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We introduce a sum-of-squares SDP hierarchy approximating the ground-stateenergy from below for quantum many-body problems, with a natural quantumembedding interpretation. We establish the connections between our approach andother variational methods for lower bounds, including the variationalembedding, the RDM method in quantum chemistry, and the Anderson bounds.Additionally, inspired by the quantum information theory, we propose efficientstrategies for optimizing cluster selection to tighten SDP relaxations whilestaying within a computational budget. Numerical experiments are presented todemonstrate the effectiveness of our strategy. As a byproduct of ourinvestigation, we find that quantum entanglement has the potential to capturethe underlying graph of the many-body Hamiltonian.

Quantum variational embedding for ground-state energy problems: sum of squares and cluster selection