Learning Coulomb Potentials and Beyond with Fermions in Continuous Space

TL;DR

The paper develops a modular framework for learning external potentials in continuum-space free-fermion Hamiltonians, addressing challenges from infinite dimensionality and unbounded propagation. It introduces data-acquisition protocols based on localized initial states and local averages that enable precise recovery of Coulomb centers and charges, first for a single center and then for multiple centers, with provable error and time bounds. The approach extends to learning general potential classes by expressing V in a basis and solving conditioned linear systems, with error propagation controlled by matrix conditioning and regularity assumptions. Continuum Lieb-Robinson bounds underpin the error analysis, allowing quantum-mechanical propagation to be bounded in the continuum setting. The resulting toolkit provides scalable, principled methods for characterizing nuclear charges and positions and offers a unified framework for learning a broad class of continuous-space potentials in quantum chemistry and related fields.

Abstract

We present a modular algorithm for learning external potentials in continuous-space free-fermion models including Coulomb potentials in any dimension. 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 thus produces an unbounded speed of information propagation. Our framework addresses these difficulties through novel optimization methods or information-propagation bounds in combination with a priori regularity assumptions on the external potential. The resulting algorithm provides a unified and robust approach that covers both Coulomb interactions and other classes of physically relevant potentials. One possible application is the characterization of charge and position of nuclei and ions in quantum chemistry. Our results thus lay the foundation for a scalable and generalizable toolkit to explore fermionic systems governed by continuous-space interactions.

Figures (7)

• Figure 1: Depiction of the initial states for the three different pairs $(\alpha,\beta) = (0,1),(0,2),(1,2)$ on a grid of 6 $\times$ 6 boxes. The boxes are divided into triples (see first bold box, indexed by the first box at ${\mathbf{j}} =(0,0)$), and for each triple we consider the superposition of the vacuum with the state containing particles with rescaled profile $f$ in the boxes $\alpha$ and $\beta$ (shown in red).
• Figure 2: Demonstration of the sequential scheme, in which only one triplet is fixed, then \ref{['protocol']} is executed, and the process continues with the next triplet. This approach is only efficient when the number of boxes is small.
• Figure 3: Visualization in two dimensions of a possible optimizer $\widehat{{\mathbf{j}}}_y$ and its neighborhood $\partial B_{\hat{{\mathbf{j}}}_y}$, in the context of the partition shown on the left and the construction of the defining vectors of the matrix $\mathbf{A}$ on the right.
• Figure 4: Visualization of the construction in three dimensions. Red indicates the box $\widehat{{\mathbf{j}}}_y$ representing the estimated location of $y$, gray denotes the neighborhood $\partial B_{\hat{{\mathbf{j}}}_y}$, and blue highlights the boxes used to construct the matrix $\mathbf{A}$. We are interested in the distances between $y$ and the midpoints of the blue boxes.
• Figure 5: The graphic visualizes the different regions for the $\operatorname{supp}(g)$, $rv$, and the distance in between.

Theorems & Definitions (37)

• Theorem 2.1
• Remark 2.2: Precision vs. pollution
• Theorem 3.1
• proof : Proof-sketch
• Remark 3.2
• Theorem 3.3
• proof : Proof sketch
• Theorem 4.1
• proof
• Remark 4.2