SPAᴴM: the spectrum of approximated hamiltonian matrices representations

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 Schrödinger 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 (SPAᴴM) as an alternative pathway to construct quantum machine learning representations through leveraging the foundation of the electronic Schrödinger equation itself: the electronic Hamiltonian. As the Hamiltonian encodes all quantum chemical information at once, SPAᴴM 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 SPAᴴM representations built from the eigenvalues of a hierarchy of well-established and readily-evaluated “guess” Hamiltonians. These SPAᴴM representations are particularly compact and efficient for kernel evaluation and their complexity is independent of the number of different atom types in the database

启于物理学原理的分子表征构成了应用于解决化学问题的基于相似性学习的基石。尽管此类表征在概念与数学结构上呈现出丰富的多样性，它们却共享一个共同的哲学基础：均依赖于决定电子薛定谔方程形式的分子信息。现存的表征形式千差万别，从原子类型和位置的非线性函数，到原子密度和势场，直至直接注入机器学习架构的复杂量子化学对象。在本研究中，我们提出了一种名为“近似哈密顿矩阵谱”（Spectrum of Approximated Hamiltonian Matrices, 简称SPAᴴM）的替代途径，旨在通过利用电子薛定谔方程本身的基石——电子哈密顿量——来构建量子机器学习表征。由于哈密顿量一次性编码了所有量子化学信息，SPAᴴM表征不仅能够区分不同的分子和构象，还能够区分不同的自旋、电荷和电子状态。作为概念验证，我们在此聚焦于由一系列已建立且易于评估的“猜测”哈密顿量的本征值构建的高效SPAᴴM表征。这些SPAᴴM表征在核评估方面尤为紧凑高效，且其复杂性不依赖于数据库中不同原子类型的数量。