## 3 Qrws with Qudit Coin Constructed by Householder Reflection
### Robustness of the Coin for Different Functions \(\zeta(\phi)\)
It can be clearly seen that the curves on both figures show the same behavior as, for example, the similar shape of the different lines (the solid green, dotted blue, and dot-dashed red corresponding to \(\alpha=\alpha_{ML},\alpha=-(2\pi)^{-1},\alpha=0\) in Equation (18), respectively, and the dashed teal line--to  
Figure 4: (Color online) The probability to find a solution for coin sizes 5 (**a**) and 9 (**b**) in the case of different functions \(\zeta(\phi)\). The red dot-dashed line corresponds to Equation (16), the blue dotted line—to Equation (18) with \(\alpha=-1/(2\pi)\), the teal dashed line—to Equation (17), the solid green line—to Equation (18) with \(\alpha\) obtained by ML. Equation (17)). However, there are some asymmetries in the curves from the ML model. The lines in Figure 5c represent predictions of the neural network for QRWS algorithm with walk coin size \(m=16\). By comparing both ML extrapolations for \(m=11\) (Figure 5b) and \(m=16\) (Figure 5c), it is visible that all lines became steeper and there is an increase in the high probability \(p(\phi,m)\) plateau with the increase of \(m\), an observation supported by the MC simulations for lower coin register size (Figure 4). Due to higher computational time and memory demands, for coin size 12, we calculated fewer points for each of the probability curves (see Appendix B). However, those points were enough to confirm our predictions:  
1. Reducing the width of the curves with increasing of \(m\);
2. The suggested nonlinear dependence between angles (18) gives the highest stability of the algorithm. Worst performance is when \(\zeta=const\). Those results give us the premise to argue that the QRWS built with an alternative walk coin, proposed in [51] and thoroughly studied in this work, will give more robust implementation of the quantum algorithm for large coin register. Based on our numerical results, we predict that with increasing the coin size, the curves for probability \(p(\phi)\) given by Equation (18) with \(\alpha=-1/\left(2\pi\right)\) and \(\alpha=\alpha_{ML}(m)\) will become indistinguishable. Those curves will become steeper and closer to trapezoids. Simulations of the algorithm with coin constructed using relation (18) result in higher \(p_{max}\) in comparison with the standard construction (with constant phase factor \(\zeta=\pi\) as in Equation (17)). In Table 1, are shown the maxima of probability \(p(\phi,\zeta(\phi),m)\), for each of the functions \(\zeta(\phi)\) (Equation (16), Equation (17), and Equation (18) with \(\alpha=-1/\left(2\pi\right)\) and \(\alpha=\alpha_{ML}(m)\)) and coin size \(\phi\epsilon[0,2\pi]\). As the coin size increases, the probability to find the solution becomes higher. In the case of Equations (16) and (17) for coin size greater than 3 the maximum is always at the point with coordinates \(\phi=\pi\) and \(\zeta=\pi\) (corresponding to Grover coin). However, \(p_{max}\) is not always at this point if Equation (18) is used. For example, when \(\alpha=-1/2\pi\), relation (18) becomes \(\zeta=-2\phi+3\pi+\left(-1/\left(2\pi\right)\right)\sin(2\phi)\). It gives higher \(p(\phi)\) in comparison to the \begin{table}
\begin{tabular}{l c c c c c c c c} \hline \hline
**Line\textbackslash{}Coin Size** & **4** & **5** & **6** & **7** & **8** & **9** & **10** & **11** \\ \hline Equation (17) & 0.3906 & 0.4137 & 0.4117 & 0.4022 & 0.4344 & 0.4272 & 0.4334 & 0.4414 \\ \hline Equation (18) \& \(\alpha=0\) & 0.3906 & 0.4137 & 0.4117 & 0.4022 & 0.4344 & 0.4272 & 0.4334 & 0.4414 \\ \hline Equation (18) \& \(\alpha=(-2\pi)^{-1}\) & 0.3921 & 0.4137 & 0.4117 & 0.4082 & 0.4344 & 0.4279 & 0.4354 & 0.4414 \\ \hline Equation (18) \& \(\alpha=\alpha_{ML}\) & 0.3921 & 0.4137 & 0.4117 & 0.4093 & 0.4344 & 0.4277 & 0.4344 & 0.4414 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Maxima of probability to find solution by QRWS for different coin size when relation (17) is used are shown in the first row.