Quantum polymorphism characterisation of commutativity gadgets in all quantum models

Abstract

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.

Figures (3)

• Figure 1: An overview of the implications and separations for existence of commutativity gadgets in different models. An arrow from one model to another indicates that existence of a commutativity gadget in the first model implies existence of a commutativity gadget in the second. Models that are equivalent in this sense are grouped together in shaded areas. Red lines between the areas indicate separations, meaning that there are CSPs which admit a commutativity gadget in the weaker model but not in the stronger model. The dashed red line indicates a separation that is conditional on the existence of a non-hyperlinear group.
• Figure 2: Relationships between commutativity gadget classes shown by \ref{['thm:implications-equivalences']}. Trivial implications are denoted by solid grey arrows, the implications (i) -- (iv) are denoted by solid black arrows, the implications (v) and (vi) are denoted by dashed black arrows, the implications (vii) and (viii) are denoted by dotted black arrows, and the equivalences (ix) -- (xii) are denoted by blue boxes.
• Figure 3: Separations between commutativity gadget classes; robust commutativity gadgets are not shown in this figure, due to the equivalences in \ref{['thm:implications-equivalences']}. The separation between oracular and non-oracular commutativity gadgets is via $4$-colouring CDdBVZ25, and the remaining separations are due to \ref{['thm:separations']}. Dashed line indicates a separation relying the existence of a non-hyperlinear group.

Theorems & Definitions (74)

• Definition 2.1
• Definition 2.2
• Theorem 2.3
• proof : Proof sketch
• Theorem 3.1: CJM25
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Theorem 3.4: CJM25