Quantum Polymorphisms and Commutativity Gadgets

2025-11-28

AI Summary

This paper investigates quantum polymorphisms of relational structures in nonlocal games, aiming to characterize the existence of commutativity gadgets and establish a quantum complexity theory for entangled constraint satisfaction problems (CSPs). Method: It introduces and systematically applies the notion of quantum polymorphisms for the first time, integrating tools from quantum information theory, nonlocal game analysis, algebraic polymorphisms, and non-oracular quantum homomorphisms to construct an algebraic framework tailored to entangled strategies. Contributions: (1) A complete classification of necessary and sufficient conditions for the existence of commutativity gadgets on arbitrary relational structures—generalizing prior results restricted to Boolean domains; (2) Development of a quantum Galois connection theory for entangled CSPs under quantum homomorphisms; (3) Proof of undecidability for entangled CSPs parameterized by odd cycles, establishing a new complexity benchmark for CSPs in quantum-enhanced computational models.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Machine Learning: Quantum Machine Learning
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Large-scale security measurements

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.

Problem

• Characterizing commutativity gadgets for relational structures using quantum polymorphisms
• Proving undecidability of entangled CSPs parameterized by odd cycles
• Establishing quantum Galois connections for non-oracular homomorphisms

Innovation

• Introducing quantum polymorphisms for non-local games
• Classifying commutativity gadgets for relational structures
• Establishing quantum Galois connection for entangled CSPs