## 2 Improved Quantum Particle Swarm Optimization
### Quantum Particle Swarm Optimization  
Let \(\varphi_{i,j}(t)=c_{1}(t)r_{1,j}(t)/\left[(c_{1}(t)r_{1,j}(t)+c_{2}(t)r_{2,j}(t))\right]\), and in the practical application of the algorithm, \(c_{1}=c_{2}\) is generally taken, so that:  
\[\varphi_{i,j}(t)\,\sim U(0,1),1\leq j\leq N \tag{10}\]  
Then particle \(i\) is attracted by point \(\mathbf{p}(t)=\left[p_{i,1}(t)\quad p_{i,2}(t)\quad\cdots\quad p_{i,n}(t)\right]\) during its iterative convergence. The parameter \(\mathbf{p}(t)\) is a random value between \(\mathbf{P}_{i}(t)\) and \(\mathbf{P}_{g}(t)\), and Equation (8) can be rewritten, as shown in Equation (11):  
\[p_{i,j}(t)=\varphi_{i,j}(t)P_{i,j}(t)+(1-\varphi_{i,j}(t))P_{g,j}(t) \tag{11}\]  
In the iterative process, the particles keep approaching and finally reach point \(\mathbf{p}\). This process can be regarded as the existence of attraction potential in the space where the particle swarm is located, which causes the particle to gravitate towards point \(\mathbf{p}\). Therefore, the random distribution of particles in space is avoided, and the convergence of the whole algorithm is guaranteed. Under the mechanism of attraction potential, quantum particle swarm optimization (QPSO) is reconstructed based on the \(\delta\) potential well in quantum theory to further improve the convergence and accuracy of the algorithm. According to [PERSON]'s principle, the observer cannot determine the position and velocity of particles at the same time. The velocity information and position information of particles are described by wave function \(\psi(\mathbf{r},t)\) and defined \(\mathbf{r}=(x,y,z)\) as the position of particles, as shown in Formulas (12) and (13). \[|\psi(\mathbf{r},t)|^{2}dxdydz=Qdxdydz \tag{12}\]  
Figure 1: Flow chart of the standard particle swarm optimization algorithm. \[\int_{-\infty}^{+\infty}|\psi(\mathbf{r},t)|^{2}dxdydz=\int_{-\infty}^{+\infty}Qdxdydz=1 \tag{13}\]  
where \(\mathbf{Q}\) is the probability density function. The position distribution of particles in quantum space satisfies the Schrodinger equation, as shown in Equation (14). \[i\hbar\frac{\partial}{\partial}\psi(\mathbf{r},t)=\hat{H}\psi(\mathbf{r},t)=[-\frac{ \hbar^{2}}{2m}\
abla^{2}+V(\mathbf{r})]\psi(\mathbf{r},t) \tag{14}\]  
where \(\hbar\) is the Planck constant, \(\hat{H}\) is the Hamilton operator, \(V(r)\) is the potential field distribution in the space that the particle is located, and \(m\) is the particle mass. Generally speaking, the motion of a single particle in one-dimensional space is analyzed. The particle only has the degree of freedom in the x direction. Let the one-dimensional point of attraction be \(p_{i}=p\), \(\mathbf{r}=x\), and define the potential field \(V(x)\) in the space where the particle is located, as shown in Equation (15). \[V(x)=-\gamma\delta(x-p) \tag{15}\]  
where \(\delta\) is the Dirac function. Let \(\chi=x-p\), according to the spatial potential field distribution as shown in Equation (14), and rewrite the Hamilton operator as:  
\[\hat{H}=-\frac{\hbar^{2}}{2m}\frac{\partial^{2}}{\partial\chi^{2}}-\gamma \delta(\chi) \tag{16}\]  
Set the particle in a steady state, and its probability density is independent of time, then:  
\[\psi(\chi,t)=\psi(\chi)e^{-iEt/\hbar} \tag{17}\]  
The probability density function of the particle satisfies the Schrodinger equation as shown in Equation (18):  
\[i\hbar\frac{\partial}{\partial}\psi(\chi)e^{-iEt/\hbar}=[-\frac{\hbar^{2}}{2 m}\frac{\partial^{2}}{\partial\chi^{2}}-\gamma\delta(\chi)]\psi(\chi)e^{-iEt/\hbar} \tag{18}\]  
The wave function of a particle in one-dimensional space is:  
\[p_{i,j}(t)=\varphi_{i,j}(t)P_{i,j}(t)+(1-\varphi_{i,j}(t))P_{g,j}(t) \tag{11}\]0  
Let \(L=\frac{\hbar^{2}}{\gamma m}\), then the probability density function of the particle appearing at the designated point on the \(x\) axis is:  
\[p_{i,j}(t)=\varphi_{i,j}(t)P_{i,j}(t)+(1-\varphi_{i,j}(t))P_{g,j}(t) \tag{11}\]1  
Then its probability distribution function is:  
\[F(\chi)=\int_{-\infty}^{\chi}Qd\chi=\int_{-\infty}^{\chi}\frac{1}{L}e^{-2|\chi |/L}d\chi=\left\{