$$J = \begin{bmatrix} \frac{\partial f\_1}{\partial x} & \frac{\partial f\_1}{\partial y} & \frac{\partial f\_1}{\partial z} \\ \frac{\partial f\_2}{\partial x} & \frac{\partial f\_2}{\partial y} & \frac{\partial f\_2}{\partial z} \\ \frac{\partial f\_3}{\partial x} & \frac{\partial f\_3}{\partial y} & \frac{\partial f\_3}{\partial z} \end{bmatrix} \tag{4}$$

$$J = \begin{bmatrix} -\theta\_1 & \theta\_1 & 0\\ \theta\_2 - Z & -1 & -\mathbf{x} \\ y & \mathbf{x} & -\theta\_3 \end{bmatrix} \tag{5}$$

$$J = \begin{bmatrix} \delta\_{x1} & \delta\_{y1} & \delta\_{z1} \\ \delta\_{x2} & \delta\_{y2} & \delta\_{z2} \\ \delta\_{x3} & \delta\_{y3} & \delta\_{z3} \end{bmatrix} \tag{6}$$

The *i*th equation's *x* variation component is *δxi*. Column sums are the *x*, *y*, and *z* coordinates of the evolving variant. The rows represent the vector coordinates of the original *x*, *y*, and *z* variations. Linear equations:

$$
\begin{bmatrix}
\dot{\delta}\_{x1} & \dot{\delta}\_{y1} & \dot{\delta}\_{z1} \\
\dot{\delta}\_{x2} & \dot{\delta}\_{y2} & \dot{\delta}\_{z2} \\
\dot{\delta}\_{x3} & \dot{\delta}\_{y3} & \dot{\delta}\_{z3}
\end{bmatrix} = \begin{bmatrix}
\frac{\partial f\_1}{\partial x} & \frac{\partial f\_1}{\partial y} & \frac{\partial f\_1}{\partial z} \\
\frac{\partial f\_2}{\partial x} & \frac{\partial f\_2}{\partial y} & \frac{\partial f\_2}{\partial z} \\
\frac{\partial f\_3}{\partial x} & \frac{\partial f\_3}{\partial y} & \frac{\partial f\_3}{\partial z}
\end{bmatrix} \begin{bmatrix}
\delta\_{x1} & \delta\_{y1} & \delta\_{z1} \\
\delta\_{x2} & \delta\_{y2} & \delta\_{z2} \\
\delta\_{x3} & \delta\_{y3} & \delta\_{z3}
\end{bmatrix} \tag{7}
$$

$$
\begin{bmatrix}
\dot{\delta}\_{x1} & \dot{\delta}\_{y1} & \dot{\delta}\_{z1} \\
\dot{\delta}\_{x2} & \dot{\delta}\_{y2} & \dot{\delta}\_{z2} \\
\dot{\delta}\_{x3} & \dot{\delta}\_{y3} & \dot{\delta}\_{z3}
\end{bmatrix} = \begin{bmatrix}
\theta\_2 - Z & -1 & -\chi \\
y & \chi & -\theta\_3
\end{bmatrix} \begin{bmatrix}
\delta\_{x1} & \delta\_{y1} & \delta\_{z1} \\
\delta\_{x2} & \delta\_{y2} & \delta\_{z2} \\
\delta\_{x3} & \delta\_{y3} & \delta\_{z3}
\end{bmatrix} \tag{8}
$$

#### 2.1.3. Chaos Synchronization

Chaos synchronization occurs when two (or more) chaotic systems (identical or nonidentical) adapt a characteristic of their motion to the same behavior, owed to force or coupling. This includes trajectories and phase locking. Complete, projective, and antiphase synchronization have been explored [29]. These three synchronization types are usually of interest for master-slave configurations, i.e., two connected systems. However, for a more general case of networks, the less regular synchronization regimes such as multi-clustering and synchronization of groups of nodes are of relevance. See [30,31] for more details.

Complete synchronization means having equivalent state variables over time. Generalized synchronization for master-slave systems implies a functional relation between connected chaotic oscillators, *x*2(*t*) = *F*[*x*1(*t*)].:

1. Complete Synchronization

Considering the following master and slave systems:

$$
\dot{\mathbf{x}} = \theta(\mathbf{x}),
\tag{9}
$$

$$
\dot{y} = \psi(y) + \mu(\mathbf{x}, y),
\tag{10}
$$

State vectors *<sup>x</sup>*, *<sup>y</sup>* <sup>∈</sup> <sup>R</sup> is the vector controller for *<sup>f</sup>*, *<sup>g</sup>*: <sup>R</sup><sup>n</sup> <sup>→</sup> <sup>R</sup>n. The system error dynamics are:

$$x(t) = y(x) - x(t),\tag{11}$$

The systems are said to be in complete synchronization if:

$$\lim\_{t \to \infty} \|e(t)\| = 0 \tag{12}$$

#### 2. Anti-Phase Synchronization

In this type, given the same master-slave systems, the error dynamics for the systems are defined as:

$$e(t) = y(x) + x(t) \tag{13}$$

The systems are said to be in anti-synchronization if Equation (12) is satisfied.

#### 3. Projective Synchronization

In this type, given the same master-slave systems, the error dynamics for the systems are defined as:

$$e(t) = y(x) - ax(t) \tag{14}$$

where α = 0 is the constant, called a scaling factor. The systems are said to be in projective synchronization if Equation (12) is satisfied. By setting appropriate values for α, synchronized systems may be scaled to desired levels and proportionally grow. Complete synchronization and anti-synchronization are specific examples of projective synchronization where α = 1 and α = −1. Greater mathematical complexity and chaos characterize the Lorenz map because of its higher dimension. As one-dimensional chaotic maps need fewer computing processes, they are better suited for applications that need to run with minimal latency. More basic chaotic maps, however, have serious security flaws. This shortcoming arises because of the restricted chaotic range, reduced chaotic complexity, and accelerated rate of degradation of dynamic behavior [32,33].

Several approaches for chaotic synchronization have been presented. Active nonlinear control and adaptive mode control have been widely employed for synchronization in recent literature [29]. Based on the Lyapunov stability theory, active nonlinear control has gained popularity in recent years. Adaptive control assumes that there is a controller with a fixed structure and complexity for each potential plant parameter value, which can achieve the required performance with suitable controller parameter values. All these strategies are not applicable if the parameters of the chaotic system are unknown. Chaos control and synchronization focus on estimating the unknown parameters of chaotic dynamical systems. Parameter identification may be transformed into a multi-dimensional optimization problem using an objective function [34–36].

#### 2.1.4. Chaotic Maps

Chaotic maps are differential equations that describe chaotic discrete dynamics [18]. Chaos can only be detected in deterministic, continuous systems with a three-dimensional phase space or more. Low-dimensional chaotic systems are resource-efficient. The logistic map is a typical low-dimensional system [37]. Chaos is degenerative in these systems. It is hard to give the output sequence a long period. High-dimensional chaotic systems are more nonlinear. However, they have the drawbacks of excessive resource consumption and low-speed performance. Therefore, a large-period, high-dimensional, digital chaotic system with high speed and minimal resources is needed. Chen, Rossler, and Henon are 3D chaotic systems utilized in wireless communication [38].

#### 1. Chen Chaotic System

Chen identified a classical chaotic attractor in a basic 3D system [38]:

$$\begin{cases} \dot{\mathbf{x}} = a(\mathbf{y} - \mathbf{x})\\ \dot{\mathbf{y}} = (c - a)\mathbf{x} - \mathbf{x}\mathbf{z} - c\mathbf{y} \\ \dot{z} = x\mathbf{y} + bz \end{cases} \tag{15}$$

*x*, *y*, and *z* are state variables, whereas *a*, *b*, and *c* are parameters. Chen chaotic-based encryption relies on secret keys. An invader cannot guess the wireless key. As Chen chaotic systems are sensitive to beginning circumstances and system characteristics, two near-initial conditions lead to diverse paths, as shown in Figure 1a.

**Figure 1.** 3D view of (**a**) Chen chaotic, (**b**) Rossler chaotic, and (**c**) Henon chaotic map.

#### 2. Rossler Chaotic System

Rossler is a basic chaotic dynamical system with one non-linear term with standard system equations [39]:

$$\begin{cases} \dot{x} = a(y - x) \\ \dot{y} = x + ay \\ \dot{z} = b + xz - cz \end{cases} \tag{16}$$

*x*, *y*, and *z* are state variables; a and b are fixed; and *c* is the control parameter. Rossler attractor parameters are *a* = 0.2, *b* = 0.2, and *c* = 5.7. Figure 1b shows the Rossler chaotic attractor. This system is the minimum for continuous chaos for at least three reasons: (1) Its phase space has minimal dimensions, (2) Nonlinearity is minimal because there is a single quadratic term, and (3) It generates a chaotic attractor with a single lobe, unlike the Lorenz attractor, which has two.

3. Henon Chaotic System

The Henon chaotic map is a chaotic discrete-time dynamical system. The map simplifies the Lorenz model's Poincare portion. The plane will either approach the Henon odd attractor or diverge to infinity.

$$\begin{cases} \dot{\mathbf{x}} = a - \left( y^2 + bz \right) \\ \dot{y} = x \\ \dot{z} = y \end{cases} \tag{17}$$

The Henon chaotic map parameters are *a* = 1.4 and *b* = 0.3. The conventional Henon map is chaotic (Figure 1c).

#### 2.1.5. Quantum Fruit Fly Optimization Algorithm

Optimizing means picking the best element (based on some criteria) from a group of options or finding the least or maximum output for an experiment [34]. Heuristic methods are intelligent search strategies that speed up the process of obtaining a satisfying or near-optimal solution in bio-inspired procedures. A heuristic approach is simpler than an analytical one. However, precision is lost. Metaheuristics are iterative processes that help identify near-optimal solutions. Metaheuristics combine heuristic approaches to improve their performance [10,40]. Recent metaheuristic algorithms include the FOA [41]. FOA is inspired by fruit fly foraging. FOA has fewer adjusting parameters, less computational quantity, and offers great global search and convergence abilities. FOA is two-phased. The first step is smelling. In this phase, flies travel toward food by smelling it. Second phase begins when they are closer to the food supply: the vision stage. The fruit flies utilize their eyesight to come closer to the food. This phase repeats until the fruit fly eats the food. The steps of FOA include [42,43]:


$$\begin{cases} X\_i = X\_{\text{axis}} + \text{Random Value } R\_1\\ Y\_i = Y\_{\text{axis}} + \text{Random Value } R\_2 \end{cases} \tag{18}$$

(3) As the food's location is unknown, the distance (*Dist*) to the origin is inferred before calculating the decision value of smell concentration (*S*).

$$\begin{cases} Dist\_i = \sqrt{X\_i^2 + Y\_i^2} \\ S\_i = \frac{1}{Dist\_i} \end{cases} \tag{19}$$

(4) The smell concentration decision value (*S*) is inserted in the Fitness function to calculate the fruit fly's *Smelli*.

$$Snell\_i = Function(\mathbb{S}\_i) \tag{20}$$

(5) Determine the fruit fly swarm's strongest smell (seek for the maximum value)

$$[bestSmell\quad bestIndex] = \max(Small)\tag{21}$$

(6) Using the best smell concentration and *x*, *y* coordinates, the fruit fly swarm flies to the position.

$$\begin{cases} Smelllbest = bestSmell\\ X\_-axis = X(bestimdex) \\ Y\_-axis = Y(bestimdex) \end{cases} \tag{22}$$

(7) If the smell concentration is better than the previous iteration of smell concentration, execute Step 6.

Quantum theory assigns the fruit fly swarm to move in quantum space. The delta potential well model increases the uncertainty that fruit flies recognize and migrate to food. All quantum objects have wave-like features and may be in several locations at once; hence, they are characterized in quantum theory by the wave function (*x*, *t*), rather than by their position *x* and velocity *v*. A location's likelihood of hosting the item in quantum space is determined by the strength of the wave function at that location, as shown below in module form [15].

$$\|\psi(\mathbf{x},t)\|^2 d\mathbf{x}dydz = \mathcal{Q}d\mathbf{x}dydz\tag{23}$$

*Qdxdydz* is the object's probability of appearing at (*x*, *y*, *z*) at time *t*. Thus, |*ψ*(*x*, *t*)| <sup>2</sup> is the probability density function meeting the equation:

$$\int\_{-\infty}^{+\infty} |\psi|^2 dx dy dz = \int\_{-\infty}^{+\infty} Q dx dy dz = 1 \tag{24}$$

Schrödinger's equation describes object motion in quantum physics.

$$i\hbar\frac{\partial}{\partial t}\psi(X,t) = \hat{\mathcal{H}}\psi(X,t) \tag{25}$$

$$
\hat{\mathbf{H}} = -\frac{\hbar}{2m}\nabla^2 + V(t) \tag{26}
$$

 is the Planck Constant, Hˆ is the Hamiltonian operator, *m* is the object mass, and *V*(*t*) denotes the potential field of the object. Fruit flies search for food in the delta potential well, where they move in quantum space. Quantum behavior replaces fruit fly foraging and random search in quantum space. Both fruit fly smell and vision become more uncertain, increasing population diversity. One-dimensional space was used for simplicity. If food source location is *x*, its potential energy in the one-dimensional delta potential well is:

$$V(\mathbf{x}) = -\gamma \delta(\mathbf{x} - \rho\_{\text{axis}}) = -\gamma \delta(y) \tag{27}$$

where the location of the fruit fly swarm, *ρaxis*, is in the center of the delta potential well. According to Schrödinger's equation, the following normalized wave function can be obtained:

$$
\psi(y) = \frac{1}{\sqrt{L}} e^{-|y|/L} \tag{28}
$$

*L* is the delta potential well length. Thus, the probability density function is:

$$Q(y) = \left|\psi(y)\right|^2 = \frac{1}{L}e^{-|y|/L} \tag{29}$$

This equals

$$y = \pm \frac{L}{2} \ln \frac{1}{u} \tag{30}$$

*u* is a random number (0, 1). Thus, we can determine the fruit fly's food source location:

$$\propto = \rho\_{\text{axis}} \pm \frac{L}{2} \ln \frac{1}{u} \tag{31}$$

The model assumes that a 1D delta potential well is on each dimension at the swarm center attractor point, and osphresis-based search has quantum properties. The fruit fly's quantum-behaved foraging is shown by the wave function, not randomly. The employed QFOA model included swarm location initialization, osphresis-based search, and visionbased search. The employed QFOA model used quantum-behaved searching instead of random osphresis-based searching. In the osphresis-based search process, *Mosp*, new food

source locations (*Xaxis*, *Yaxis*) were generated in the delta potential well. FOA's quantumbehaved searching mechanism is:

$$\begin{cases} X\_i = X\_{axis} \pm \frac{L\_{xj}}{2} \ln \frac{1}{r\_x} \\ Y\_i = Y\_{axis} \pm \frac{L\_{yj}}{2} \ln \frac{1}{r\_y} \end{cases} \tag{32}$$

where *i* = 1, 2, ... , *Mosp*, *rx* and *ry* are random [0, 1] values. *Lx*,*<sup>i</sup>* and *Ly*,*<sup>i</sup>* are delta potential well characteristic lengths of the corresponding dimension, determined by the fruit fly's last search location, based on their olfactory senses.

$$\begin{cases} L\_{x,i} = 2b|X\_{axis} - X\_i| \\ L\_{y,i} = 2b|Y\_{axis} - Y\_i| \end{cases} \tag{33}$$

*i* is the iteration number and *b* controls the quantum searching range.

$$b = b\_1 \log \text{sig} \left( 10 \cdot \left( 0.5 - \frac{\mathcal{g}}{\mathcal{G}\_{\text{max}}} \right) \right) + b\_2 \tag{34}$$

*b*<sup>1</sup> and *b*<sup>2</sup> restrict the value range to *b* ∈ [*b*2, *b*<sup>1</sup> + *b*2].
