Quantum Polymorphisms and the Complexity of Quantum Constraint Satisfaction

TL;DR

This paper develops an algebraic framework for quantum constraint satisfaction problems using quantum polymorphisms, characterizes their properties, and proves certain quantum CSPs are undecidable.

Contribution

It introduces quantum polymorphisms, establishes a Galois connection, and fully characterizes commutativity gadgets, advancing the understanding of quantum CSP complexity.

Findings

Quantum CSPs parameterized by odd cycles are undecidable.

Quantum polymorphisms relate to quantum relational structures.

Full classification of quantum polymorphisms for certain languages is achieved.

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.