## 4 Numerical Results  
### Region of Stability for Different Coin Sizes  
During the experiments, there always are unavoidable time variations of the experimental setup's parameters, such as in the frequency and the shape of the laser pulses controlling ions in the trap. In order to investigate the robustness of QRWS algorithm's implementation we will study the half-length \(\varepsilon=(\varepsilon_{-}+\varepsilon_{+})/2\) of the interval \(\Delta\) (defined in Section 3.2). This quantity will allow us to quantitatively evaluate the stability of the algorithm and make comparisons for different register sizes and functional dependencies between the coin parameters. Here we show results for the width \(\varepsilon\) of each of the functions (16), (17), and Equation (18) with \(\alpha=-1/(2\pi)\) and \(\alpha=\alpha_{ML}(m)\), where \(m\in[2,16]\). The solid lines correspond to results from MC simulation, and dashed lines--to prognosis given by ML. In Appendix F, an explanation of ML methods used and their drawbacks are given. Different colors and markers correspond to different dependence between phases in the walk coin. The red curve with 4-pointed star marker corresponds to Equation (16), the teal with 3-pointed star marker--to Equation (17), and the blue with 5-pointed star marker and the green with 2-pointed star marker--to Equation (18) with \(\alpha=-1/(2\pi)\) and \(\alpha=\alpha_{ML}(m)\) correspondingly. We remark that MC points for \(m=11\) were not enough to obtain good ML estimation of \(\alpha\). This is why we used \(\alpha_{ML}(m=11)\) predicted with neural network model in QRWS algorithm simulation. This approach gives significantly better estimation for \(\varepsilon\) in the case \(m=11\). In Figure 6 are shown examples for the values of \(\varepsilon\) in the cases when \(p(\phi\in(\phi_{max}-\varepsilon_{-},\phi_{max}+\varepsilon_{+}))\geq 0.9 \times p_{max}\) and analogically for \(0.7\times p_{max}\):  
The value of \(\varepsilon\), depends on the coin size and on the function used to construct the coin. Monte Carlo simulation of QRWS using coins of size \(m\leq 3\) have completely different behavior compared with the larger coins as can be seen in Appendix A in Figure 11. For those cases, the Grover coin gives very low probability to find solution. This is the reason to exclude MC simulations of QRWS with coin size \(m\leq 3\) from further considerations. In case of Equation (17) and for \(m=4\) the length \(\varepsilon\) is low and it continues to decrease with increasing coin size. The curve from Equation (16) for small coins have high value of the width \(\varepsilon\), however it decreases fast with increasing the coin size. For large coin we  
Figure 6: (Color online) Change of the area width \(\varepsilon\), having probability to find solution equal to percentage of the maximum probability \(p_{max}\), with increase of the coin size \(m\). (**a**) shows the case for \(90\%\times p_{max}\) and (**b**)—for \(70\%\times p_{max}\). The red curve with 4-pointed star marker corresponds to Equation (16), the teal with 3-pointed star marker—to Equation (17), and the blue with 5-pointed star marker and the green with 2-pointed star marker—to Equation (18) with \(\alpha=-1/(2\pi)\) and \(\alpha=\alpha_{ML}(m)\) correspondingly. The solid lines correspond to values obtained by MC simulations, dashed to—results prognosed by ML. predict that they will have similar behavior and Equation (17) will act like Equation (16) for smaller coin. So, the linear approximation is not good enough. Equation (18) with \(\alpha=0\) coincides with Equation (16). However, other values of \(\alpha\) are much more promising. The probability \(p\) for coins constructed with functional dependence \(\zeta(\phi)\), given by Equation (18), did not seem to decrease if the coin size increases. The curve corresponding to \(\alpha=\alpha_{ML}(m)\) obtained by ML gives a slightly larger \(\varepsilon\) and shows slightly better behavior with increase of the coin size compared to \(\alpha=-1/(2\pi)\).