## 3 Methodology
### Transmission
#### 3.1.2 System Parameter Optimization  
This section, deals with the optimization of the system parameters, such as the laser power, temperature control, and filter settings from \(\tilde{J}\), using the 2E-HRIO method to reduce the decoherence rate and support long-distance quantum communication. The conventional Elk Herd Optimizer (EHO) helped maintenance of a good balance between exploration and exploitation. However, there was the likelihood of EHO suffering from stagnation, with a decrease in the population diversity over iterations, leading to premature convergence. To solve this issue, Elitist Random Immigrants (ERI) was utilized for the preservation of the best-performing individuals and to introduce a fixed or adaptive number of new elk with random characteristics into the herd. To start with, the populations were initialized. Here, the laser power, temperature control, and filter settings were considered as the population. Following that, the Elk Herd matrix was modeled with the position of the population size \(P\) and the number of iterations \((I)\). The population matrix \(\tilde{E}\) is represented as follows:  
\[\tilde{E}= \begin{bmatrix}e_{j1}^{b1}&e_{j2}^{b1}&\cdots&e_{jn}^{b1}\\ e_{j1}^{b2}&e_{j2}^{b2}&\cdots&e_{jn}^{b2}\\ \vdots&\vdots&\cdots&\vdots\\ e_{j1}^{b^{p}}&e_{j2}^{b^{p}}&\cdots&e_{jn}^{b^{p}}\end{bmatrix}  \tag{4}\]  
where \(n\) is the total number of elk and \(e_{j}^{b}\) is the \((j^{th})\) solution of the \((e^{b})\) elk. The position is randomly initialized as follows:  
\[e_{j}^{b}=\hat{I}_{j}+r^{*}\times(\hat{u}_{j}-\hat{I}_{j}) \tag{5}\]  
where \((r^{*})\) represents a random number that ranges from 0 to 1, and \(\hat{u}_{j}\) and \(\hat{I}_{j}\) are the upper and lower bounds, respectively. Following this, the fitness function was estimated for the population. In this work, a high key rate \(K_{h}\), high Signal-to-Noise-Ratio (SNR) \(S_{n}r\), low decoherence rate \(D_{l}\), and low quantum Bit Error Rate (BER) \(Q_{a}\) were considered to constitute the fitness function of the population represented in Equation (6):  
\[\Psi=\{K_{h},S_{nr},D_{l},Q_{a}\} \tag{6}\]  
where \(\Psi\) represents the fitness function of the population. * Phase 1: Rutting Season: The total number of families is determined during rutting season by calculating the bull rate \(Y^{\prime\prime\prime}\), as mentioned below. \[\lambda=|Y^{\prime\prime\prime}\times P|\] (7) where \((\lambda)\) is the total number of families. The bulls were subsequently selected from \(\tilde{E}\) based on their fitness scores. The elks exhibiting the highest fitness values, originating from the apex of \(\tilde{E}\), were classified as bulls and were enumerated as follows: \[\lambda^{*}=argmin(\Psi(e^{b}))\qquad b\in(1,2,...,\lambda)\] (8) The bull was represented by \(e^{b}\), while the set of bulls trying to establish families was represented by \((\lambda^{*})\). Following this, the selection probability was determined for each bull \(e^{b}\) in \((\lambda^{*})\) according to its absolute fitness value. The next step involved the use ofthe following equation for the assignment of harems to bulls based on their selection probability (9):  
\[A=\frac{\Psi(e^{b})}{\sum_{\vartheta=1}^{\lambda}\Psi(e^{\vartheta})} \tag{9}\] \[\vartheta=P-\lambda \tag{10}\]  
where \((A)\) represents the selection probability derived from the fitness value, and \((\vartheta)\) represents the harems allocated to each bull, decided through roulette-wheel selection based on the bull's index. * Phase 2: Calving Season: During the calving season, the characteristics passed down from the mother harem and father bull were the primary influences on the offspring. In a family of bulls, a calf is considered reproduced and represented by Equation (11) if its index is the same as that of the father bull.