  Categorical quantum mechanics and the Wolfram model offer distinct but
complementary approaches to studying the relationship between diagrammatic
rewriting systems over combinatorial structures and the foundations of physics;
the objective of the present article is to begin elucidating the formal
correspondence between the two methodologies in the context of the ZX-calculus
formalism of Coecke and Duncan for reasoning diagrammatically about linear maps
between qubits. After briefly summarizing the relevant formalisms, and
presenting a categorical formulation of the Wolfram model in terms of adhesive
categories and double-pushout rewriting systems, we illustrate how the
diagrammatic rewritings of the ZX-calculus can be embedded and realized within
the broader context of Wolfram model multiway systems, and illustrate some of
the capabilities of the software framework (ZXMultiwaySystem) that we have
developed specifically for this purpose. Finally, we present a proof (along
with an explicitly computed example) based on the methods of Dixon and
Kissinger that the multiway evolution graphs and branchial graphs of the
Wolfram model are naturally endowed with a monoidal structure based on rulial
composition that is, furthermore, compatible with the monoidal product of
ZX-diagrams.
