# Advantages of Density in Tensor Network Geometries for Gradient-Based Training
## 4 Results  
The training described in the previous sections reveals that the connectivity of the tensor network structure directly affects the maximum precision that they can reach. In Figure 4, we show how the infidelities for simulations of a system of \(n=12\) sites approach 0, and thus improve, as we increase the bond dimension of the structure. They plateau when the maximal bond dimension \(\chi_{S}\) is reached, which is to be expected from the point of view of the complexity of the quantum state, but it proves that the redundant information does not help the training procedure. In the inset, we can see how this effect correlates with the maximum distance between nodes in the structure, as highlighted in Figure 1. This means that structures that are more dense, in the sense of proximity between nodes, perform better in the training. This is the case even if the bond dimension is high enough to represent the target state perfectly (\(\chi=\chi_{S}\)), for which ideally, the infidelity should decrease to machine precision for any structure. We also see that increasing the amount of information that the TN can store, which we do by using bond dimensions \(\chi\) beyond \(\chi_{S}\), does not help the training and, in fact, hurts the success rate, as we see below. For the PEPS, we observe that the first data point with good convergence happens for a large \"_size of largest tensor_\". This is due to the structure of the PEPS, as the largest tensor grows as \(\chi^{4}\). Since the bond dimension must be an integer, this number jumps from being small to \(\chi^{4}>2^{n}\). For a tensor in a tree structure, each bond dimension \(\chi\) is bounded by the number of nodes it connects to, meaning that the largest tensor _is_ bounded by \(2^{n}\), explaining the vertical alignment of the points. In Figure 5, we show a training with larger geometries (\(n=16\)) and plot the infidelities against bond dimension. The denser structures achieve high fidelity for low bond dimensions, whereas those that are less dense need to reach \(\chi=\chi_{S}\). Beyond underlining the previous conclusions on infidelity vs. structure density, we can see that even with 400 trials, the MPS training fails to converge, showing its shortcomings in this kind of training and agreeing with current research around barren plateaus (discussed in Section 2.5). The convergence of the training with \(\chi=128\) and \(\chi\geq 256\) shows that this is not due to the structure not being able to store enough information, but to the training process. In Figure 5b, we plot the number of trainings that cross the threshold of \(\mathcal{I}=10^{-3}\), seeing that they decrease with bond dimension. For the most powerful structures (dense, star1, and star2), the decrease is not meaningful. For the rest, the decrease is more pronounced in the regime where \(\chi>\chi_{S}=256\), showing that the additional information in the structure makes the training harder, even though, as seen above, this does not translate into improved infidelity. The compactification approach shown in Figure 3 is not shown in the previous plots, as doing so with \(\chi=\chi_{S}\) leads to a single large tensor and there would be no difference between structures. To visualize the effect, we need to test it with reduced bond dimensions, but for random quantum states, this would lead to very high infidelities. We can, however, use the hidden data structure of Figure 2 to prepare targets with reduced complexity equivalent to a limited \(\chi\), and perform the training using that same \(\chi<\chi_{S}\). In Figure 6, we show the results of such training and see that for each geometry, the corresponding compact version trains better by either achieving the same precision at a smaller bond dimension or reaching lower infidelities at the same bond dimension. This is visible, as the dashed lines (for compact TN) always run closer to the origin, below the solid lines (for regular TN) on the y axis and to the left on the x axis.