Noncommutative constraint satisfaction problems (NC-CSPs) are higher-dimensional operator extensions of classical CSPs. Despite their significance in quantum information, their approximability remains largely unexplored. A notable example of a noncommutative CSP that is not solvable in polynomial time is NC-Max-$3$-Cut. We present a $0.864$-approximation algorithm for this problem. Our approach extends to a broader class of both classical and noncommutative CSPs. We introduce three key concepts: approximate isometry, relative distribution, and $\ast$-anticommutation, which may be of independent interest.

翻译：非交换约束满足问题（NC-CSP）是经典约束满足问题的高维算子推广。尽管其在量子信息领域具有重要意义，其近似性至今仍未得到充分研究。一个不可在多项式时间内求解的非交换约束满足问题的显著例子是NC-Max-$3$-Cut。针对该问题，我们提出了一种$0.864$近似算法。我们的方法可推广至更广泛的经典与非交换约束满足问题类别。我们引入了三个核心概念：近似等距、相对分布与$\ast$-反交换性，这些概念可能具有独立的学术价值。