The sum-of-squares hierarchy of semidefinite programs has become a common tool for algorithm design in theoretical computer science, including problems in quantum information. In this work we study a connection between a Hermitian version of the SoS hierarchy, related to the quantum de Finetti theorem, and geometric quantization of compact K\"ahler manifolds (such as complex projective space $\mathbb{C}P^{d}$, the set of all pure states in a $(d + 1)$-dimensional Hilbert space). We show that previously known HSoS rounding algorithms can be recast as quantizing an objective function to obtain a finite-dimensional matrix, finding its top eigenvector, and then (possibly nonconstructively) rounding it by using a version of the Husimi quasiprobability distribution. Dually, we recover most known quantum de Finetti theorems by doing the same steps in the reverse order: a quantum state is first approximated by its Husimi distribution, and then quantized to obtain a separable state approximating the original one. In cases when there is a transitive group action on the manifold we give some new proofs of existing de Finetti theorems, as well as some applications including a new version of Renner's exponential de Finetti theorem proven using the Borel--Weil--Bott theorem, and hardness of approximation results and optimal degree-2 integrality gaps for the basic SDP relaxation of \textsc{Quantum Max-$d$-Cut} (for arbitrary $d$). We also describe how versions of these results can be proven when there is no transitive group action. In these cases we can deduce some error bounds for the HSoS hierarchy on complex projective varieties which are smooth.

翻译：半定规划的和平方层次已成为理论计算机科学中算法设计的常用工具，包括量子信息领域的问题。本文研究了与量子德菲内蒂定理相关的埃尔米特型SoS层次结构与紧致凯勒流形（如复射影空间$\mathbb{C}P^{d}$，即$(d + 1)$维希尔伯特空间中所有纯态的集合）几何量子化之间的联系。我们证明，先前已知的HSoS舍入算法可重新表述为：将目标函数量子化以获得有限维矩阵，求其最大特征向量，然后（可能非构造性地）利用胡西米准概率分布的某个版本进行舍入。对偶地，我们通过逆序执行相同步骤恢复了大多数已知的量子德菲内蒂定理：首先用胡西米分布逼近量子态，再通过量子化获得逼近原态的可分离态。在流形上存在可迁群作用的情形中，我们给出了若干现有德菲内蒂定理的新证明，以及一些应用实例，包括利用博雷尔-韦伊-博特定理证明的雷纳指数型德菲内蒂定理新版本，以及针对\textsc{量子最大-$d$-割}基础SDP松弛的近似硬度结果与最优二阶完整性间隙（适用于任意$d$）。我们还描述了在无可迁群作用时如何证明这些结果的变体。在此类情形中，我们能够推导出光滑复射影簇上HSoS层次结构的若干误差界。