# Quantum Walk on the Generalized Birkhoff Polytope Graph

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## Abstract

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## 1. Introduction

## 2. Linear Programming Problems, Polytopes, and Polytope Graphs

## 3. TLP, the Generalized Birkhoff Polytope, and GBPG

## 4. Classical and Quantum Walk, and Their Mixing Times

#### 4.1. RW on GBPG

#### 4.2. QW on GBPG

#### 4.2.1. QW

#### 4.2.2. Limiting Distribution and Mixing Time

## 5. Simulation and Numerical Results for RW and QW on GBPG

#### 5.1. Computational Platform and Instance Generation

^{TM}7702 benchmarked at 804 TFLOPS using 4 GB of RAM per node, running Linux CentOS version 8.1. The code used to calculate average limiting distributions and mixing times ran on Python 3.8.5.

#### 5.2. Classical Mixing Time

#### 5.3. Limiting Probability Distribution (Quantum Case)

#### 5.4. Quantum Mixing Time

## 6. Conclusions and Directions for Further Research

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Classical mixing time as a function of the number of sources m for $\u03f5$ from $0.01$ to $0.1$.

**Figure 2.**Limiting distribution ${\pi}_{v}$ of a quantum walk on GBPG for $m=6$ (left panel) and $m=7$ (right panel), and initial state (15) localized on node 1.

**Figure 3.**Limiting distribution ${\pi}_{v}$ of a quantum walk on GBPG for $m=6$ (left panel) and $m=7$ (right panel), using an initial state distributed on the clique $\{1,2,3,4\}$.

**Figure 4.**Log-log plot of the total variation distance between the instantaneous and limiting distributions $\parallel p\left(t\right)-\pi \parallel $ (upper black curve), and between the average and limiting distributions $\parallel \overline{p}\left(t\right)-\pi \parallel $ (lower blue curve) as a function of time steps for GBPG for $m=7$. The equation of the straight line is $0.7761/{t}^{1.0007}$.

**Figure 5.**Quantum mixing time as a function of the number of sources m for $\u03f5$ from $0.01$ to $0.1$.

**Figure 6.**Quantum mixing time as a function of the number of sources m for $\u03f5$ from $0.01$ to $0.1$ using the alternative initial state (16).

**Figure 7.**Instantaneous mixing time to the limiting distribution as a function of m for $\u03f5$ from $0.06$ to $0.4$ using the initial state (16).

Instance | m | N | Diam. | $1-{\mathit{\lambda}}_{1}$ |
---|---|---|---|---|

1 | 2 | 2 | 1 | 2 |

2 | 3 | 6 | 3 | $1/2$ |

3 | 4 | 12 | 3 | $1/3$ |

4 | 5 | 30 | 5 | $1/4$ |

5 | 6 | 60 | 5 | $1/5$ |

6 | 7 | 140 | 7 | $1/6$ |

7 | 8 | 280 | 7 | $1/7$ |

8 | 9 | 630 | 9 | $1/8$ |

9 | 10 | 1260 | 9 | $1/9$ |

10 | 11 | 2772 | 11 | $1/10$ |

11 | 12 | 5544 | 11 | $1/11$ |

12 | 13 | 12,012 | 13 | $1/12$ |

13 | 14 | 24,024 | 13 | − |

14 | 15 | 51,480 | 15 | − |

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**MDPI and ACS Style**

Cação, R.; Cortez, L.; de Farias, I.; Kozyreff, E.; Khatibi Moqadam, J.; Portugal, R.
Quantum Walk on the Generalized Birkhoff Polytope Graph. *Entropy* **2021**, *23*, 1239.
https://doi.org/10.3390/e23101239

**AMA Style**

Cação R, Cortez L, de Farias I, Kozyreff E, Khatibi Moqadam J, Portugal R.
Quantum Walk on the Generalized Birkhoff Polytope Graph. *Entropy*. 2021; 23(10):1239.
https://doi.org/10.3390/e23101239

**Chicago/Turabian Style**

Cação, Rafael, Lucas Cortez, Ismael de Farias, Ernee Kozyreff, Jalil Khatibi Moqadam, and Renato Portugal.
2021. "Quantum Walk on the Generalized Birkhoff Polytope Graph" *Entropy* 23, no. 10: 1239.
https://doi.org/10.3390/e23101239