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arXiv:2511.23445

Quantum Polymorphisms and Commutativity Gadgets

Lorenzo Ciardo, Gideo Joubert, Antoine Mottet

quant-phcs.CCcs.LOmath.COSubmitted Nov 28, 2025• Announced Dec 1, 2025• 0 citations
v1 · Nov 28, 2025v2 · Apr 1, 2026
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Abstract

We introduce the concept of quantum polymorphisms to the complexity theory of non-local games. We use this notion to give a full characterisation of 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, a classification was only known in the Boolean case [Culf--Mastel, STOC'25]. As an application of our framework, we prove that the entangled CSP parameterised by odd cycles is undecidable. Furthermore, we establish a quantum version of Galois connection for entangled CSPs in the case of non-oracular quantum homomorphisms.

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•Jan 1, 2025•0 citations
We introduce the concept of quantum polymorphisms to the complexity theory of non-local games. We use this notion to give a full characterisation of 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, a classification was only known in the Boolean case [Culf--Mastel, STOC'25]. As an application of our framework, we prove that the entangled CSP parameterised by odd cycles is undecidable. Furthermore, we establish a quantum version of Galois connection for entangled CSPs in the case of non-oracular quantum homomorphisms.
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|A|^2∣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 KnK_nKn​ have a very special structure, wherein the noncommutative behaviour only comes from the quantum permutation group Sn+S_n^+Sn+​. 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.
Information-processing tasks modelled by homomorphisms between relational structures can witness quantum advantage when entanglement is used as a computational resource. We prove that the occurrence of quantum advantage is determined by the same algebraic structure (known as the polymorphism minion) that captures the complexity of CSPs. We investigate the connection between the minion of quantum advantage and other known minions controlling CSP tractability and width. In this way, we make use of complexity results from the algebraic theory of CSPs to characterise the occurrence of quantum advantage in the case of graphs, and to obtain new necessary and sufficient conditions in the case of arbitrary relational structures.
Information-processing tasks modelled by homomorphisms between relational structures can witness quantum advantage when entanglement is used as a computational resource. We prove that the occurrence of quantum advantage is determined by the same type of algebraic structure (known as a minion) that captures the polymorphism identities of CSPs and, thus, CSP complexity. We investigate the connection between the minion of quantum advantage and other known minions controlling CSP tractability and width. In this way, we make use of complexity results from the algebraic theory of CSPs to characterise the occurrence of quantum advantage in the case of graphs, and to obtain new necessary and sufficient conditions in the case of arbitrary relational structures.
Information-processing tasks modelled by homomorphisms between relational structures can witness quantum advantage when entanglement is used as a computational resource. We prove that the occurrence of quantum advantage is determined by the same type of algebraic structure (known as a minion) that captures the polymorphism identities of CSPs and, thus, CSP complexity. We investigate the connection between the minion of quantum advantage and other known minions controlling CSP tractability and width. In this way, we make use of complexity results from the algebraic theory of CSPs to characterise the occurrence of quantum advantage in the case of graphs, and to obtain new necessary and sufficient conditions in the case of arbitrary relational structures.
Constraint satisfaction problems (CSPs) with operator assignments to the variables provide a well-structured setting to study the decision complexity of the entangled value of classes of nonlocal games. Due to the CSP dichotomy theorem, the complexity of constraint satisfaction problems with classical assignments can be fully understood by studying the symmetries of the CSP, in terms of the polymorphisms of the underlying relational structure. In this work, we show that the polymorphism-based reductions between CSPs can be generalised to gap-preserving reductions between entangled CSPs based on an entangled analogue of the polymorphisms. This reduction allows us to show undecidability of entangled graph colouring with more than three colours, a problem that has proved resistant to prior hardness reductions based on commutativity gadgets.
We investigate quantum relaxations of two key decision problems in computer science: the constraint satisfaction problem (CSP) and the structure isomorphism problem. CSP asks whether a homomorphism exists between two relational structures, while structure isomorphism seeks an isomorphism between them. In recent years, it has become increasingly apparent that many special cases of CSP can be reformulated in terms of the existence of perfect classical strategies in non-local games, a key topic of study in quantum information theory. These games have allowed us to study quantum advantage in relation to many important decision problems, such as the k-colouring problem, and the problem of solving binary constraint systems. Abramsky et al. (2017) have shown that all of these games can be seen as special instances of a non-local CSP game. Moreover, they show that perfect quantum strategies in this CSP game can be viewed as Kleisli morphisms of a graded monad on the category of relational structures, which they dub the quantum monad. In this way, the quantum monad provides a categorical characterisation of quantum advantage for the non-local CSP game. In this work we solidify and expand the results of Abramsky et al., answering several of their open questions. Firstly, we compare the definition of quantum graph homomorphisms arising from this work with an earlier definition of the concept due to Mančinska and Roberson and show that there are graphs which exhibit quantum advantage under one definition but not the other. Our second contribution is to extend the results of Abramsky et al. which only hold in the tensor product framework of quantum mechanics to the commuting operator framework. Next, we study a non-local structure isomorphism game, which generalises the well-studied graph isomorphism game. We show how the construction of the quantum monad can be refined to provide categorical semantics for quantum strategies in this game. This results in a category where morphisms coincide with quantum homomorphisms and isomorphisms coincide with quantum isomorphisms.