Machine Learning Discovers Minimal Representations of Fermionic Ground States in -site Models with Latent Dimensions
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The challenge of representing the complex quantum states of many-particle systems limits progress in fields like materials science and quantum chemistry. Felix Frohnert, Emiel Koridon, and Stefano Polla, all from Universiteit Leiden, have developed a new unsupervised method that discovers remarkably compressed representations of these states. Their approach uses a neural network to identify the smallest possible ‘latent space’ needed to accurately describe the ground state of a system, effectively matching the number of essential characteristics. This breakthrough circumvents a fundamental problem in quantum simulations and allows scientists to directly minimise energy within this simplified space, promising more efficient and accurate modelling of complex quantum materials. Applying an autoencoder neural network to data from Fermi-Hubbard models, the researchers identified minimal latent spaces that accurately represent the ground state manifold, reducing the computational cost associated with storing and manipulating these complex states. This approach effectively captures essential correlations within the ground state, enabling accurate reconstruction of the wave function from the compressed latent representation and paving the way for scalable simulations of strongly correlated fermionic systems central to understanding materials with emergent quantum phenomena. The reconstruction quality of this method reaches a threshold at a latent dimensionality of L-1, matching the system’s intrinsic degrees of freedom. The trained decoder functions as a differentiable variational ansatz, allowing direct energy minimization within the latent space. Importantly, this approach circumvents the challenges of ensuring physical validity, as the learned manifold implicitly restricts optimization to physically plausible quantum states. The exponential growth of the Hilbert space with system size presents a fundamental challenge in quantum many-body physics, rendering direct simulation intractable for all but the smallest systems, but approximate methods offer a viable path forward.
Hubbard Model Ground States Lie on Low-Dimensional Manifold This research demonstrates that the ground states of the Hubbard model, a standard model in condensed matter physics, lie on a surprisingly low-dimensional manifold. Specifically, the intrinsic dimensionality of the ground state manifold is approximately L-1, where L is the number of lattice sites in the system. This means the ground state can be accurately represented with far fewer parameters than previously expected. Unsupervised autoencoders, a type of neural network, prove highly effective at learning these compressed representations, encoding essential information into a low-dimensional latent space. The dimensionality of L-1 is critical; below this dimension, reconstruction quality degrades significantly, while above it, the latent space becomes unstable. This compressed latent space serves as a variational ansatz for energy optimization, allowing researchers to efficiently find approximate ground states by minimizing energy in the latent space and decoding back to the original system.
The team also successfully applied this compression approach to two-body reduced density matrices, which contain more detailed information about ground state correlations. The intrinsic dimensionality remained L-1, suggesting that the additional correlation information does not require extra latent dimensions. Systems with degeneracies in their ground states, such as those with odd numbers of sites, exhibit more complex latent space structures and slightly worse reconstruction quality, highlighting the need for the autoencoder to account for underlying symmetries and degeneracies.
This research has several important implications, including the potential for efficient quantum simulations, the discovery of new quantum phases, improved variational algorithms, and a deeper understanding of quantum complexity.
Learned Latent Spaces Capture Quantum Ground States This research demonstrates an unsupervised machine learning framework capable of discovering compressed representations of complex quantum many-body ground states. Applying an autoencoder neural network to Fermi-Hubbard models, the researchers identified minimal latent spaces that accurately reconstruct the system’s state, with the dimensionality of these spaces aligning with the system’s intrinsic degrees of freedom. This approach bypasses limitations associated with representing quantum states, by implicitly restricting optimization to physically plausible configurations within the learned latent space. Analysis of the learned representations reveals a clear hierarchy of feature importance, determined by examining the influence of different physical observables on the latent coordinates. Results consistently show that density and on-site interaction terms dominate the latent representation across varying system sizes, while nearest-neighbor correlations play a comparatively minor role. This suggests the autoencoder prioritizes local quantities directly related to the system’s potential and interactions when constructing its compact description. Systems with an odd number of sites, tested as a means of exploring the limits of the compression approach, exhibit slightly higher reconstruction loss compared to systems with even numbers of sites, indicating a potential area for future refinement. 👉 More information 🗞 Learning Minimal Representations of Fermionic Ground States 🧠 ArXiv: https://arxiv.org/abs/2512.11767 Tags: