  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.
