Commutativity gadgets provide a technique for lifting classical reductions between constraint satisfaction problems to quantum-sound reductions between the corresponding nonlocal games. We develop a general framework for commutativity gadgets in the setting of quantum homomorphisms between finite relational structures. Building on the notion of quantum homomorphism spaces, we introduce a uniform notion of commutativity gadget capturing the finite-dimensional quantum, quantum approximate, and commuting-operator models. In the robust setting, we use the weighted-algebra formalism for approximate quantum homomorphisms to capture corresponding notions of robust commutativity gadgets. Our main results characterize both non-robust and robust commutativity gadgets purely in terms of quantum polymorphism spaces: in any model, existence of a commutativity gadget is equivalent to the collapse of the corresponding quantum polymorphisms to classical ones at arity $|A|^2$, and robust gadgets are characterized by stable commutativity of the appropriate weighted polymorphism algebra. We use this characterisation to show relations between the classes of commutativity gadget, notably that existence of a robust commutativity gadget is equivalent to the existence of a corresponding non-robust one. Finally, we prove that quantum polymorphisms of complete graphs $K_n$ have a very special structure, wherein the noncommutative behaviour only comes from the quantum permutation group $S_n^+$. Combining this with techniques from combinatorial group theory, we construct separations between commutativity-gadget classes: we exhibit a relational structure admitting a finite-dimensional commutativity gadget but no quantum approximate gadget, and, conditional on the existence of a non-hyperlinear group, a structure admitting a quantum approximate commutativity gadget but no commuting-operator gadget.

翻译：交换性小工具提供了一种将约束满足问题之间的经典归约提升为相应非局部博弈之间量子可靠归约的技术。我们发展了一个通用框架，用于有限关系结构之间量子同态设定中的交换性小工具。基于量子同态空间的概念，我们引入了一种统一的交换性小工具概念，涵盖了有限维量子、量子近似和交换算子模型。在鲁棒设定中，我们使用近似量子同态的加权代数形式来捕捉相应的鲁棒交换性小工具概念。我们的主要结果纯粹用量子多态性空间来表征非鲁棒和鲁棒交换性小工具：在任何模型中，交换性小工具的存在等价于相应量子多态性在元数为$|A|^2$时坍缩为经典多态性，而鲁棒小工具则由相应加权多态性代数的稳定交换性来表征。我们利用这一表征揭示了交换性小工具类别之间的关系，特别是鲁棒交换性小工具的存在等价于相应非鲁棒小工具的存在。最后，我们证明了完全图$K_n$的量子多态性具有非常特殊的结构，其中非交换行为仅来自量子置换群$S_n^+$。结合组合群论技术，我们构造了交换性小工具类别之间的分离：展示了一个关系结构，它允许有限维量子小工具但不允许量子近似小工具；并且在存在非超线性群的条件下，展示了一个允许量子近似交换性小工具但不允许交换算子小工具的结构。