Motivated by non-local games and quantum coloring problems, we introduce a graph homomorphism game between quantum graphs and classical graphs. This game is naturally cast as a “quantum–classical game,” that is, a non-local game of two players involving quantum questions and classical answers. This game generalizes the graph homomorphism game between classical graphs. We show that winning strategies in the various quantum models for the game is an analog of the notion of non-commutative graph homomorphisms due to Stahlke [IEEE Trans. Inf. Theory 62(1), 554–577 (2016)]. Moreover, we present a game algebra in this context that generalizes the game algebra for graph homomorphisms given by Helton et al. [New York J. Math. 25, 328–361 (2019)]. We also demonstrate explicit quantum colorings of all quantum complete graphs, yielding the surprising fact that the algebra of the four coloring game for a quantum graph is always non-trivial, extending a result of Helton et al. [New York J. Math. 25, 328–361 (2019)].

arXiv:2009.07229Journal of Mathematical PhysicsNov 1, 202220 citations

Composing classical and quantum relaxations of CSP and structure isomorphism

Mohammad Karamlou

This thesis investigates relaxations of two key decision problems in computer science: the constraint satisfaction problem (CSP) and the structure isomorphism problem. CSP asks whether a homomorphism exists between two relational structures, while structure isomorphism seeks an isomorphism between them. By adopting a categorical approach, we add to a growing list of results showing that relaxations of these problems, studied in universal algebra, finite model theory, and quantum information can all be captured abstractly using (co)monads. In particular, we will show that distribution minion monads capture many important minion tests from universal algebra, and that the quantum monad can be used to capture quantum homomorphisms and isomorphisms from quantum information. These results add to the already well-established connection between game comonads and Spoiler-Duplicator games from finite model theory. A key focus is the exploration of distributive laws, the existence of which enables the combination of comonads and monads, potentially leading to novel relaxations of CSP and structure isomorphism that integrate ideas from different domains. We establish a no-go theorem, showing that game comonads like the Ehrenfeucht-Fraïssé and pebbling comonads do not distribute over many important distribution minion monads, such as the basic linear programming and arc-consistency monads. On the positive side, we identify sufficient conditions under which a distributive law between game comonads and distribution minion monads exists. Neither our no-go theorems nor our positive results apply to the quantum monad. Thus, it remains an open question whether any game comonads distribute over the quantum monad. In addition, we prove several novel results related to quantum advantage. Firstly, we compare two existing definitions of quantum graph homomorphisms, and show that there are graphs where quantum advantage manifests under one definition but not the other. Secondly, we extend existing characterisations of tensor product strategies for the non-local CSP and structure isomorphism games to the more general class of commuting operator strategies. Finally, we introduce quantum versions of one-sided Spoiler-Duplicator games and show that quantum advantage is possible in such games. This result suggests that there is scope for adapting further tools from finite model theory to improve our understanding of quantum homomorphisms and quantum isomorphisms.

Jan 1, 20250 citations