We introduce a sum-of-squares SDP hierarchy approximating the ground-state energy from below for quantum many-body problems, with a natural quantum embedding interpretation. We establish the connections between our approach and other variational methods for lower bounds, including the variational embedding, the RDM method in quantum chemistry, and the Anderson bounds. Additionally, inspired by the quantum information theory, we propose efficient strategies for optimizing cluster selection to tighten SDP relaxations while staying within a computational budget. Numerical experiments are presented to demonstrate the effectiveness of our strategy. As a byproduct of our investigation, we find that quantum entanglement has the potential to capture the underlying graph of the many-body Hamiltonian.

翻译：我们引入了一种平方和SDP层级结构，从下方近似量子多体问题的基态能量，并赋予其自然的量子嵌入解释。我们建立了该方法与其他下界变分方法之间的联系，包括变分嵌入、量子化学中的RDM方法以及安德森界。此外，受量子信息理论启发，我们提出了高效策略以优化簇选择，从而在计算预算内收紧SDP松弛。通过数值实验验证了该策略的有效性。作为研究的副产品，我们发现量子纠缠具有捕获多体哈密顿量潜在图结构的能力。