Symmetry-invariant quantum machine learning force fields

TL;DR

The paper addresses the challenge of trainability and scalability in variational quantum learning models for generating molecular force fields. It introduces symmetry-invariant quantum learning models (siVQLM) that embed rototranslational and permutational symmetries via SU(2)/SO(3), $S_n$, and reflection symmetries, using $G$-invariant observables and $G$-equivariant encodings. The authors construct siVQLMs for LiH, H2O, and an H2O dimer, showing substantially improved energy and force predictions, robustness to noisy training labels, and beneficial effects from controlled symmetry-breaking. These results suggest that geometric quantum machine learning offers a viable pathway to scalable, accurate molecular force-field generation and can be integrated with classical methods for larger-scale simulations.

Abstract

Machine learning techniques are essential tools to compute efficient, yet accurate, force fields for atomistic simulations. This approach has recently been extended to incorporate quantum computational methods, making use of variational quantum learning models to predict potential energy surfaces and atomic forces from ab initio training data. However, the trainability and scalability of such models are still limited, due to both theoretical and practical barriers. Inspired by recent developments in geometric classical and quantum machine learning, here we design quantum neural networks that explicitly incorporate, as a data-inspired prior, an extensive set of physically relevant symmetries. We find that our invariant quantum learning models outperform their more generic counterparts on individual molecules of growing complexity. Furthermore, we study a water dimer as a minimal example of a system with multiple components, showcasing the versatility of our proposed approach and opening the way towards larger simulations. Our results suggest that molecular force fields generation can significantly profit from leveraging the framework of geometric quantum machine learning, and that chemical systems represent, in fact, an interesting and rich playground for the development and application of advanced quantum machine learning tools.

Figures (8)

• Figure 1: Overview of the work: we design invariant quantum learning models for a set of relevant molecular symmetries, obtaining -- upon input of simple Cartesian coordinates -- invariant predictions for potential energy surfaces and force fields to be employed in molecular dynamics simulations.
• Figure 2: Symmetry-invariant models for diatomic (LiH) and triatomic ($\text{H}_2 \text{O}$) molecules: (a) Translationally invariant inputs are obtained by fixing the origin of the Cartesian coordinate system. For LiH, we choose $\vec{x}_1 = -\vec{x}_2$, while for H2O $\vec{x}_O =\vec{0}$ and, in general, $\vec{x}_1 \neq -\vec{x}_2$, where $\vec{x}_1, \vec{x}_2$ are the coordinates of the hydrogen atoms. (b) The siVQLM for a single molecule of LiH and H2O. The equivariant embedding layers (blue) encode $\mathcal{X}=(\vec{x}_1, \vec{x}_2)$ via \ref{['eq:LiH-EQNN-encoding']}. Importantly, note that this embedding depends on a trainable parameter $\alpha_\text{enc}\in\mathds{R}$. The equivariant trainable layers (red) are given by \ref{['eq:RH_four']} and \ref{['eq:h2O_trainable']}. In the case of H2O, the trainable layer is extended with the red dashed operator and an optional symmetry-breaking layer (yellow). Finally, the invariant observable $\mathcal{O}$ (orange) is measured.
• Figure 3: Symmetry-invariant model for the example of two $\text{H}_2 \text{O}$ molecules. (a) Translationally invariant inputs are obtained by fixing the origin of the Cartesian coordinate system. The oxygen atoms lie on opposite sites of the origin, i.e., $\vec{x}_O^1 = -\vec{x}_O^2$. (b) The siVQLM for $\text{H}_2 \text{O}$ dimer. The equivariant embedding layers (blue) encode $\mathcal{X}=(\vec{x}_i)_{i=1}^6=(\vec{x}_O^1, \vec{x}_H^1, \vec{x}_H^1, \vec{x}_O^2, \vec{x}_H^2, \vec{x}_H^2)$ via \ref{['eq:H2O_dimer_encoding']}, where the embedding depends on a atom-type $a$ specific trainable parameter $\alpha_{\text{enc},{a}}\in\mathds{R}$. The equivariant trainable layers (red) are given by \ref{['eq:h2O_dimer_trainable']} and extended by a symmetry-breaking layer (yellow). In the end, the invariant observable $\mathcal{O}$ (orange) is measured.
• Figure 4: Energy prediction (a) and force prediction (b) for LiH. The predictions by the generic VQLM (red crosses) and the siVQLM (blue crosses) are compared to the exact energy/forces (black solid line). The force prediction in (b) by the generic VQLM is given with respect to internal coordinates, while the siVQLM directly predicts atomic forces with respect to Cartesian coordinates.
• Figure 5: Training on noisy energy labels for LiH. The energy (black) and force (blue) prediction of the VQLM and siVQLM are compared for increasing amounts of noise $E_\text{std}$. In the case of the original VQLM, the loss is taken for the relevant test data set as before in the exact case. The inset shows the noisy energy labels for $E_\text{std}=0.05$.