Hello I would like to clarify something about the process of encoding a graph into a GBS device. In paper such as this one it is said that once you have the doubled adjacency matrix \tilde{A} of the graph you can map that matrix to covariance matrix \Sigma of a pure Gaussian state through a series of matrix multiplications: \begin{equation} \Sigma = Q - \mathbb{I}_{2M}/2, \quad Q = (\mathbb{I} - X\tilde{A})^{-1}, \quad X = \begin{bmatrix} 0 & \mathbb{I}_{M}\\ \mathbb{I}_{M} & 0 \end{bmatrix} \end{equation}.\\ It is then said that the GBS device can be programmed according to this matrix \Sigma.

However in this paper the Takagi-Autonne decomposition of the adjacency matrix is taken to program the gates of the GBS. I would like to know if these methods are independent of each other? Can either one of them be used? And is there a reference that explains how to program the GBS device according to the matrix \Sigma as I can’t seem to find one in the first paper?