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

• URL: http://arxiv.org/abs/2603.05524v1
• Date: Mon, 23 Feb 2026 22:41:21 GMT
• Title: Direct Variational Calculation of Two-Electron Reduced Density Matrices via Semidefinite Machine Learning
• Abstract summary: We introduce a data-driven framework for approxing the convex set of $N$-representable two-electron reduced density matrices (2-RDMs)<n>We use a data-driven approximation to its boundary from molecular data to improve the set defined by low-order positivity constraints.<n>The resulting semidefinite machine learning approach drives a direct variational calculation of the 2-RDM.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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