## 3 Results and Discussion
### Predicting Charge Localization in Neuromorphic QIs  
We generate two groups of systems, corresponding to narrow (\(R_{0}=10\) nm) and wide (\(R_{0}=20\) nm) leads. A typical structure is depicted in Figure 3. Starting with the case \(N=1\), which also corresponds to a non-interacting system with two particles up to a scaling factor, we observe the charge accumulating in one single peak. This is located in the wider region positioned at the crossing of the incoming and the outgoing leads, where the quantum well is wider. As no Coulomb interaction is present in this case, the charge distribution is concentrated in one peak, corresponding to \(\phi_{0,\uparrow}\) (equivalently \(\phi_{0,\downarrow}\)) as the kinetic energy is minimized. However, when two interacting particles are present, the ground state charge distribution is more extended towards the input leads, as the Coulomb repulsion drives the charge in regions with stronger confinement. In effect, a trade-off between the kinetic and potential energy settles the ground state density: the higher confinement increases the kinetic energy but allows a larger charge separation, reducing the Coulomb energy. In this case, higher-order orbitals \(\phi_{i,\downarrow s}\) are present in the Slater determinants. This trend is further enhanced for \(N=3\) and \(N=4\), where the electronic charge becomes distributed in the entire dendritic structure. For a large number of electrons, a more even distribution is  
Figure 2: Workflow summarizing the steps in the design of the QIs: From a computational perspective, the confinement potentials of the neuromorphic structures give the charge density maps (CDMs) inside the device, and this is determined by the exact diagonalization method. The tunneling current maps (TCMs) result as convolutions from the charge densities. Using the cGAN approach, the TCMs are deconvoluted into CDMs, while mappings between confinement potentials and CDMs can be obtained in the same way. In a prototypical experimental device, the TCMs can be measured and can be fed in the machine learning approach in order to efficiently optimize the device structure. The cGAN approach is based on two coupled system of networks, the generator and the discriminator, which process pairs of images: the generator provides an output image based on the input, while the discriminator compares the generated image with the reference one. expected as we approach a mean-field description. Comparing the narrow- and wide-lead systems, we can see a larger spread in the latter, as the weaker confinement leads to a smaller kinetic energy. The predicted charge densities are illustrated comparatively with the reference ones. The most simple case corresponds to the non-interacting case (\(N=1\)), where the rather localized charge density is well reproduced. The other extreme corresponds to a large number of particles, where the charge density is more uniformly distributed. The intermediate cases, like \(N=2\) and \(N=3\), are more difficult to predict, as more pronounced maxima are visible in the reference data, which can be localized in the branches of the dendritic structures. This localization occurs particularly at the bending regions. For \(N=4\) particles, the particle density covers most of the quantum well, and it is therefore easier to predict. In order to assess the overall accuracy, we employed the \(R^{2}\) coefficient of determination for the test set. The results are indicated in Figure 4 for both narrow- and wide-lead systems. As previously discussed, the best accuracies are found for \(N=1\), with \(R^{2}>0.95\). The worst performance is not surprisingly obtained for \(N=2\), with \(R^{2}\simeq 0.75\) (narrow QIs) and \(R^{2}\simeq 0.87\) (wide QIs). Gradual improvements are observed for \(N=3\) and \(N=4\). Also, the larger dispersion of the particle density in the wide-lead systems induces a better prediction accuracy compared to the narrow-lead systems.