# Continuous Variables Graph States Shaped as Complex Networks: Optimization and Manipulation
## 3 Discussion  
We have shown that, for CV cluster states with complex graphical representation the quality, measured as the mean of the variance of the nullifiers, can be better optimized when the quantity of entanglement links increases. This is probably due to the fact that, if the number of links is higher, there are more available ways, given by the newly introduced links, of distributing the larger values of the initial squeezing. We have analyzed in the quantum regime different complex shapes corresponding to different models of real-world networks. In the Barabasi-Albert and Erdos-Renyi models, regardless of the topology, clusters with a similar average degree \(\langle k\rangle\) have a comparable overall quality. The average degree \(\langle k\rangle\) in complex graphs could thus be used as a benchmark for the quality of the state implemented with the optimization protocol. Moreover, analyzing \"small-world\" networks that evolve from a regular network to a random network as their characteristic parameter \(p_{WS}\) increases, we found that randomness in the structure is detrimental to the quality of the state.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Graph** & **Between A and B** & **Same Team** \\ \hline
6-node grid & Yes & No \\
8-node grid & No & No \\
10-node grid & Yes & No \\ “X” & Yes & No \\ “Y” & Yes & No \\ Fully-connected & No & Yes \\ “Z” & No & No \\ Dual-rail & No & No \\ \hline \hline \end{tabular}
\end{table}
Table 4: Results on the possibility to create a quantum communication channel for the graphs of Figure 6, between nodes belonging to different teams A and B and between nodes belonging to the same team.  
Figure 6: Graphs analyzed with the aim of creating an EPR channel between [PERSON] (green) and [PERSON] (blue) (or eventually between nodes of the same team).  
The optimization procedure can be also used to concentrate the entanglement between two given nodes. Global optimizations of cluster states via linear optics unitaries can always be realized via analytical procedures.  
On the contrary, if the cluster is distributed to different locations and the players want to reshape it via local operations (quantum routing) with the aim of performing quantum communication protocols, we need to use a numerical procedure in order to deal with the larger number of constraints. In this case, we have used a Derandomized Evolutionary Strategy (DES) algorithm with a suitable fitness function, the one of Equation (11), in order to find solutions to generate an EPR state between two arbitrary nodes of a network shared between two parties, which can perform only local linear optics operations.  
We have studied small networks and we have found that it is possible to create EPR channels between two nodes of the two different teams for some network shapes, while creating an EPR channel from two nodes of the same team has never been possible, except for the case of the fully connected network. Thus, except for this last case, it has never been possible, by using only local linear optics operations, to disconnect two nodes of one player from the ones of the other player. It has to be stressed that, in the cases where no solution is found, we cannot conclude with certainty that the solution does not exist, as the DES algorithm can be stuck in a local minimum in the parameters space.  
In future works, we will investigate quantum routing operations when complex clusters are distributed between several parties by also adding, when linear optics operations are not sufficient, quadrature measurements. The measurement of the \(q\) quadrature of a mode of the cluster allows in fact for node removal, while the measurement of \(p\) is useful in wire shortening [22]: both can be used in order to cut the residual edges after the optimization via linear optics operations.