## 4 Simulation and Results
These numerical results are depicted in box plots, which are derived from 2000 iterations, with the observed QBER set at four distinct values: 2%, 4%, 7%, and 10%. The specific values from these box plots are elaborated in Table 2, which details the performance and underlying statistical distribution of each QBER estimation method. \begin{table}
\begin{tabular}{l c c c c} \hline \hline \multirow{2}{*}{**QBER Estimation Methods**} & **5\% Random** & **10\% Random Key** & **Syndrome est.** & **Syndrome est.** \\  & **Key Sampling** & **Sampling** & **(N = 64,800 & (\(N_{sb}\) = 16,200 \\  & & **bits/\(R_{C}^{(max)}\) = 9/10)** & **Bits/\(R_{C}^{(max)}\) = 8/9)** \\ \hline \multicolumn{5}{c}{**(a) Observed QBER: 2\%**} \\ \hline Mean accuracy (\%) & 99.8092 & 99.8685 & 99.9545 & 99.9035 \\ Mean estimated QBER (\(q_{est}\)) & 0.019962 & 0.020019 & 0.019994 & 0.020039 \\ Mean squared error (MSE) & 5.7996 \(\times\) 10\({}^{-6}\) & 2.7454 \(\times\) 10\({}^{-6}\) & 3.2805 \(\times\) 10\({}^{-7}\) & 1.4677 \(\times\) 10\({}^{-6}\) \\ Median & 0.020062 & 0.020062 & 0.019988 & 0.019978 \\ Interquartile range (IQR) & 0.003395 & 0.002315 & 0.001104 & 0.002225 \\ Mean number of outliers & 0.0090 & 0.0070 & 0.0065 & 0.0085 \\ \hline \multicolumn{5}{c}{**(b) Observed QBER: 4\%**} \\ \hline Mean accuracy (\%) & 99.7352 & 99.8195 & 99.9220 & 99.8443 \\ Mean estimated QBER (\(q_{est}\)) & 0.040069 & 0.040069 & 0.039987 & 0.040081 \\ Mean squared error (MSE) & 1.1270 \(\times\) 10\({}^{-5}\) & 5.1504 \(\times\) 10\({}^{-6}\) & 9.6916 \(\times\) 10\({}^{-7}\) & 3.8172 \(\times\) 10\({}^{-6}\) \\ Median & 0.040123 & 0.039969 & 0.039984 & 0.040027 \\ Interquartile range (IQR) & 0.004630 & 0.003241 & 0.001599 & 0.003413 \\ Mean number of outliers & 0.0115 & 0.0085 & 0.0120 & 0.0050 \\ \hline \multicolumn{5}{c}{**(c) Observed QBER: 7\%**} \\ \hline Mean accuracy (\%) & 99.6501 & 99.7682 & 99.8574 & 99.7451 \\ Mean estimated QBER (\(q_{est}\)) & 0.070037 & 0.070037 & 0.070031 & 0.070105 \\ Mean squared error (MSE) & 1.9198 \(\times\) 10\({}^{-5}\) & 8.4687 \(\times\) 10\({}^{-6}\) & 3.1647 \(\times\) 10\({}^{-6}\) & 1.0266 \(\times\) 10\({}^{-5}\) \\ Median & 0.070062 & 0.070062 & 0.069934 & 0.070078 \\ Interquartile range (IQR) & 0.006173 & 0.003858 & 0.002734 & 0.005078 \\ Mean number of outliers & 0.0060 & 0.0125 & 0.0080 & 0.0060 \\ \hline \multicolumn{5}{c}{**(d) Observed QBER: 10\%**} \\ \hline Mean accuracy (\%) & 99.5781 & 99.7186 & 99.7808 & 99.6005 \\ Mean estimated QBER (\(q_{est}\)) & 0.100013 & 0.099960 & 0.100085 & 0.100068 \\ Mean squared error (MSE) & 2.7643 \(\times\) 10\({}^{-5}\) & 1.2399 \(\times\) 10\({}^{-5}\) & 7.6506 \(\times\) 10\({}^{-6}\) & 2.4914 \(\times\) 10\({}^{-5}\) \\ Median & 0.100000 & 0.100000 & 0.1000027 & 0.099976 \\ Interquartile range (IQR) & 0.007407 & 0.005093 & 0.003965 & 0.007457 \\ Mean number of outliers & 0.0070 & 0.0070 & 0.0120 & 0.0050 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Numerical results of random key sampling with 5% and 10% sampling rates and the syndrome estimation using \(N\) = 64,800 bits with \(R_{C}^{(max)}\) = 9/10 and \(N_{sb}\) = 16,200 bits with \(R_{C}^{(max)}\) = 8/9. The results are presented for the observed QBER values: (**a)** 2%, (**b) 4%, (**c) 7%, and (**d)** 10%. In Table 2, the mean squared error (MSE) for all observed QBER values (\(q_{\text{obs}}\)) are calculated using:  
\[\text{Mean accuracy}\;(\%)\;=\;\frac{1}{n}{\sum_{i=1}^{n}{\left[{\left({1-{{{{ \left|{q_{est}}}_{i}}-{q_{obs}}}\right|}}\right)\ \times\ 100}\right]}}, \tag{21}\]  
and  
\[\text{Mean squared error}\;(\text{MSE})\;=\;\frac{1}{n}{\sum_{i=1}^{n}{\left({{ q_{est}}_{i}}-{q_{obs}}_{i}\right)^{2}}}, \tag{22}\]  
where \(n\) denotes the number of iterations. Based on the performance and statistical metrics presented in Figure 4 and Table 2, the proposed syndrome-based QBER estimation with \(N=64,800\) bits demonstrates the highest accuracy and the smallest mean squared error (MSE) for all observed QBER values (\(q_{\text{obs}}\)).