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

TL;DR

This paper introduces a sum-of-squares SDP hierarchy for approximating quantum ground-state energies, linking it to existing methods and proposing optimized cluster selection strategies inspired by quantum information theory.

Contribution

It presents a novel SDP hierarchy with a quantum embedding interpretation and efficient cluster optimization strategies for better lower bounds in quantum many-body problems.

Findings

Effective SDP hierarchy for ground-state energy approximation.

Cluster selection strategies improve relaxation tightness.

Quantum entanglement captures the structure of the Hamiltonian.

Abstract

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.