  We study the problem of counting the number of homomorphisms from an input
graph $G$ to a fixed (quantum) graph $\bar{H}$ in any finite field of prime
order $\mathbb{Z}_p$. The subproblem with graph $H$ was introduced by Faben and
Jerrum [ToC'15] and its complexity is subject to a growing series of research
articles, e.g. the work of Focke, Goldberg, Roth, and Zivn\'y [SIDMA'21] and
the work of Bulatov and Kazeminia [STOC'22], subsequent to this article's
conference version. Our contribution is threefold. First, we introduce the
study of quantum graphs to the study of modular counting homomorphisms. We show
that the complexity for a quantum graph $\bar{H}$ collapses to the complexity
criteria found at dimension 1: graphs. Second, in order to prove cases of
intractability we establish a further reduction to the study of bipartite
graphs. Lastly, we establish a dichotomy for all bipartite
($K_{3,3}\backslash\{e\}$, ${domino}$)-free graphs by a thorough structural
study incorporating both local and global arguments. This result subsumes all
results on bipartite graphs known for all prime moduli and extends them
significantly. Even for the subproblem with $p$ equal to $2$, this establishes
new results.
