The noncommutative sum-of-squares (ncSoS) hierarchy was introduced by Navascues--Pironio--Acin as a sequence of semidefinite programming relaxations for approximating values of "noncommutative polynomial optimization problems," which were originally intended to generalize quantum values of nonlocal games. Recent work has started to analyze the hierarchy for approximating ground energies of local Hamiltonians, initially through rounding algorithms which output product states for degree-2 ncSoS. Some rounding methods are known which output entangled states, but they use degree-4 ncSoS. Based on this, Hwang--Neeman--Parekh--Thompson--Wright conjectured that degree-2 ncSoS cannot beat product state approximations for quantum max-cut and gave a partial proof relying on a conjectural generalization of Borrell's inequality. In this talk we will describe the ncSoS hierarchy and a family of Hamiltonians (called the quantum rotor model in condensed matter literature or lattice O(n) vector model in QFT) over an infinite-dimensional local Hilbert space, and sketch a proof that a degree-2 ncSoS relaxation approximates the ground energy better than any product state.
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