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

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 a computational cost comparable to two-positivity calculations. Applications to the potential energy curves of C₂^2-, N₂, O₂²⁺, CO, NO⁺, and CN^- 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.