Learning Minimal Representations of Fermionic Ground States

We introduce an unsupervised machine-learning framework that discovers optimally compressed representations of quantum many-body ground states. Using an autoencoder neural network architecture on data from $L$-site Fermi-Hubbard models, we identify minimal latent spaces with a sharp reconstruction quality threshold at $L-1$ latent dimensions, matching the system's intrinsic degrees of freedom. We demonstrate the use of the trained decoder as a differentiable variational ansatz to minimize energy directly within the latent space. Crucially, this approach circumvents the $N$-representability problem, as the learned manifold implicitly restricts the optimization to physically valid quantum states.