Oliver Siebert

We present a modular algorithm for learning external potentials in continuous-space free-fermion models including Coulomb potentials. Compared to the lattice-based approaches, the continuum presents new mathematical challenges: the state space is infinite-dimensional and the Hamiltonian contains the Laplacian, which is unbounded in the continuum and produces an unbounded speed of information propagation. Our framework addresses these difficulties through novel optimization methods and information-propagation bounds in combination with a priori regularity assumptions on the external potential. The resulting algorithm provides a unified and robust approach to learn both Coulomb interactions and other classes of physically relevant potentials, like trigonometric polynomials or general smooth functions. One possible application is to learn the charge and position of nuclei and ions distributed in continuous space as in quantum chemistry. Our results thus lay the foundations for a scalable and generalizable toolkit to explore fermionic systems governed by continuous-space interactions. Moreover, we provide numerical evidence supporting the classical post-processing algorithm of our Coulomb algorithm.