# Percolation Distribution in Small-World Quantum Networks
## 1 Introduction  
With the continuous research on quantum communication, the structure and scale of quantum networks are changing dramatically. Quantum networks are no longer limited to regular network shapes but are developing toward large-scale, long-range, multi-user complex networks [1; 2], which is determined by the future research and application prospects of quantum networks. The properties of quantum communication itself, such as Entanglement Swapping, also make it have better transmission performance than regular quantum networks under complex networks. Entanglement Swapping is the cornerstone of large-scale quantum networks. Relying on quantum repeaters in the network, Entanglement Swapping is able to overcome the exponential fading of quantum pairs over distance in the channel, allowing any two end nodes in the network to be connected. The connection between two nodes represents a pair of entangled qubits shared by these two nodes. Ideally, this pair of qubits would be in singlet state \(|\rho(\theta)\rangle=\cos(\theta)|00\rangle+\sin(\theta)|11\rangle\), where \(0\leq\theta\leq\frac{\pi}{4}\) so that complete transmission of information (\(p=1\)) is ensured [3]. However, in practical applications, the qubits in singlet state are affected by the environment, storage and channel noise, thus turning into a purely partially entangled state, making the transmission success probability \(p<1\), resulting in unstable transmission of information. In complex quantum networks, the above problems can be amplified geometrically, causing serious impact on communication. Therefore, how to convert partially entangled state to singlet state (maximally entangled state) via LOCC (Local Operations and Classical Communications) while building complex quantum networks is our key concern [4]. Similar to the traditional networks, quantum networks can be described in terms of graphs with similar properties. Specifically, a quantum network can be represented as \(G(V,E)\), where \(V=1,\ldots\ldots,N\) is the set of nodes and \(E\) is the set of edges contained in \(V\times V\), representing the communication nodes (including quantum relay nodes and endnodes) and the entangled connections between nodes in a quantum network, respectively. The entangled connections are in a singlet state in the ideal case and in a purely partially entangled state in the non-ideal case. For any node in the network, the number of nodes with which it has entangled connections is called the degree of the node, denoted as \(k\)[5]. Since we choose a small-world quantum network for research, the graphs under study also have a large clustering coefficient and a short average distance. Percolation is a common phenomenon in complex networks: There is a certain value \(p_{0}\) for the connection probability \(p\) between nodes in the network. When the connection probability is less than this value \((p<p_{0})\), the nodes are slowly connected to form numerous small-scale connection clusters, and when the connection probability exceeds this value \((p>p_{0})\), a Giant Connected Component (GCC), which basically runs through the whole network, will rapidly appear in the network. This certain value \(p_{0}\) is called the percolation threshold. For complex quantum networks, the percolation phenomenon occurs during the conversion from a partially entangled state to a maximally entangled state. When the conversion probability exceeds the percolation threshold of the network, we can quickly build a nearly fully connected quantum network, and this process is called Classical Entanglement Percolation (CEP) [6]. Studies of percolation in classical complex networks have shown that the structure of the network affects the value of the percolation threshold to a large extent. Similarly, in complex quantum networks, as the edge reconnection occurs after entanglement swapping between the nodes and newly connected edges replace the original ones, both the structure of the network and the percolation threshold of the network are changed.