## 4 Proposed Protocol
### Multi-Hop Routing Approaches
#### 4.3.2 Quantum Annealing  
Different from the simulated annealing algorithm which jumps over the barrier to eliminate the local optimum, the quantum annealing algorithm introduces a penetration \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**No.** & \(f_{CH_{1}}\) & \(f_{CH_{2}}\) & \(f_{CH_{3}}\) & \(f_{CH_{4}}\) & \(CH_{probability}\) \\ \hline
1 & VL & F & N & L & \(B_{ave}\) \\
2 & VL & F & M & L & \(B_{ave}\) \\
3 & VL & F & F & L & P \\
4 & VL & F & N & M & P \\
5 & VL & F & M & M & P \\
6 & VL & F & F & M & VP \\ \(\vdots\) & \(\vdots\) & \(\vdots\) & \(\vdots\) & \(\vdots\) & \(\vdots\) \\
73 & M & N & N & L & S \\
74 & M & N & M & L & \(A_{ave}\) \\
75 & M & N & F & L & A \\
76 & M & N & N & M & A\({}_{ave}\) \\
77 & M & N & M & M & A \\
78 & M & N & F & M & \(B_{ave}\) \\ \(\vdots\) & \(\vdots\) & \(\vdots\) & \(\vdots\) & \(\vdots\) & \(\vdots\) \\
130 & VH & N & N & M & VS \\
131 & VH & N & M & M & VS \\
132 & VH & N & F & M & \(A_{ave}\) \\
133 & VH & N & N & H & S \\
134 & VH & N & M & H & \(A_{ave}\) \\
135 & VH & N & F & H & \(A_{ave}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Optimized fuzzy rule table for CH selection. field into the quantum system and generates quantum wave motion to penetrate the potential barrier itself, then reaches the lowest energy state [29]. In the early stage of annealing, the kinetic energy term is relatively large, providing a large disturbance that can fully traverse the entire solution space. The kinetic energy term is gradually reduced, and in the later stage of annealing, the kinetic energy term is gradually reduced to 0. The comparison of the ways of crossing the barrier is shown in Figure 5.  
In other words, the state of the minimum potential energy of the system is reached. The evolution of the quantum system under the action of potential and kinetic energy can be described by the following Schrodinger Equation (8):  
\[i\hbar\frac{\partial}{\partial r}\Psi(r,t)=\hat{H}\Psi(r,t) \tag{8}\]  
where \(\hbar\) is Planck's constant and \(\hat{H}\) is the Hamiltonian operator. \[\hat{H}=-\frac{\hbar^{2}}{2m}\
abla^{2}+V(r) \tag{9}\]  
where \(m\) is the particle mass, \(\
abla\) is the gradient operator, \(V(r)\) is the potential field of the particle. In practice, the complexity of Equation (9) increases exponentially with the increase in problem complexity, so we adopt the PIMC (path-integral Monte Carlo) method to simulate the quantum annealing process. The two-dimensional stochastic Ising model is employed to elucidate the PIMC process [37]:  
As shown in Figure 6, each arrow \(a_{i}\in\{+1,-1\}\) represents a distinct magnetic needle with varying spin magnetism. The collective spin orientation of all these magnetic needles determines the overall magnetization. The interaction between adjacent magnetic needles and environmental thermal noise influences the behavior of each needle. Thermal noise interference intensifies as temperature increases, leading to random magnetism in the magnetic needles. However, during the annealing process characterized by a gradual decrease in temperature, neighbor interactions become more prominent. Eventually, when the temperature reaches a sufficiently low level, many tiny magnetic needles in a uniform direction exhibit magnetism. At this time, the system energy is minimal. The Hamiltonian function of the transverse field Ising model is formulated as  
\[H_{p}(t)=\sum_{i=1}^{N}h_{i}\sigma_{i}\ ^{Z}+\sum_{i,j=1}^{N}J_{i,j}\sigma_{i}\ ^{Z}\sigma_{j}\ ^{Z} \tag{10}\]  
where \(h_{i}\) represents the external local field, which is not considered in this paper for the sake of simplicity and is assumed to be 0. \(J_{i,j}\) denotes the coupling strength between quantum bit \(i\), \(j\); \(\sigma_{i}\ ^{Z}\) and \(\sigma_{j}\ ^{Z}\) denote the Pauli operator Z spin matrices at point \(i\), \(j\). The concept of spin can be interpreted as a minute magnetic needle positioned in a specific state, oscillating  
Figure 5: Difference between SA and QA.