Approximating Quantum Ground States via Machine-Learned N-Representability Constraints

The N-representability problem concerns the necessary and sufficient conditions for a k-particle density matrix to be a valid reduced density matrix (RDM) of some global N-particle state. A notable application of this problem is minimizing a fermionic Hamiltonian's energy over the set of valid two-particle RDMs. The ground-state RDM then lies on this convex set's boundary. However, identifying this set is computationally intractable, even for a quantum computer. Traditional approaches run a semidefinite program (SDP) with a subset of analytically derived N-representability constraints. Here, we address the problem using machine learning, tailoring to many-electron systems relevant to quantum chemistry and condensed matter. We apply supervised learning to approximate the convex boundary, which subsequently serves as a new RDM constraint for the ground-state problem. With sufficient training, our method can enable faster and more accurate solutions compared to standard SDP approaches.