Quantum Advantage and CSP Complexity
Lorenzo Ciardo
TL;DR
This workDevelops an algebraic framework for quantum advantage in constraint satisfaction problems by introducing the quantum minion Q_H and proving that quantum advantage is characterized by minion-homomorphisms to polymorphism minions. It establishes a triptych of results: (i) for dim(H) ≤ 2, Q_H collapses to the dictator minion, (ii) for dim(H) ≥ 3, Q_H remains distinct and links quantum advantage to CSP hardness and unbounded width, and (iii) Q_H maps to the SDP/minion S, tying quantum advantage to SDP-based tractability and yielding a graph-level dichotomy (non-bipartite graphs have quantum advantage when dim(H) ≥ 3). The framework is extended to promise CSPs, showing that quantum advantage for pairs of structures corresponds to promise-CSP complexity via SDP/CLP relaxations, and reveals separations between single-structure and two-structure quantum advantages. Overall, the paper demonstrates that quantum resources induce a CSP-algebraic landscape parallel to classical tractability questions, with precise conditions governed by minion relations and well-known CSP dichotomies.
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.