# A Machine Learning Study of High Robustness Quantum Walk Search Algorithm with Qudit Householder Coins

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## Abstract

**:**

## 1. Introduction

## 2. Quantum Random Walk Search with an Alternative Walk Coin

#### 2.1. Quantum Random Walk Search Algorithm—Quantum Circuit

#### 2.2. Walk Coin by a Householder Reflection and an Additional Phase Multiplier

## 3. Qrws with Qudit Coin Constructed by Householder Reflection

#### 3.1. Monte Carlo Simulations of Qrws

#### 3.2. Robustness of the Coin for Different Functions $\zeta \left(\varphi \right)$

- (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 $\zeta =const$.

## 4. Numerical Results

#### 4.1. Region of Stability for Different Coin Sizes

#### 4.2. Analysis of Algorithm’S Robustness

#### 4.3. Dependence between Alpha and Coin Size

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

QRW | Quantum Random Walk |

QRWS | Quantum Random Walk Search |

HR | Householder Reflection |

ML | Machine Learning |

DNN | Deep Neural Network |

Eq. | Equation |

## Appendix A. Monte Carlo Simulations for Different Coin Size

**Figure A1.**(Color online) Monte Carlo simulation of the probability to find solution p of QRWS algorithm with a coin constructed by a generalized HR and a phase multiplier. The coordinate axes correspond to the phases $\varphi $ and $\zeta $. In the first row are images for coin size $m=2,3,4$ (

**a**–

**c**), on the second—for $m=5,6,7$ (

**d**–

**f**), and on the third—for $m=8,9,10$ (

**g**–

**i**). Higher probability is shown with darker color.

## Appendix B. Curves for Different Qudit Size

**Figure A2.**(Color online) Probability to find solution $p(\varphi ,m)$ for different coin size $2\le m\le 10$ and $m=$ 12. In the first row are images for coin size $m=2,3$ (

**a**,

**b**), on the second—for $m=4,5$ (

**c**,

**d**), on the third—for $m=6,7$ (

**e**,

**f**), on the fourth—for $m=8,9$ (

**g**,

**h**), and on the fifth—for $m=10,12$ (

**i**,

**j**). Different curves correspond to different relations between coin parameters: the red dot-dashed line—to Equation (16), the blue dotted line—to Equation (18) with $\alpha =-1/\left(2\pi \right)$, the teal dashed line—to Equation (17), the green line—to Equation (18) with $\alpha $ obtained by ML.

## Appendix C. Region of Stability for Different Coin Sizes—Practical Consideration

**Figure A3.**(Color online) Change of the width of the area with probability to find a solution greater than $0.37$ (

**a**) and greater than $0.31$ (

**b**). The red curve (4-pointed star marker) corresponds to Equation (16), the teal (3-pointed star marker) to Equation (17) and the blue (5-pointed star marker) and the green (2-pointed star marker) to Equation (18) with $\alpha =-1/\left(2\pi \right)$ and $\alpha ={\alpha}_{ML}\left(m\right)$ correspondingly.

## Appendix D. Area of Stability of QRWS as Function of α and ϕ

**Figure A4.**(Color online) Probability to find solution p as function of $\varphi $ and the parameter $\alpha $ in the non-linear Equation (18). The solid green line corresponds to $\alpha ={\alpha}_{ML}\left(m\right)$, the dash-doted blue—to $\alpha =-1/\left(2\pi \right)$ and dashed teal—to $\alpha =0$. The lighter colors show higher probability and darker—lower. The scale is between 0 and ${p}_{max}$, and the contours are at intervals $5\%\times {p}_{max}$. The probabilities larger than $95\%\times {p}_{max}$ are depicted with the lightest color. Each picture corresponds to a different value of $m\in [2,10]$. In the first row from left to right are given the images for $m=2,3,4$ (

**a**–

**c**), on the second—$m=5,6,7$ (

**d**–

**f**), and on the last—$m=8,9,10$ (

**g**–

**i**).

**Figure A5.**(Color online)Probability to find solution p as function of $\varphi $ and the parameter $\alpha $ in the non-linear Equation (18). The solid green line corresponds to $\alpha ={\alpha}_{ML}\left(m\right)$, the dash-doted blue—to $\alpha =-1/\left(2\pi \right)$ and dashed teal—to $\alpha =0$. The lighter colors show higher probability and darker—lower. The scale is from 0 to ${p}_{max}$, and the contours are at fixed values of p (between 0.037 and 0.37). Each picture corresponds to a different value of $m\in [2,10]$. In the first row from left to right are given the images for $m=2,3,4$ (

**a**–

**c**), on the second—$m=5,6,7$ (

**d**–

**f**), and on the last—$m=8,9,10$ (

**g**–

**i**).

## Appendix E. Robustness of the Modified Quantum Random Walk Algorithm for Different Size of the Walk Coin

**Figure A6.**(Color online) Root-mean-square deviation of the probability $p\left(\varphi ,\alpha ,m\right)$, given with Equation (19), for walk coin of size $m=2,3,4$ (

**a**–

**c**), $m=5,6,7$ (

**d**–

**f**), and $m=8,9,10$ (

**g**–

**i**). The dark central area represents high stability region of the quantum algorithm for small deviations of the algorithm’s parameters $\varphi $ and $\alpha $. The dashed gray, the dash-dotted blue, and the solid green lines correspond to $\alpha =0$, $\alpha =-1/\left(2\pi \right)$, and $\alpha ={\alpha}_{ML}$.

**Figure A7.**(Color online) Relative stability of the QRWS algorithm with walk coin size $m=2,3,4$ (

**a**–

**c**), $m=5,6,7$ (

**d**–

**f**), and $m=8,9,10$ (

**g**–

**i**), given by the ratio of the root-mean-square uncertainty with and without the proposed optimization of the coin. The chosen ranges of the parameter $\varphi $ correspond to the high probability central peaks of $p(\varphi ,\zeta =\pi ,\phantom{\rule{4pt}{0ex}}m)$ (Equation (17)). The dashed gray, dash-dotted blue, and the solid green lines correspond to $\alpha =0$, $\alpha =-1/\left(2\pi \right)$, and $\alpha ={\alpha}_{ML}$.

## Appendix F. Deep Network Model and Machine Learning Predictions

**Figure A8.**(Color online) Scheme of the Deep neural network used for prediction of the probability $p(\phi ,\zeta ,m\ge 11)$.

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**Figure 2.**(Color online) Probability to find solution for coin sizes $m=5$ (

**a**) and $m=9$ (

**b**). The probability p is plotted as a function of the angles $\varphi $ and $\zeta $.

**Figure 3.**(Color online) Probability to find solution $p(\varphi ,\zeta ,m)$ for coin sizes $m=11$ (

**a**,

**b**) and $m=16$ (

**c**). The image (

**a**) shows Monte Carlo simulation results and images (

**b**,

**c**) show the ML predictions.

**Figure 4.**(Color online) The probability to find a solution for coin sizes 5 (

**a**) and 9 (

**b**) in the case of different functions $\zeta \left(\varphi \right)$. The red dot-dashed line corresponds to Equation (16), the blue dotted line—to Equation (18) with $\alpha =-1/\left(2\pi \right)$, the teal dashed line—to Equation (17), the solid green line—to Equation (18) with $\alpha $ obtained by ML.

**Figure 5.**(Color online) The probability to find a solution $p(\varphi ,m)$ for different functions $\zeta \left(\varphi \right)$: the red dot-dashed line corresponds to Equation (16), the blue dotted line—to Equation (18) with $\alpha =-1/\left(2\pi \right)$, the teal dashed line—to Equation (17), the green line—to Equation (18) with $\alpha $ obtained by ML. (

**a**) shows results from Monte Carlo simulations for coin size $m=11$, (

**b**)—ML predictions for $m=11$, and the one on (

**c**)—ML predictions for $m=16$.

**Figure 6.**(Color online) Change of the area width $\epsilon $, 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 $\alpha =-1/\left(2\pi \right)$ and $\alpha ={\alpha}_{ML}\left(m\right)$ correspondingly. The solid lines correspond to values obtained by MC simulations, dashed to—results prognosed by ML.

**Figure 7.**(Color online) Probability of QRWS algorithm to find solution $p\left(\varphi ,\alpha ,m\right)$ as a function of $\alpha $ and $\varphi $ 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, $\alpha =-1/\left(2\pi \right)$—dash-dotted blue line, and the solid green line correspond to ${\alpha}_{ML}\left(m\right)$ given in Table 2.

**Figure 8.**(Color online) R.m.s. deviation of the probability $p\left(\varphi ,\alpha ,m\right)$, given by Equation (19), for walk coin of size $m=5$ (on (

**a**)) and $m=9$ (on (

**b**)). The dark central area represents high robustness of the quantum algorithm for small deviations of the algorithm’s parameters. The dashed gray, dash-dotted blue, and the solid green lines correspond to $\alpha =0$, $\alpha =-1/\left(2\pi \right)$, and $\alpha ={\alpha}_{ML}$.

**Figure 9.**(Color online) Relative stability of the QRWS algorithm with walk coin size $m=5$ (

**a**) and $m=9$ (

**b**), given by the ratio of the root-mean-square uncertainty with and without the proposed optimization of the coin (Equation (19) and Equation (20), respectively). The chosen ranges of the parameter $\varphi $ correspond to the high probability central peaks of $p(\varphi ,\zeta =\pi ,\phantom{\rule{4pt}{0ex}}m)$ (Equation (17)). The dashed gray, the dash-dotted blue, and the solid green lines are obtained from Equation (18) with $\alpha =0$, $\alpha =-1/\left(2\pi \right)$, and $\alpha ={\alpha}_{ML}\left(m\right)$.

**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 ${\alpha}_{1}\left(m\right)$, an equal weight of all data points was suggested. The dot-dashed green line ${\alpha}_{2}\left(m\right)$ 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 ${\alpha}_{3}\left(m\right)$ (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 ${\alpha}_{ML}\left(11\right)$ is higher than the weight of ${\alpha}_{ML}\left(20\right)$). The horizontal solid line figure corresponds to $\alpha \left(m\right)=-1/\left(2\pi \right)=const$.

**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. Analogically, p corresponding to Equation (18), with $\alpha =0$ is shown in the second row, and with $\alpha =-1/\left(2\pi \right)$—on the third row. The fourth row corresponds to Equation (18) with $\alpha $ obtained by ML.

Line∖Coin Size | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 |
---|---|---|---|---|---|---|---|---|

Equation (17) | 0.3906 | 0.4137 | 0.4117 | 0.4022 | 0.4344 | 0.4272 | 0.4334 | 0.4414 |

Equation (18) & $\alpha =0$ | 0.3906 | 0.4137 | 0.4117 | 0.4022 | 0.4344 | 0.4272 | 0.4334 | 0.4414 |

Equation (18) & $\alpha ={(-2\pi )}^{-1}$ | 0.3921 | 0.4137 | 0.4117 | 0.4082 | 0.4344 | 0.4279 | 0.4354 | 0.4414 |

Equation (18) & $\alpha ={\alpha}_{ML}$ | 0.3921 | 0.4137 | 0.4117 | 0.4093 | 0.4344 | 0.4277 | 0.4344 | 0.4414 |

**Table 2.**Values of the parameter ${\alpha}_{ML}\left(m\right)$ in Equation (18) for QRWS algorithm’s walk coin size $2\le m\le 20$. The primed values are derived from ML predictions, and the remaining come from Monte Carlo simulations.

m | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
---|---|---|---|---|---|---|---|

$-{\alpha}_{ML}$ | 0.558 | 0.552 | 0.142 | 0.155 | 0.163 | 0.209 | 0.206 |

m | 9 | 10 | 11 | 11 | 12 | 13 | 14 |

$-{\alpha}_{ML}$ | 0.185 | 0.168 | 0.150 | 0.170${}^{\prime}$ | 0.179${}^{\prime}$ | 0.180${}^{\prime}$ | 0.203${}^{\prime}$ |

m | 15 | 16 | 17 | 18 | 19 | 20 | |

$-{\alpha}_{ML}$ | 0.225${}^{\prime}$ | 0.197${}^{\prime}$ | 0.205${}^{\prime}$ | 0.206${}^{\prime}$ | 0.216${}^{\prime}$ | 0.223${}^{\prime}$ |

**Table 3.**Values of the parameters a, b, and c for the fitting functions ${\alpha}_{i}\left(m\right),i=1,2,3$ (Equation (21)). The bolded last line represents the most reliable fit.

a | b | c | |
---|---|---|---|

${\alpha}_{1}\left(m\right)$ | $4.85\times {10}^{-5}$ | $-4.78\times {10}^{-3}$ | $-1.41\times {10}^{-1}$ |

${\alpha}_{2}\left(m\right)$ | $2.71\times {10}^{-4}$ | $-9.82\times {10}^{-3}$ | $-1.20\times {10}^{-1}$ |

${\alpha}_{3}\left(m\right)$ | $\mathbf{1}.\mathbf{17}\times {\mathbf{10}}^{-\mathbf{4}}$ | $-\mathbf{6}.\mathbf{24}\times {\mathbf{10}}^{-\mathbf{3}}$ | $-\mathbf{1}.\mathbf{35}\times {\mathbf{10}}^{-\mathbf{1}}$ |

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**MDPI and ACS Style**

Tonchev, H.; Danev, P.
A Machine Learning Study of High Robustness Quantum Walk Search Algorithm with Qudit Householder Coins. *Algorithms* **2023**, *16*, 150.
https://doi.org/10.3390/a16030150

**AMA Style**

Tonchev H, Danev P.
A Machine Learning Study of High Robustness Quantum Walk Search Algorithm with Qudit Householder Coins. *Algorithms*. 2023; 16(3):150.
https://doi.org/10.3390/a16030150

**Chicago/Turabian Style**

Tonchev, Hristo, and Petar Danev.
2023. "A Machine Learning Study of High Robustness Quantum Walk Search Algorithm with Qudit Householder Coins" *Algorithms* 16, no. 3: 150.
https://doi.org/10.3390/a16030150