LAMP-QDF: Local Attention Map for Potential in Quantum Deep Field

The Hohenberg--Kohn (HK) theorem establishes a one-to-one mapping between the ground-state electron density and the external potential. Quantum Deep Field (QDF) architectures parameterize latent molecular fields within a Linear Combination of Atomic Orbitals (LCAO) framework, but typically approximate the density-to-potential mapping using a pointwise functional. We introduce LAMP-QDF, which replaces this pointwise map with a distance-aware graph-attention operator. This local representation provides a more consistent approximation of the HK inverse while preserving the density-generation capabilities of QDF. On the QM9 dataset, when trained on molecules with $N_{\text{Atoms}} \leq 14$, LAMP-QDF reduces atomization-energy MAE by $9.1\%$ in the interpolation test regime ($N_{\text{Atoms}} \leq 14$), achieving $1.167\pm0.044$~kcal/mol versus $1.284\pm0.026$~kcal/mol for QDF. In the size-extrapolation regime ($15 \leq N_{\text{Atoms}}$), LAMP-QDF achieves $2.553\pm0.107$~kcal/mol versus $3.024\pm0.100$~kcal/mol for QDF, a $15.6\%$ reduction. Across the interpolation test set, LAMP-QDF also reduces the reconstruction MSE of the regularized Gaussian nuclear-potential target by $92.4\%$.