## 4 Numerical Results
### Dependence between Alpha and Coin Size  
In this section, we show results from in depth analysis of the parameter \(\alpha\) introduced in our modification of QRWS algorithm for hypercube. Its value depends on the increase in the robustness of the studied modification of QRWS algorithm. Due to imperfections of the experimental setup, maximal theoretical probability to find a solution is unachievable. However, for many applications, probability equal to or higher than \(0.9\times p_{max}\) is sufficient. Based on this, we define the optimal value of alpha as the one that corresponds to a curve having the largest width at \(p=0.9\times p_{max}\). The optimal values of \(\alpha\) for each coin size \(m\), namely \(\alpha_{ML}(m),2\leq m\leq 20\) are given in Table 2. For coin size \(2\leq m\leq 11\), they are extracted from the numerical simulations' data and for \(11\leq m\leq 20\) (primed values)--from predictions of a deep network model, trained with the Monte Carlo datapoints for \(2\leq m\leq 10\). A detailed explanation of the DNN models used in the paper is given in Appendix F. In the case of coin of size \(11\), we give two relatively close values of \(\alpha_{ML}\). They are derived independently, the first one from the MC simulations' data, and the second primed \(\alpha_{ML}\) comes from ML predictions. The value of \(\alpha_{ML}(11)\) obtained by MC has lower credibility because the simulated data points are much less than the ones for other values of \(m\). The value \(\alpha=-1/(2\pi)\simeq-0.159\) lies close to all \(\alpha_{ML}(m),m\geq 4\) which justifies its use as a benchmark throughout the paper. Figure 9: (Color online) Relative stability of the QRWS algorithm with walk coin size \(m=5\) (**a**) and \(m=9\) (**b**), given by the ratio of the root\(-\)mean\(-\)square uncertainty with and without the proposed optimization of the coin (Equation (19) and Equation (20), respectively). The chosen ranges of the parameter \(\phi\) correspond to the high probability central peaks of \(p(\phi,\zeta=\pi,\ m)\) (Equation (17)). The dashed gray, the dash-dotted blue, and the solid green lines are obtained from Equation (18) with \(\alpha=0,\alpha=-1/(2\pi)\), and \(\alpha=\alpha_{ML}(m)\). The values of \(\alpha_{ML}\) extracted only from Monte Carlo simulations' data for QRWS with coin size \(m=2\div 10\) are not enough to make any reasonable analysis of the behavior of this parameter for bigger coin size. However, by combining those results with the values of alpha obtained from the trained DNN models, we are able to make some predictions. To narrow the possible fitting functions, the following assumptions will be made. First, due to the completely different behavior of \(p(\phi,m)\) for \(m=2,3\) (see for example Figures 12 and 13), \(\alpha_{ML}(m=2)\) and \(\alpha_{ML}(m=3)\) are not included in the fitting procedure. Second, as explained in the text, the nonlinear term in Equation (18) could be considered to be a small perturbation to the linear functional dependence, so we expect that \(\alpha\) will not become too large in absolute value for very large coin size m (\(|\alpha(\infty)|<\alpha_{\infty}\), where \(\alpha_{\infty}\) is a finite positive real number). An implication of the above statement is that if \(\alpha_{ML}(m)\
eq const\) (the simulations support this inequality), then \(\alpha(m)\) should not be a linear function. Finally, the obtained values of \(\alpha\) for different \(m\), shown in Figure 10 (the teal dots are from MC data and the orange triangles represent ML predictions) are less than 20, so the fitting function has to be as simple as possible. We used a quadratic function and considered three scenarios. In the first scenario, we obtained \(\alpha_{1}(m)\) (the dotted blue line in Figure 10) by suggesting that all points have equal weight.