量子转子模型的平方和松弛分析
Analysis of sum-of-squares relaxations for the quantum rotor model
AI总结：
针对具有无限维局域空间的量子转子模型，证明 2 次 ncSoS 松弛在基态能量逼近上可优于任意乘积态，反驳相关猜想的适用范围。
摘要：
非交换平方和（ncSoS）层级由 Navascués-Pironio-Acín 提出，是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，其最初旨在推广非局部博弈的量子值。近期工作已开始分析该层级在逼近局域哈密顿量基态能量方面的应用，最初是通过针对应用于 Quantum Max-Cut 的 2 次 ncSoS 输出乘积态的舍入算法。已知一些输出纠缠态的舍入方法，但它们使用 4 次 ncSoS。基于此，Hwang-Neeman-Parekh-Thompson-Wright 猜想 2 次 ncSoS 在 Quantum Max-Cut 上无法胜过乘积态逼近，并给出了依赖于 Borrell 不等式的一个猜想性推广的部分证明。在本工作中，我们考虑一族哈密顿量（在凝聚态文献中称为量子转子模型，在量子场论中称为格点 $O(k)$ 向量模型），其具有无限维局域 Hilbert 空间 $L^{2}(S^{k-1})$，并证明 2 次 ncSoS 松弛对基态能量的逼近优于任何乘积态。
英文摘要：
The noncommutative sum-of-squares (ncSoS) hierarchy was introduced by Navascués-Pironio-Acín 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.
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