Universality of Graph Homomorphism Games and the Quantum Coloring Problem

We show that quantum graph parameters for finite, simple, undirected graphs encode winning strategies for all possible synchronous non-local games. Given a synchronous game \(\mathcal {G}=(I,O,\lambda )\) with \(|I|=n\) and \(|O|=k\), we demonstrate what we call a weak \(*\)-equivalence between \(\mathcal {G}\) and a 3-coloring game on a graph with at most \(3+n+9n(k-2)+6|\lambda ^{-1}(\{0\})|\) vertices, strengthening and simplifying work implied by Ji [16] for winning quantum strategies for synchronous non-local games. As an application, we obtain a quantum version of Lovász’s reduction [21] of the k-coloring problem for a graph G with n vertices and m edges to the 3-coloring problem for a graph with \(3+n+9n(k-2)+6mk\) vertices. Moreover, winning strategies for a synchronous game \(\mathcal {G}\) can be transformed into winning strategies for an associated graph coloring game, where the strategies exhibit perfect zero knowledge for an honest verifier. We also show that, for “graph of the game” \(X(\mathcal {G})\) associated with \(\mathcal {G}\) from Atserias et al. [1], the independence number game \(\text {Hom}(K_{|I|},\overline{X(\mathcal {G})})\) is hereditarily \(*\)-equivalent to \(\mathcal {G}\), so that the possibility of winning strategies is the same in both games for all models, except the game algebra. Thus, the quantum versions of the chromatic number, independence number and clique number encode winning strategies for all synchronous games in all quantum models.