## 4 Numerical Results
### Dependence between Alpha and Coin Size
\(\alpha_{2}(m)\) (the dot-dashed green line in Figure 10) is the fitting function, when the alpha values from the DNN model are considered to be less reliable (the fitting weights were set two times smaller) than the ones from Monte Carlo simulations data. The third, more well-grounded scenario, is an intermediate between the previous two. To obtain the function \(\alpha_{3}(m)\) (the dashed red line in Figure 10) we have suggested that the ML predictions are more reliable for smaller coin size. This is due to the fact that the training of ML models is performed with data points with coin size \(4\leq m\leq 10\). To numerically achieve this, we have set weight coefficients for \(\alpha_{ML}(m),m=11\div 20\) that decrease with increasing of \(m\). The parameter values for the fitting functions (Equation (21)) in all of the studied scenarios are given in Table 3.  
\[\alpha_{i}(m)=am^{2}+bm+c,\qquad i=1,2,3 \tag{21}\]  
All three lines explained above and shown in Figure 10, representing the behavior of \(\alpha_{ML}\) as a function of the coin size in QRWS algorithm, meet the restrictions explained earlier in the section. The horizontal solid line plotted on the figure corresponds to \(\alpha(m)=-1/(2\pi)=const\). All the functions \(\alpha=\alpha(m)\) suggest a relatively large value of \(\alpha_{ML}\) for coin registers with dimension even higher than 20. This translates to quantum \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \(m\) & \(2\) & \(3\) & \(4\) & \(5\) & \(6\) & \(7\) & \(8\) \\ \hline \(-\alpha_{ML}\) & \(0.558\) & \(0.552\) & \(0.142\) & \(0.155\) & \(0.163\) & \(0.209\) & \(0.206\) \\ \hline \(m\) & \(9\) & \(10\) & \(11\) & \(11\) & \(12\) & \(13\) & \(14\) \\ \hline \(-\alpha_{ML}\) & \(0.185\) & \(0.168\) & \(0.150\) & \(0.170^{\prime}\) & \(0.179^{\prime}\) & \(0.180^{\prime}\) & \(0.203^{\prime}\) \\ \hline \(m\) & \(15\) & \(16\) & \(17\) & \(18\) & \(19\) & \(20\) & \\ \hline \(-\alpha_{ML}\) & \(0.225^{\prime}\) & \(0.197^{\prime}\) & \(0.205^{\prime}\) & \(0.206^{\prime}\) & \(0.216^{\prime}\) & \(0.223^{\prime}\) & \\ \hline \end{tabular}
\end{table} 
Table 2: Values of the parameter \(\alpha_{ML}(m)\) in Equation (18) for QRWS algorithm’s walk coin size \(2\leq m\leq 20\). The primed values are derived from ML predictions, and the remaining come from Monte Carlo simulations. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & \(a\) & \(b\) & \(c\) \\ \hline \(\alpha_{1}(m)\) & \(4.85\times 10^{-5}\) & \(-4.78\times 10^{-3}\) & \(-1.41\times 10^{-1}\) \\ \(\alpha_{2}(m)\) & \(2.71\times 10^{-4}\) & \(-9.82\times 10^{-3}\) & \(-1.20\times 10^{-1}\)algorithm able to search in a database with an arbitrary topology consisting of \(2^{m}\) entries. The functions \(\alpha_{i}(m),i=1,2,3\) given here, would allow physical implementation of the QRWS algorithm with increased robustness for any practically useful size of the database. In this section, the relation between \(\alpha\) and the coin size of QRWS algorithm has been thoroughly studied. This parameter introduced as a factor giving the nonlinearity in the functional dependence between walk coin angles \(\phi\) and \(\zeta\), leads to an improvement in the stability of our modification of the random walk search algorithm. Here, by imposing a few reasonable restrictions to the possible fitting functions, three relations \(\alpha=\alpha(m)\) were numerically derived.