## 4 Simulation and Results
\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/ & (\(N_{\text{sb}}\) =16,200/ \\  & & \(R_{C}^{(max)}\) = 9/10)** & \(R_{C}^{(max)}\) = 8/9)** \\ \hline \multicolumn{5}{c}{**(a) Low error range: observed QBER 1.00–3.50\%**} \\ \hline Mean accuracy (\%) & 99.7985 & 99.8618 & 99.9498 & 99.9006 \\ Standard deviation (SD) & 0.1589 & 0.1050 & 0.0417 & 0.0803 \\ Lower error bar (mean—SD) (\%) & 99.6396 & 99.7568 & 99.9081 & 99.8203 \\ Upper error bar (mean + SD) (\%) & 99.9574 & 99.9668 & 99.9915 & 99.9809 \\ Mean squared error (MSE) & 6.5846 \(\times\)\(10^{-6}\) & 3.0130 \(\times\)\(10^{-6}\) & 4.2539 \(\times\)\(10^{-7}\) & 1.6325 \(\times\)\(10^{-6}\) \\ \hline \multicolumn{5}{c}{**(b) Middle error range: observed QBER 3.51–7.00\%**} \\ \hline Mean accuracy (\%) & 99.6941 & 99.7939 & 99.8984 & 99.8025 \\ Standard deviation (SD) & 0.2342 & 0.1609 & 0.0807 & 0.1545 \\ Lower error bar (mean—SD) (\%) & 99.4599 & 99.6330 & 99.8177 & 99.6480 \\ Upper error bar (mean + SD) (\%) & 99.9283 & 99.9548 & 99.9791 & 99.9570 \\ Mean squared error (MSE) & 1.4840 \(\times\)\(10^{-5}\) & 6.8321 \(\times\)\(10^{-6}\) & 1.6833 \(\times\)\(10^{-6}\) & 6.2887 \(\times\)\(10^{-6}\) \\ \hline \multicolumn{5}{c}{**(c) High error range: observed QBER 7.01–11.00\%**} \\ \hline Mean accuracy (\%) & 99.6168 & 99.7283 & 99.8092 & 99.6556 \\ Standard deviation (SD) & 0.2935 & 0.2003 & 0.1490 & 0.2698 \\ Lower error bar (mean—SD) (\%) & 99.3233 & 99.5280 & 99.6602 & 99.3858 \\ Upper error bar (mean + SD) (\%) & 99.9103 & 99.9286 & 99.9582 & 99.9254 \\ Mean squared error (MSE) & 2.3293 \(\times\)\(10^{-5}\) & 1.1395 \(\times\)\(10^{-5}\) & 5.8601 \(\times\)\(10^{-6}\) & 1.9135 \(\times\)\(10^{-5}\) \\ \hline \multicolumn{5}{c}{**Sifted key size after channel error estimation (bits)**} \\ \hline \(N=64\),800/\(N_{sb}=16\),200 bits & 61,560/15,390 & 58,320/14,580 & 64,800 & 16,200 \\ (100\% sifted key) & (95\%) & (90\%) & (100\%) & (100\%) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Simulation results of random key sampling with 5% and 10% rates, compared with the syndrome estimation using \(N=64\),800 bits with \(R_{C}^{(max)}=9/10\) and \(N_{sb}=16\),200 bits with \(R_{C}^{(max)}=8/9\) in three observed QBER ranges: (**a**) low error range of 1.00–3.50%, (**b**) moderate error range of 3.51–7.00%, and (**c**) high error range of 7.01–11.00%. Although the syndrome-based QBER estimation using the subblock length of \(N_{sb}\) = 16,200 bits does not outperform the 10% random key estimation in terms of mean accuracy and MSE within the high error range of 7.01-11.00%, as shown in Figure 5 and Table 3, it still achieves notable efficacy with a mean accuracy of 99.6556% and relatively low MSE of 1.9135 \(\times\) 10\({}^{-5}\). These results affirm its capability for accurate error estimation in the variability of observed error rates. Crucially, the maximum QBER threshold \(q_{threshold}\) for the syndrome-based estimation was set at 0.25. This threshold corresponds to the maximum QBER typically encountered during an intercept-and-resend attack in the BB84 QKD protocol. Additionally, this experimental setting ensures that the estimated QBER maintains  
Figure 5: Comparison of various QBER estimation methods in terms of the mean accuracy (%) and the mean squared error (MSE) of estimated QBER (\(q_{est}\)) from 2000 iterations. The comparison of QBER estimation methods encompasses random key sampling with 5% and 10% sampling rates, as well as the syndrome estimation with a block length of \(N\) = 64,800 bits and \(N_{sb}\) = 16,200 bits. These results are obtained from simulations over three observed QBER ranges: (**a**) low error rates of 1.00–3.50%, (**b**) middle error rates of 3.51–7.00%, and (**c**) high error rates of 7.01–11%. an accuracy above 95%.