## 5 Conclusions  
In this paper, we studied a modified QRWS algorithm with improved stability. It uses an alternative walk coin constructed by a HR and an additional phase multiplier. In this work, the more general qudit walk coin is studied. An implementation of the modified QRWS algorithm with qudits has the potential to have numerous advantages. For example, qudits' robustness against noise, combined with the results presented in this paper, could make the quantum algorithm extremely stable to deviations in various experimental settings. Monte Carlo simulations of the QRWS algorithm for different sizes of the coin register were conducted. This allowed us to study in more detail the behavior of the proposed modification with increasing edge register's dimensions. It also enabled us to make extrapolations for the QRWS algorithm's stability at larger coin sizes. We have shown that, if a proper relation between the coin parameters is maintained, the quantum algorithm becomes much more robust, and this property remains true for every size of the quantum algorithm registers studied in the paper. We investigated the behavior of practically important quantities, such as the width of the area, resulting in high probability to find a solution and the maximum of this probability, for different coin parameters and relations between them. The dependence of nonlinear parameters (introduced in  
Figure 10: (Color online) Dependence of the parameter \(\alpha_{ML}\) on the QRWS algorithm’s coin size \(m\). By fitting the values of \(\alpha_{ML}\) obtained from MC simulations’ data (teal dots) and ML predictions (orange triangles) three fits were derived. For the dotted blue line \(\alpha_{1}(m)\), an equal weight of all data points was suggested. The dot-dashed green line \(\alpha_{2}(m)\) is the fitting function, when the alpha values from the DNN model are considered to be ss reliable (the fitting weights were set two times smaller) than the ones from the Monte Carlo simulations’ data. To obtain the function \(\alpha_{3}(m)\) (the dashed red line) we have suggested that the ML predictions are more reliable for points close to the training area (i.e., the weight of \(\alpha_{ML}(11)\) is higher than the weight of \(\alpha_{ML}(20)\)). The horizontal solid line figure corresponds to \(\alpha(m)=-1/(2\pi)=const\). the relation between coin phases) on the coin size was also studied by Monte Carlo and ML methods. Optimal functional dependencies and parameter values are derived for a walk coin of size up to one qudit with 11 states. Even more, by using ML methods, we made extrapolations of the mentioned quantities for larger sizes of the coin. From an analysis of the Monte Carlo data and ML predictions, we show that the QRWS algorithm, with the proposed alternative walk coin, demonstrates high robustness to deviations of its parameters. We calculated the stability of the proposed alternative construction of the QRWS algorithm. The numerical results show that there exists a wide area in the space spanned by the walk coin's parameters with extremely low root-mean-square uncertainty of the probability of QRWS algorithm to find a solution. The relative uncertainty of our modification of the quantum algorithm to the uncertainty of QRWS algorithm with a standard walk coin has been studied. We show that the former is more robust to coin parameter's deviations and also that the relative stability increases for a larger coin register. These authors contributed equally to this work. All authors have read and agreed to the published version of the manuscript. This research was funded by Bulgarian National Science Fund under Grant _KP-_06-_M_48/2/26.11.2020. The datasets generated during and analyzed in the current study are available from the corresponding author on reasonable request. The work on this paper was supported by the Bulgarian National Science Fund under Grant _KP-_06-_M_48/2/26.11.2020. The authors declare no conflict of interest. The following abbreviations are used in this manuscript: