SPA$^\mathrm{H}$M: the Spectrum of Approximated Hamiltonian Matrices representations

TL;DR

This paper introduces SPA$^ ext{H}$M, a novel quantum machine learning representation derived from the eigenvalues of approximate Hamiltonian matrices, capturing comprehensive electronic information efficiently for molecules and their states.

Contribution

It proposes SPA$^ ext{H}$M as a new molecular representation based on Hamiltonian eigenvalues, offering a compact, efficient, and state-sensitive alternative to existing descriptors.

Findings

SPA$^ ext{H}$M effectively distinguishes molecules, conformations, and electronic states.

The representation is compact and computationally efficient for kernel evaluations.

Complexity is independent of the number of atom types.

Abstract

Physics-inspired molecular representations are the cornerstone of similarity-based learning applied to solve chemical problems. Despite their conceptual and mathematical diversity, this class of descriptors shares a common underlying philosophy: they all rely on the molecular information that determines the form of the electronic Schr\"odinger equation. Existing representations take the most varied forms, from non-linear functions of atom types and positions to atom densities and potential, up to complex quantum chemical objects directly injected into the ML architecture. In this work, we present the Spectrum of Approximated Hamiltonian Matrices (SPA$^\mathrm{H}$M) as an alternative pathway to construct quantum machine learning representations through leveraging the foundation of the electronic Schr\"odinger equation itself: the electronic Hamiltonian. As the Hamiltonian encodes all quantum chemical information at once, SPA$^\mathrm{H}$M representations not only distinguish different molecules and conformations, but also different spin, charge, and electronic states. As a proof of concept, we focus here on efficient SPA$^\mathrm{H}$M representations built from the eigenvalues of a hierarchy of well-established and readily-evaluated "guess" Hamiltonians. These SPA$^\mathrm{H}$M representations are particularly compact and efficient for kernel evaluation and their complexity is independent of the number of different atom types in the database.