Quantum Advantage and CSP Complexity

TL;DR

This paper explores the relationship between quantum advantage in information processing tasks and the algebraic structures that determine the complexity of constraint satisfaction problems (CSPs), revealing new insights into when quantum resources provide computational benefits.

Contribution

It establishes that quantum advantage is governed by the same algebraic structures (minions) that determine CSP complexity, linking quantum information theory with algebraic CSP theory.

Findings

Quantum advantage is characterized by specific algebraic structures (minions).

Connections between quantum advantage minions and CSP tractability are identified.

New necessary and sufficient conditions for quantum advantage in relational structures are derived.

Abstract

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.