  Adapting a definition of Aaronson and Ambainis [Theory Comput. 1 (2005),
47--79], we call a quantum dynamics on a digraph "saturated Z-local" if the
nonzero transition amplitudes specifying the unitary evolution are in exact
correspondence with the directed edges (including loops) of the digraph. This
idea appears recurrently in a variety of contexts including angular momentum,
quantum chaos, and combinatorial matrix theory. Complete characterization of
the digraph properties that allow such a process to exist is a long-standing
open question that can also be formulated in terms of minimum rank problems. We
prove that saturated Z-local dynamics involving complex amplitudes occur on a
proper superset of the digraphs that allow restriction to the real numbers or,
even further, the rationals. Consequently, among these fields, complex numbers
guarantee the largest possible choice of topologies supporting a discrete
quantum evolution. A similar construction separates complex numbers from the
skew field of quaternions. The result proposes a concrete ground for
distinguishing between complex and quaternionic quantum mechanics.
