# Continuous Variables Graph States Shaped as Complex Networks: Optimization and Manipulation
## 2 Results  
### Background: Cluster States and Complex Networks  
Ideal cluster states are built starting from a set of modes of light, which are placed in the zero-momentum eigenstate \(|0\rangle_{p}\), i.e., infinitely squeezed vacuum states along the momentum quadrature \(p\). Entangling \(C_{Z}=\exp(\imath\hat{q}_{i}\otimes\hat{q}_{j})\) gates are then applied between couples of modes \(i\) and \(j\), according to a given configuration, which can be represented by a graph. A graph is defined by a set \(\{\mathcal{V},\mathcal{E}\}\), where \(\mathcal{V}\) is the set of nodes (vertices) and \(\mathcal{E}\) are the edges. In the graphical representation of the cluster, the nodes represent different modes of light, while the edges between the nodes are the entanglement correlations given by the application of the \(C_{Z}\) gates. The graph is characterized by the adjacency matrix \(V\) whose elements \(V_{ij}\) are set to 1 when an entangling gate has been applied between the two nodes \(i\) and \(j\), and 0, otherwise. An ideal cluster state of \(N\) modes with adjacency matrix \(V\) is given by  
\[|\Psi_{C}\rangle=\prod_{1\leq i<j<N}\exp(\imath V_{ij}q_{i}\otimes q_{j})|0 \rangle_{p}^{\otimes N}. \tag{1}\]  
Cluster states are characterized by a particular set of operators, called _nullifiers_, that read  
\[\hat{\delta}_{i}=\hat{p}_{i}-\sum_{j\in\mathcal{N}(i)}\hat{q}_{j},\quad\text{ such that}\quad\hat{\delta}_{i}\,|\Psi_{C}\rangle=0, \tag{2}\]where \(\hat{\delta}_{i}\) and \(\mathcal{N}(i)\) denote respectively the nullifier and the set of the nearest neighbours of the \(i\)-th node. Cluster states are eigenstates of the nullifiers with zero eigenvalue, so, for an ideal cluster, the following condition holds: \(\Delta^{2}\delta_{i}=0.\) In addition, being Gaussian states, cluster states are fully described by mean values of quadratures and covariance matrices [29]. By defining the vector \(\hat{X}=\{\hat{q}_{1},...,\hat{q}_{N},\hat{p}_{1},...,\hat{p}_{N}\}\), cluster states are identified by mean values \(\langle\hat{X}_{i}\rangle=0\) and covariance matrix \(\sigma\) with elements \(\sigma_{ij}=(1/2)\langle(\hat{X}_{i}\hat{X}_{j}+\hat{X}_{j}\hat{X}_{i})\rangle.\)  
As already said, in a realistic situation, CV cluster states are always imperfect, due to the impossibility of reaching an infinite amount of squeezing to obtain the zero momentum eigenstate. Approximated clusters, when used in measurement-based quantum computing protocols, at each step of the computation, introduce a certain amount of noise that can be quantified with the variance of the nullifiers of the cluster [22]. Therefore, the smaller the value of \(\Delta^{2}\delta_{i}\) is for every node, the better the quality of cluster is. Instead of acting with \(C_{Z}\) gates on multimode squeezed vacuum states, as in the original formulation, cluster states can be implemented via linear optics transformations on the multimode squeezed vacuum state [30; 31; 32]. This is the implementation that is pursued in experimental setups [18; 19; 20; 21], as it is easier and less costly than the implementation of \(C_{Z}\), which requires online squeezing. Linear optics manipulations are described by unitary transformations \(U\) on the creation and annihilation operators of the different light modes, so that the annihilation operator of the mode \(i\) is transformed as \(\hat{a}_{i}-\sum_{i}U_{ji}\hat{a}_{i}\). This corresponds to orthogonal symplectic transformations \(S\) acting on the quadratures, which transform the covariance matrix as \(\sigma\to 5\sigma S^{\mathrm{T}}\). To implement a cluster with adjacency matrix \(V\), we can apply the following unitary operation:  
\[U=(\mathrm{I}+\imath V)(V^{2}+\mathrm{I})^{-\frac{1}{2}}O, \tag{3}\]  
where \(\mathrm{I}\) is the identity matrix and \(O\) is an arbitrary real orthogonal matrix. This orthogonal matrix provides supplementary \(N(N-1/2)\) degrees of freedom that can be exploited to optimize given properties of the cluster [25].