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Quantum Polymorphisms and Commutativity Gadgets

Lorenzo Ciardo, Gideo Joubert, Antoine Mottet

TL;DR

This work develops a quantum-algebraic framework for entangled CSPs by introducing quantum polymorphisms and commutativity gadgets. A central result is a precise equivalence: a CSP language admits commutativity gadgets if and only if its quantum polymorphisms coincide with the quantum closure of its classical polymorphisms. Using this, the authors prove that odd cycles induce undecidable entangled CSPs and extend the framework to Boolean structures and non-oracular variants via a quantum Galois-type connection, yielding logspace reductions. The methodology quantizes classical reductions and provides a unified lens on contextuality phenomena in CSP reductions, with broad implications for the complexity of quantum Constraint Satisfaction problems.

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.

Quantum Polymorphisms and Commutativity Gadgets

TL;DR

This work develops a quantum-algebraic framework for entangled CSPs by introducing quantum polymorphisms and commutativity gadgets. A central result is a precise equivalence: a CSP language admits commutativity gadgets if and only if its quantum polymorphisms coincide with the quantum closure of its classical polymorphisms. Using this, the authors prove that odd cycles induce undecidable entangled CSPs and extend the framework to Boolean structures and non-oracular variants via a quantum Galois-type connection, yielding logspace reductions. The methodology quantizes classical reductions and provides a unified lens on contextuality phenomena in CSP reductions, with broad implications for the complexity of quantum Constraint Satisfaction problems.

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.

Paper Structure

This paper contains 15 sections, 40 theorems, 90 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Contributions
  3. Structure of the paper
  4. Preliminaries
  5. Relational structures
  6. Quantum CSPs
  7. Quantum polymorphisms and reductions
  8. Warm-up: Quantum polymorphisms of cliques
  9. Intermezzo: Contextuality and bifurcations
  10. Quantum polymorphisms of odd cycles
  11. The non-oracular setting
  12. Quantum polymorphisms of Boolean structures
  13. The case of structures without a majority polymorphism
  14. The minimal contextual clone
  15. Acknowledgments

Key Result

Lemma 2

Let $Q\colon\mathbb{X}\to_{H}\mathbb{A}$ and $Q'\colon\mathbb{X}\to_{H'}\mathbb{A}$ be quantum homomorphisms. Then $Q\oplus Q'$ is a quantum homomorphism $\mathbb{X}\to_{H\oplus H'}\mathbb{A}$.

Figures (3)

  • Figure 1: A bifurcation of length $2$. Solid lines mean non-orthogonality.
  • Figure 2: A bifurcation of length $\omega$. Solid lines mean non-orthogonality.
  • Figure 3: Simplified illustration of $\mathbb{B}^4$. Black lines indicate the poset $S_{10}$. The blue (resp. red) dotted lines show parts of $S_{00}$ (resp. $S_{11}$); this relation additionally relates $(0,0,0,0)$ (resp. $(1,1,1,1)$) to everything and relates two nodes when they are a bit-flips of each other (see coloring).

Theorems & Definitions (95)

  • Definition 1: Entangled CSP
  • Lemma 2
  • proof
  • Definition 3
  • Lemma 4
  • proof
  • Definition 5
  • Lemma 6: abramsky2017quantum
  • proof
  • Lemma 7
  • ...and 85 more