# Percolation Distribution in Small-World Quantum Networks
## 4 Concurrence Percolation in Small-World Quantum Networks
As shown in Figure 10, the number of nodes, reconnection probability, and average degree are the same for both WS quantum networks. It is assumed that all connected edges \(|\varphi(\theta)\rangle\) in these two networks have \(\theta=0.2\frac{\pi}{4}\) and percolation from any node in the network will have little effect on the final obtained. Take the ConPT method as an example: We start from ConPT \(0\to 2\) in sequence with series-parallel rule and SM rule. \(a\) obtained \((1,3)\theta_{SC}=0.063\frac{\pi}{4}\) \begin{table}
\begin{tabular}{c c c} \hline \hline
**Rules** & **Classical** & **ConPT** \\ \hline series & \(p=p_{1}p_{2}\cdots\) & \(c=c_{1}c_{2}\cdots\) \\ parallel & \(p=1-(1-p_{1})(1-p_{2})\cdots\) & \(\frac{1+\sqrt{1-c^{2}}}{2}=\max\{\frac{1}{2},\frac{1+\sqrt{1-c^{2}_{1}}}{2} \frac{1+\sqrt{1-c^{2}_{2}}}{2}\cdots\}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Connectivity rules. while \(b\) obtained \(\theta_{SC}=0.016\frac{\pi}{4}\). For the same parameters, different network topologies also obtain different \(\theta_{SC}\), which affects the percolation threshold. In order to analyze the effect of disconnection-reconnection probability \(p_{re}\) on the percolation threshold in WS quantum networks, \(p_{re}=0.3,0.5,0.8\) are selected. As shown in Figure 11, when \(p_{re}=0.3\), \(C_{th}=0.347\), \(p_{th}=0.55\) and when \(p_{re}=0.8\), \(C_{th}=0.233\), \(p_{th}=0.39\). With the increase in \(p_{re}\), the percolation threshold of the network decreases gradually whether the network is percolated using the ConPT or classical method. The effect of the average degree on percolation threshold in a WS quantum network is also considered. Take a WS quantum network with an average degree of \(4\) (\(\bar{k}=4\)) as an example, as shown in Figure 12, \(C_{th}=0.405\), \(p_{th}=0.589\). Compared to a WS quantum network with an average degree of \(6\) (\(\bar{k}=6\)):  
\[\Delta C=\frac{C_{th(\bar{k}=4)}-C_{th(\bar{k}=6)}}{C_{th(\bar{k}=4)}}=37.5 \%,\ \Delta p=\frac{p_{th(\bar{k}=4)}-p_{th(\bar{k}=6)}}{p_{th(\bar{k}=4)}}=31.1\%. \tag{4}\]  
It can be seen that for WS quantum networks, the percolation threshold obtained after ConPT method has a greater rate of change compared to that after classical method. For an Erdos-Renyi (ER)) network, an average degree of \(6\) (\(\bar{k}=6\)) is selected. By simulation, we can restore the percolation threshold using the classical method \(p_{th}=\frac{1}{\bar{k}}=0.167\). While using the ConPT method, \(C_{th}=0.085\) is obtained, and the percolation threshold is reduced by \(\gamma=\frac{p_{th}-C_{th}}{p_{th}}=49\%\). It is shown in Figure 13 that the ConPT method can take advantage of concurrency and perform entanglement metric on quantum states to effectively reduce the percolation threshold of the network. Figure 10: Two kinds of topology networks (\(N=4\), \(p_{re}=0.5\), \(\bar{k}=2\)). (a) Ring topology; (b) Mesh topology. Figure 9: Comparison between classical percolation and ConPT: the percolation threshold using classical method (red vertical line) and the percolation threshold using ConPT method (blue vertical line) can be obtained by Monte Carlo simulation. Figure 11: Effects of (**a**) \(p_{re}=0.3\) and (**b**) \(p_{re}=0.8\) on the percolation threshold of WS quantum networks. Figure 12: ConPT and classical percolation in a WS quantum network with average degree \(\bar{k}=4\).