## 4 Numerical Results
### Analysis of Algorithm's Robustness
Here, \(\alpha_{ML}(m),2\leq m\leq 10\) is the optimal value of the parameter \(\alpha\) in Equation (18) obtained by fitting that expression to the numerical datapoints for QRWS with walk coin size m. Their values are given in Table 2.  
The results for walk coin size 5 (Figure 8a) and 9 (Figure 8b) are graphically represented in a logarithmic scale. The actual values of the root-mean-square deviations \(\sigma_{\alpha}\) and \(\sigma_{\phi}\) depend on the particular physical realization of the algorithm, thus we have performed the computations with a fixed conservative value for both r.m.s. deviations \(\sigma_{\alpha}=\sigma_{\phi}=0.1\). As can be seen from the figures, there is a relatively large central area where the r.m.s. deviation of the probability \(\sigma_{p}\) is less than 0.01, reaching levels \(\sigma_{p}\leq 10^{-4}\) in the innermost dark region. This behavior is the same for the quantum algorithm using a walk coin with five and nine states. The corresponding images in Figure A6 in Appendix E for coin of size \(2\leq m\leq 10\) show similar characteristics. The above analysis confirms that the alternative walk coin, constructed by a HR and an additional phase, improves the QRWS algorithm, not only to obtain high probability \(p(\phi,\alpha,m)\) for large interval of \(\phi\), but also to have extremely high stability to changes in both parameters of the coin (\(\varphi\) and \(\alpha\)). Figure 7: (Color online) Probability of QRWS algorithm to find solution \(p(\phi,\alpha,m)\) as a function of \(\alpha\) and \(\phi\) for walk coin of size \(m=5\) (**a**) and \(m=9\) (**b**). The horizontal lines represent simulation of QRWS algorithm with \(\alpha=0\)—dashed gray line, \(\alpha=-1/(2\pi)\)—dash-dotted blue line, and the solid green line correspond to \(\alpha_{ML}(m)\) given in Table 2.  
It is useful to study how the robustness of the proposed construction of the QRWS algorithm compares to the stability of QRWS when the walk coin is built according to Equation (17)--corresponding to the case when the parameter \(\zeta\) is not controlled as in [44] (if \(\phi=\pi\) the Grover coin is obtained). We have calculated the r.m.s. of the probability for the latter case:  
\[\sigma_{p^{\prime}}^{i}=\frac{1}{p^{\prime i}}\Big{|}\frac{\partial p^{\prime i }}{\partial\phi^{i}}\sigma_{\phi}(\phi^{i}-\pi)\Big{|}. \tag{20}\]  
Here \(\sigma_{\phi}=0.1\) and \(p^{\prime}(\phi,m)\) is the probability to find a solution corresponding to the curves \(\zeta=\pi\) given by Equation (17) (depicted with the dashed teal lines on both images of Figure 4). In Figure 9, the ratios \(\sigma_{p}/\sigma_{p^{\prime}}\) are plotted at the logarithmic scale (similarly to the case of \(\sigma_{p}^{ij}\) in Equation (19)) and the division is performed for every value of \(\alpha_{j}\) (i.e., \(\sigma_{p}^{ij}/\sigma_{p^{\prime}}^{i}\mid_{j=const}\)), where the number of points on the coordinate axes are \(1\leq i\leq 100\) and \(1\leq j\leq 250\). Again, each node of the grid defined by the variables \(\phi\) and \(\alpha\) is indexed by a pair of numbers \(i\) and \(j\). Here, \(\phi_{min}\leq\phi\leq\phi_{max}\) and \(-1.5\leq\alpha\leq 1\). The angles \(\phi_{min}\) and \(\phi_{max}\) are chosen in such a way that the intervals in \(\phi\) correspond to the high probability central peaks of \(p(\phi,\zeta=\pi,\ m)\) given by Equation (17) and plotted in Figure 4 (teal dashed lines). Outside this region, the probability \(p(\phi,\zeta=\pi,\ m)\) of the standard QRWS algorithm is very low and comparison of the robustness is not relevant to our study. In Figure 9a,b, the results are shown for coin size \(m=5\) and \(m=9\), respectively. We will make remarks on two important points. First, the alternative QRWS shows stability \(\geq\mathcal{O}(10^{2})\) in comparison to the most commonly used walk coin.