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方法，用于近似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。近期研究开始分析该层级在近似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，用于作用于量子最大割问题的度-2 ncSoS。已知一些输出纠缠态的舍入方法，但它们使用了度-4 ncSoS。基于此，Hwang-Neeman-Parekh-Thompson-Wright推测度-2 ncSoS无法超越量子最大割问题的乘积态近似，并给出了一个部分证明，该证明依赖于Borrell不等式的猜测性推广。在本工作中，我们考虑一类哈密顿量（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），其局部希尔伯特空间为无限维$L^{2}(S^{k - 1})$，并证明度-2 ncSoS松弛对基态能量的近似优于任何乘积态。