Direct Variational Calculation of Two-Electron Reduced Density Matrices via Semidefinite Machine Learning

TL;DR

This paper presents a novel data-driven semidefinite machine learning method for directly calculating two-electron reduced density matrices, improving accuracy over traditional methods without requiring higher-order positivity constraints.

Contribution

It introduces a vertex-based approximation of the 2-RDM set using neural networks and semidefinite programming, enhancing variational calculations in quantum chemistry.

Findings

Achieves more accurate 2-RDMs and energies compared to traditional two-positivity methods.

Demonstrates systematic improvements on molecular potential energy curves.

Close agreement with complete active space configuration interaction results.

Abstract

We introduce a data-driven framework for approximating the convex set of $N$-representable two-electron reduced density matrices (2-RDMs). Traditional approaches characterize this set through linear matrix inequalities that define its supporting hyperplanes. Here, we instead learn a vertex-based approximation to its boundary from molecular data and use this information to improve the set defined by low-order positivity constraints, without explicitly constructing higher-order conditions. The resulting semidefinite machine learning approach -- combining an input convex neural network with semidefinite programming -- drives a direct variational calculation of the 2-RDM with enhanced accuracy at computational cost comparable to two-positivity calculations. Applications to the potential energy curves of ${\rm C}_2^{2-}$, ${\rm N}_2$, and ${\rm O}_2^{2+}$ demonstrate these systematic improvements as well as close agreement with complete active space configuration interaction results. Overall, semidefinite machine learning interweaves data-driven boundary information with semidefinite positivity constraints to yield more accurate energies and 2-RDMs without explicit higher-order positivity conditions.