# Percolation Distribution in Small-World Quantum Networks
## 2 Percolation in Small-World Quantum Networks  
A general quantum network composed of quantum nodes and connections are shown in Figure 1. We consider a Watts-Strogatz (WS) small-world quantum network, as shown in Figure 2, where any connected edge in the network is composed of a pure partially entangled state, where \(|\varphi(\theta)\rangle=\cos(\theta)|00\rangle+\sin(\theta)|11\rangle(0\leq \theta\leq\frac{\pi}{4})\). \(|\psi(\theta)\rangle\) can be converted to a maximally entangled state \(\left|\varphi(\frac{\pi}{4})\right|\) with singlet conversion probability (SCP) \(p=min(1,2\sin^{2}\theta)\) by LOCC. \(\left|\varphi(\theta)^{\otimes 2}\right\rangle\) is composed of two pure partially entangled states \(|\varphi(\theta)\rangle\), and its optimal SCP is \(p_{1}=min(1,2p-\frac{p^{2}}{2})\)[8]. Similar to the classical WS operation, the connected edges in the network are converted to maximally entangled state with a certain probability, and the connected edges (maximally entangled state) can be established between any two nodes in the connected cluster through entanglement swapping [3].  
Before quantum percolation, quantum preprocessing will be performed [8; 9], and the basic operation of quantum preprocessing is q-swap [10], which performs an entanglement swapping on nodes with a degree of \(q\) in the network (shown in Figure 3). If an edge \(\ket{\varphi}\) has already undergone an entanglement swapping, then the newly generated edge is no longer \(\ket{\varphi}\), so it cannot participate in the swapping operation again. Its SCP is identical to \(\ket{\varphi}\). Different network structures change the percolation threshold of the network, therefore, quantum networks can change the network structure through quantum operations compared to classical ones [10].  
In regular networks, such as two-dimensional square lattice networks, triangular networks, cellular networks, etc., the nodes that perform preprocessing have the same degree and have no influence on each other when performing QEP. Unlike regular networks [4], most of the nodes in a WS quantum network have different degrees, and since  
Figure 1: A general quantum network. Each black dot represents a qubit and each edge represents a pure partially entangled state. The blue circles, which include multiple qubits, are nodes of the network. By performing Bell State measurement (BSM), new entanglement can be constructed between two qubits and new edges are formed between two nodes.  
Figure 2: Entanglement percolation of a \(10\)-node WS small-world quantum network. (**a**): A nearest-neighbor coupled network with four qubits per node, which are entangled with the qubits on the four neighbor nodes. Each edge in the network is a pure partially entangled state. (**b**): A WS small-world quantum network is established and a disconnect–reconnect operation is performed with probability \(p_{r}=0.6\). (**c**): The partially entangled states that form the connected edges in the WS small-world quantum network are converted to the maximally entangled states by LOCC with probability \(p_{r}=0.6\), and all the connected edges in the obtained network are maximally entangled state [6]. (**d**): An entanglement swapping is performed to disconnect the connected edges \((3,5)\)\((1,3)\) and connect \((1,5)\). A communication link is established between nodes \(1\) and \(5\) (red line).  
the edges after q-swap cannot be operated again, the q-swap of one node has an impact on neighboring nodes. WS small-world quantum networks have shorter average distances and larger clustering coefficients [11].