# Continuous Variables Graph States Shaped as Complex Networks: Optimization and Manipulation
## 2 Results
### Creating a Quantum Channel between Nodes by Manipulating Existing Networks
The transformation acting on the whole set of quadratures of the modes then reads  
\[S= \begin{pmatrix}\text{Re}(U_{A})&0&-\text{Im}(U_{A})&0\\ 0&\text{Re}(U_{B})&0&-\text{Im}(U_{B})\\ \text{Im}(U_{A})&0&\text{Re}(U_{A})&0\\ 0&\text{Im}(U_{B})&0&\text{Re}(U_{B})\end{pmatrix} , \tag{6}\]  
where \(U_{A}\) and \(U_{B}\) are two unitary matrices parametrized respectively by \(n^{2}\) and \(p^{2}\) parameters [40]. A method for the generation of numerically random unitary matrices is presented in [41]. If we define \(\sigma_{1}\) as the covariance matrix of the cluster we are given and \(\sigma_{2}\) as the covariance matrix of the cluster we obtain after the transformation,  
\[\sigma_{2}=S\sigma_{1}S^{T} \tag{7}\]  
holds, where \(S\) is defined in Equation (6). Our goal is to find the two matrices \(U_{A}\) and \(U_{B}\), whose real and imaginary parts define \(S\), such that \(\sigma_{2}\) is of the desired form. In our case, we want \(\sigma_{2}\) to be such that, for two given nodes, we get an EPR channel. In this case, it is hard to find an analytical solution to the problem, so we use a Derandomized Evolution Strategy (DES) algorithm to explore the parameter space [42]. If \(n_{1}\) and \(n_{2}\) are the nodes out of which we want to obtain an EPR channel, we can define  
\[\sigma_{2,red}(n_{1},n_{2})= \begin{pmatrix}\sigma_{2,n_{1}}\\ \sigma_{2,n_{2}}\\ \sigma_{2,n_{2}+n+p}\\ \sigma_{2,n_{2}+n+p}\end{pmatrix}  \tag{8}\]  
as the reduced \(4\times 2(n+p)\) matrix obtained by selecting only the rows of \(\sigma_{2}\) we want to set as a quantum channel. We thus define the \(4\times 2(n+p)\) matrix \(\sigma_{channel}\) as the matrix with null entries except for  
\[\sigma_{1,n_{1}}=\sigma_{2,n_{2}}=\sigma_{3,n_{1}+n+p}=\sigma_{4, n_{2}+n+p}=\alpha, \tag{9}\] \[\sigma_{1,n_{2}+p+n}=\sigma_{2,n_{1}+p+n}=\sigma_{3,n_{2}}=\sigma _{4,n_{1}}=\beta, \tag{10}\]  
where the subscript \"channel\" has been omitted for simplicity and where \(\alpha\) and \(\beta\) are numerical values that we fix according to the squeezing we want to attain on the modes of the quantum channel. This squeezing cannot exceed the squeezing of the input states of the cluster. For simplicity, we worked with an equal value of squeezing of -10 dB, both for the input squeezed states used to implement the cluster and for the squeezing chosen for the quantum channel. We will search for the minimal value of the function  
\[f_{opt}=||\sigma_{channel}-\sigma_{2,red}(n_{1},n_{2})||, \tag{11}\]  
where \(||\cdot||\) indicates the Frobenius norm. In Table 4, we show preliminary results on different graphs with a restricted number of nodes, shown in Figure 6. For the 6-mode and 10-mode \"grid\" graphs and the graphs \"X\" and \"Y\", a result can be found for the creation of a quantum channel between [PERSON] and [PERSON]. For these structures,  
Figure 5: (**a**) a quantum network is created; (**b**) the resource is distributed to two spatially separated teams, [PERSON] and [PERSON]; (**c**) [PERSON] performs a linear optics operation \(U_{A}\) on her set of nodes and [PERSON] performs a linear optics operation \(U_{B}\) on his set of nodes to create a quantum channel out of two given nodes; (**d**) the quantum channel is established. however, it was not possible to find a solution for the creation of a quantum channel between nodes of the same team. The opposite stands for the fully-connected cluster, for which it was possible to create a channel between nodes of the same team but not between nodes of different teams. Lastly, for the graph \"Z\", which represents two 3-mode cluster states distributed to the two parties, for the dual-rail and for the 8-mode \"grid\" graph, no solution was found. As an example, the results of the fully-connected graph and of the 6-mode \"grid\" graph of Figure 6 are shown in Appendix A.