We introduce the concept of quantum polymorphisms to the complexity theory of quantum constraint satisfaction. Via this notion, we build an algebraic framework of reductions between quantum CSPs, and we establish a Galois connection between quantum polymorphism minions and quantum relational constructions. By leveraging a contextuality property of quantum polymorphisms, we fully characterise the existence of commutativity gadgets for relational structures, introduced by Ji as a method for achieving quantum soundness of classical CSP reductions. Prior to our work, only a partial classification was known for a subclass of Boolean languages and for non-Boolean languages meeting specific structural conditions [Culf--Mastel, FOCS'25]. As an application of our framework, we prove that the quantum CSPs parameterised by odd cycles and the quantum CSP expressing quantum satisfiability of Siggers clauses are undecidable.

翻译：我们将量子多态性的概念引入量子约束满足问题的复杂性理论中。借助这一概念，我们构建了量子CSP之间归约的代数框架，并建立了量子多态性微丛与量子关系构造之间的伽罗瓦连接。通过利用量子多态性的一种境遇性性质，我们完全刻画了关系结构上交换性构造的存在性——该构造由Ji提出，作为实现经典CSP归约的量子可靠性的一种方法。在此之前，仅对布尔语言的一个子类及满足特定结构条件的非布尔语言获得了部分分类结果[Culf–Mastel, FOCS'25]。作为我们框架的一个应用，我们证明了由奇圈参数化的量子CSP以及表达西格斯子句量子可满足性的量子CSP是不可判定的。