Quantum Polymorphisms and the Complexity of Quantum Constraint Satisfaction
AI Breakdown
Get a structured breakdown of this paper — what it's about, the core idea, and key takeaways for the field.
Abstract
We introduce the concept of quantum polymorphisms to the complexity theory of quantum constraint satisfaction. Via this notion, we build an algebraic framework of reductions between quantum CSPs, and we establish a Galois connection between quantum polymorphism minions and quantum relational constructions. By leveraging a contextuality property of quantum polymorphisms, we fully characterise 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, only a partial classification was known for a subclass of Boolean languages and for non-Boolean languages meeting specific structural conditions [Culf--Mastel, FOCS'25]. As an application of our framework, we prove that the quantum CSPs parameterised by odd cycles and the quantum CSP expressing quantum satisfiability of Siggers clauses are undecidable.