Quantum polymorphism characterisation of commutativity gadgets in all quantum models

TL;DR

This paper develops a comprehensive framework for commutativity gadgets in quantum models, characterizing their existence through quantum polymorphism spaces and exploring their structural properties and separations.

Contribution

It introduces a unified approach to commutativity gadgets across quantum models, linking their existence to quantum polymorphism space collapses and analyzing their structure in relation to quantum permutation groups.

Findings

Existence of a commutativity gadget is equivalent to quantum polymorphisms collapsing to classical ones.

Robust and non-robust gadgets are characterized by stable commutativity in weighted polymorphism algebras.

Constructs relational structures with specific gadget properties, including separations between classes.

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.