## 4 Numerical Results
### Analysis of Algorithm's Robustness  
The proposed in the work optimization of the QRWS algorithm is based on walk coin parameterized by two phases (see Equation (12)). We have showed in Section 3.2 that, if an optimal relation between them is maintained, the algorithm becomes more robust. Here, we will support this statement by numerical analysis of the stability of probability \(p\) to uncertainties in the walk coin parameters \(\phi\) and \(\alpha=\alpha(\phi,\zeta)\). The latter expression is derived by solving Equation (18) for \(\alpha\). In Figure 7, simulations of the probability to find a solution \(p(\phi,\alpha,m)\) are shown as a function of the angle \(\phi\), the coin size \(m=5\) (in Figure 7a), \(m=9\) (in Figure 7b), and the parameter \(\alpha\) of Equation (18). For walk coin of size \(m\geq 4\) there is a central plateau with high values of the probability \(p(\phi,\alpha,m)\). The lightest color area corresponds to \(p>0.95\times p_{max}(m)\), the second contour is at \(p=0.9\times p_{max}(m)\), etc. The values of \(p_{max}(m)\) are given in Table 1. The plateau is relatively wide not only in the horizontal axis (along \(\phi\)), but in \(\alpha\) direction too. Furthermore, this behavior remains the same for all simulated QRWS schemes with walk coin size up to 10, as could be verified from the figures illustrating the function \(p(\phi,\alpha,m),2\leq m\leq 10\) in Appendix D. The above observation shows that the gere proposed construction of the coin leads to a stable quantum algorithm against small deviations in both \(\phi\), and the parameter \(\alpha\) introduced by our scheme. The horizontal lines on both images of Figure 7 mark specific values for the angle \(\alpha\). The dashed gray, dash-dotted blue, and the solid green lines in Figure 7 correspond to \(\alpha=0\), \(\alpha=-1/(2\pi)\), and \(\alpha=\alpha_{ML}(m)\) (given in Table 2), respectively. The latter two lies close one to the other and both lie in regions with high probability p for all the simulated QRWS algorithms with walk coin size \(m\geq 4\). This shows that the value \(\alpha=-1/(2\pi)\) is relatively good and could be used in practice, even if the optimal value of \(\alpha\) is not known precisely for quantum algorithms with even larger registers. To give quantitative proof for the increased robustness of the QRWS algorithm, we make estimation of the root-mean-square (r.m.s) deviation of the probability \(p(\phi,\alpha,m)\) for a fixed uncertainty in the parameters \(\alpha\) and \(\phi\), namely \(\sigma_{\alpha}\) and \(\sigma_{\phi}\). The numerical computations of the r.m.s. deviation of the probability \(\sigma_{p}\) for each coin size \(m\) are performed at fixed points--the nodes of a square grid in the plane \((\phi,\alpha)\). Each node is indexed by a pair of numbers \(i\) and \(j\). The values of the parameter \(\phi\) are taken for 180 points equally spaced between 0 and \(2\pi\), where \(\phi^{i}=2\pi/179(i-1)\), \(i=1,\ldots,180(\phi^{0}=0,\phi^{180}=2\pi)\). Analogously, for \(\alpha\) are taken 250 discrete values, so that \(\alpha^{i}=\alpha^{min}+\left((\alpha^{max}-\alpha^{min})\right)/249(j-1)\), where \(j=1,\ldots,250\), \(\alpha^{min}=-1.5\), and \(\alpha^{max}=1\). We do not assume any correlation between these parameters and calculate the quantity:  
\[\sigma_{p}^{ij}=\frac{1}{p^{ji}}\sqrt{\left(\frac{\partial p^{ij}}{\partial \phi^{j}}\right)^{2}\sigma_{\phi}^{2}(\phi^{i}-\pi)^{2}+\left(\frac{\partial p ^{ij}}{\partial\alpha^{j}}\right)^{2}\sigma_{\alpha}^{2}(\alpha^{i}-\alpha_{ ML}(m))^{2}} \tag{19}\]  
for each discrete point \(\phi_{i}\) and \(\alpha_{j}\). We center the data around the points \((\pi,\alpha_{ML}(m))\) which lie in the middle of the region giving the most stable implementation of the quantum algorithm.