# An Improved Multi-Objective Grey Wolf Optimizer for Aerodynamic Optimization of Axial Cooling Fans
## 2 Improved Multi-Objective Grey Wolf Optimizer
### Population Initialization Based on Bloch Coordinates of Qubits  
High-quality initial populations significantly contribute to the convergence speed and solution quality of algorithms. However, the MOGWO algorithm, which employs random initialization of the population, often results in uneven population distribution. This uneven distribution can lead to reduced population diversity and adversely affect the algorithm's convergence speed. Therefore, this paper adopts the population initialization method based on Bloch coordinates of qubits. The primary principle is introduced below. In quantum computing, the smallest unit of information is the qubit. The state of a qubit can be expressed as follows [28]:  
\[|\phi\rangle=\cos(\frac{\theta}{2})\ |0\rangle+e^{i\varphi}\sin(\frac{\theta}{2}) \ |1\rangle \tag{1}\]  
where the two parameters on the right side of the equation uniquely determine a point P on the Bloch sphere. Because each point of Bloch spheres \((\theta,\ \varphi)\) corresponds uniquely to a normalized quantum state in terms of \(|\phi\rangle\). The symbols \(|0\rangle\) and \(|1\rangle\) are the standard basis vectors of quantum states in quantum mechanics, which form the standard orthogonal basis of a two-dimensional Hilbert space, as shown in Figure 1. Since any qubit corresponds to a point on the Bloch sphere, and the Bloch sphere is a three-dimensional unit sphere, the spherical coordinates of a qubit (\(|\phi\rangle\)) can also be expressed as follows:  
\[|\phi\rangle=[cos\varphi sin\theta sin\varphi sin\theta\ cos\theta]^{T} \tag{2}\]  
Figure 1: Bloch sphere representation of a qubit. Thus, grey wolf individuals can be encoded using the Bloch spherical coordinates of qubits. Let \(P_{i}\) be the \(i\)-th individual in the population, and \(n\) be the optimization dimension; then its encoding strategy is described as follows:  
\[P_{i}= \begin{bmatrix}cos\varphi_{i1}sin\theta_{i1}\\ sin\varphi_{i1}sin\theta_{i1}\\ cos\theta_{i1}\end{bmatrix} \dots \begin{bmatrix}cos\varphi_{in}sin\theta_{in} \\ sin\varphi_{in}sin\theta_{in}\\ cos\theta_{in}\end{bmatrix}  \tag{3}\]  
The parameters (\(\varphi_{ij}\) and \(\theta_{ij}\)) are random numbers in the range [0, 2\(\pi\)] and [0, \(\pi\)], respectively, and \(i=1,2,\dots,m,j=1,2,\dots n\), where \(m\) represents colony size, and \(n\) represents the number of qubits. The Bloch coordinates of each qubit can be regarded as three paratactic gene chains, each of them representing a solution; thus, an individual can simultaneously represent three solutions:  
\[\left\{ \begin{aligned} P_{i,x}=[cos\varphi_{i1}sin\theta_{i1}\dots cos \varphi_{in}sin\theta_{in}]\\ P_{i,y}=[sin\varphi_{i1}sin\theta_{i1},\dots,sin\varphi_{in}sin\theta_{in} ]\\ P_{i,z}=[cos\theta_{i1},cos\theta_{i2},\dots,cos\theta_{in}]\end{aligned} \right. \tag{4}\]  
where \(P_{i,x}\) is known as the x solution, \(P_{i,y}\) is known as the y solution, and \(P_{i,z}\) is known as the z solution. The individual encoded by the qubits is shown in Figure 2.  
Each individual contains \(3n\) Bloch coordinates of \(n\) qubits that can be transformed from the unit space \([-1,1]^{n}\) to the solution space of the continuous optimization problem. It is assumed that \(\left(x_{ij},y_{ij},z_{ij}\right)\) denote the Bloch coordinates of the \(j\)-th qubit in the \(i\)-th individual. The corresponding variables \(\left(X_{i,x}^{j},X_{i,x}^{j},X_{i,x}^{j}\right)\) in the solution space are, respectively, computed as follows:  
\[\left\{ \begin{aligned} X_{i,x}^{j}&=0.5\big{[}ub_{j} \big{(}1+x_{ij}\big{)}+lb_{j}\big{(}1-x_{ij}\big{)}\big{]}\\ X_{i,y}^{j}&=0.5\big{[}ub_{j}\big{(}1+y_{ij}\big{)}+lb_{j} \big{(}1-y_{ij}\big{)}\big{]}\\ X_{i,z}^{j}&=0.5\big{[}ub_{j}\big{(}1+z_{ij}\big{)}+lb_{j} \big{(}1-z_{ij}\big{)}\big{]}\end{aligned} \right. \tag{5}\]  
where \([lb_{j},\,ub_{j}]\) are the lower and upper bounds of the corresponding variables. Thus, three feasible solutions are obtained, and the best fitness among them is used as the encoding of the individual.