We study operator - or noncommutative - variants of constraint satisfaction problems (CSPs). These higher-dimensional variants are a core topic of investigation in quantum information, where they arise as nonlocal games and entangled multiprover interactive proof systems (MIP*). The idea of higher-dimensional relaxations of CSPs is also important in the classical literature. For example since the celebrated work of Goemans and Williamson on Max-Cut, higher dimensional vector relaxations have been central in the design of approximation algorithms for classical CSPs. We introduce a framework for designing approximation algorithms for noncommutative CSPs. Prior to this work Max-$2$-Lin$(k)$ was the only family of noncommutative CSPs known to be efficiently solvable. This work is the first to establish approximation ratios for a broader class of noncommutative CSPs. In the study of classical CSPs, $k$-ary decision variables are often represented by $k$-th roots of unity, which generalise to the noncommutative setting as order-$k$ unitary operators. In our framework, using representation theory, we develop a way of constructing unitary solutions from SDP relaxations, extending the pioneering work of Tsirelson on XOR games. Then, we introduce a novel rounding scheme to transform these solutions to order-$k$ unitaries. Our main technical innovation here is a theorem guaranteeing that, for any set of unitary operators, there exists a set of order-$k$ unitaries that closely mimics it. As an integral part of the rounding scheme, we prove a random matrix theory result that characterises the distribution of the relative angles between eigenvalues of random unitaries using tools from free probability.

翻译：我们研究约束满足问题（CSPs）的算子（即非交换）变体。这些高维变体是量子信息领域的核心研究课题，常以非局域博弈和纠缠多证明者交互证明系统（MIP*）的形式出现。CSPs的高维松弛思想在经典文献中亦具有重要意义。例如，自Goemans与Williamson关于Max-Cut的经典工作以来，高维向量松弛始终是经典CSPs近似算法设计的核心。我们提出了一个用于设计非交换CSPs近似算法的框架。在此工作之前，Max-$2$-Lin$(k)$是唯一已知可高效求解的非交换CSPs族。本文首次为更广泛的非交换CSPs建立了近似比。在经典CSPs研究中，$k$元决策变量常以$k$次单位根表示，其非交换推广对应为$k$阶酉算子。在我们的框架中，利用表示论，我们发展了一种从半定规划（SDP）松弛构造酉解的方法，这扩展了Tsirelson关于XOR游戏的先驱性工作。随后，我们引入了一种新型舍入方案将这些解转化为$k$阶酉算子。此处的主要技术创新在于一个定理：它保证对任意酉算子集合，必存在一组$k$阶酉算子与其高度近似。作为舍入方案的关键组成部分，我们证明了一个随机矩阵理论结果——利用自由概率工具刻画随机酉算子特征值之间相对角度的分布。