The noncommutative sum-of-squares (ncSoS) hierarchy was introduced by Navascu\'{e}s-Pironio-Ac\'{i}n 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 applied to Quantum Max-Cut. 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 work we consider a family of Hamiltonians (called the quantum rotor model in condensed matter literature or lattice $O(k)$ vector model in quantum field theory) with infinite-dimensional local Hilbert space $L^{2}(S^{k - 1})$, and show that a degree-2 ncSoS relaxation approximates the ground state energy better than any product state.

翻译：非交换平方和（ncSoS）层级由Navascués-Pironio-Acín引入，作为一系列半定规划松弛，用于逼近非交换多项式优化问题的值，最初旨在推广非局部游戏的量子值。近期工作开始分析该层级在逼近局部哈密顿量基态能量中的应用，最初通过输出乘积态的舍入算法应用于量子Max-Cut问题的度-2 ncSoS。已知某些输出纠缠态的舍入方法，但它们使用度-4 ncSoS。基于此，Hwang-Neeman-Parekh-Thompson-Wright猜想度-2 ncSoS无法超越量子Max-Cut的乘积态逼近，并基于Borrell不等式的猜想推广给出了部分证明。在本文中，我们考虑一类具有无限维局部希尔伯特空间$L^{2}(S^{k - 1})$的哈密顿量（在凝聚态物理文献中称为量子转子模型，或在量子场论中称为晶格$O(k)$矢量模型），并证明度-2 ncSoS松弛对基态能量的逼近优于任何乘积态。