# Dynamics of Quantum Networks in Noisy Environments

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## Abstract

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## 1. Introduction

## 2. Preliminaries

#### 2.1. Quantum Network Model

#### 2.2. Percolation Model

## 3. The Evolution of Quantum Network

#### 3.1. Analytical Framework

#### 3.2. Amplitude Damping and Phase Damping Noises

#### 3.3. Regular Quantum Networks

#### 3.4. Complex Quantum Networks

## 4. The Capacity of Quantum Networks

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

QKD | Quantum Key Distribution |

GCC | Giant Connected Component |

FCC | Finite Connected Component |

NMR | Nuclear Magnetic Resonance |

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**Figure 1.**Entanglement swapping in the quantum network. Alice and Bob can establish a quantum channel through two steps. Step $\left(a\right)$ is selecting a path from a quantum network, such as $A\to R\to B$ or $A\to {R}_{1}\to {R}_{2}\to \cdots \to {R}_{N}\to B$. Step $\left(b\right)$ is that intermediate nodes perform Bell state measurements to build a long-distance entanglement between the communication parties.

**Figure 2.**Schematic diagrams of bond percolation and site percolation. (

**a**) Bond percolation; (

**b**) Site percolation.

**Figure 3.**The fidelities of Bell states under amplitude damping and phase damping noises. For simulation, we consider the following parameters as identical: ${\tau}_{1}={\tau}_{2}={\gamma}_{1}={\gamma}_{2}=\tau $.

**Figure 6.**The distribution of the size of GCC for square quantum networks under amplitude damping and phase damping noises. The simulations were performed with $N=M\times M,M=50,100,120.$

**Figure 7.**The critical times for triangle, square and honeycomb quantum networks. The three intersection points are $({F}_{1},{T}_{1})=(0.6527,0.0810)$, $({F}_{2},{T}_{2})=(0.5,0.2117)$ and $({F}_{3},{T}_{3})=(0.3473,1.4044)$, respectively.

**Figure 8.**The graphical representation of the distribution of connected components generated by ${H}_{1}\left(x\right)$.

**Figure 9.**The distribution of the size of GCC for ER quantum networks under amplitude damping and phase damping noise.

**Figure 10.**Flow chart of finding all the paths between the source and destination nodes by using a greedy algorithm.

Lattice | Bond Percolation |
---|---|

1d-Chain | 1 |

2d-Honeycomb | 1 − 2sin($\pi $/18) ≈ 0.6527 |

2d-Square | 0.5 |

2d-Triangle | 2sin($\pi $/18) ≈ 0.3473 |

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**MDPI and ACS Style**

Zhang, C.-Y.; Zheng, Z.-J.; Fei, S.-M.; Feng, M. Dynamics of Quantum Networks in Noisy Environments. *Entropy* **2023**, *25*, 157.
https://doi.org/10.3390/e25010157

**AMA Style**

Zhang C-Y, Zheng Z-J, Fei S-M, Feng M. Dynamics of Quantum Networks in Noisy Environments. *Entropy*. 2023; 25(1):157.
https://doi.org/10.3390/e25010157

**Chicago/Turabian Style**

Zhang, Chang-Yue, Zhu-Jun Zheng, Shao-Ming Fei, and Mang Feng. 2023. "Dynamics of Quantum Networks in Noisy Environments" *Entropy* 25, no. 1: 157.
https://doi.org/10.3390/e25010157