  Since the introduction of the Hermitian adjacency matrix for digraphs,
interest in so-called complex unit gain graphs has surged. In this work, we
consider gain graphs whose spectra contain the minimum number of two distinct
eigenvalues. Analogously to graphs with few distinct eigenvalues, a great deal
of structural symmetry is required for a gain graph to attain this minimum.
This allows us to draw a surprising parallel to well-studied systems of lines
in complex space, through a natural correspondence to unit-norm tight frames.
We offer a full classification of two-eigenvalue gain graphs with degree at
most $4$, or with multiplicity at most $3$. Intermediate results include an
extensive review of various relevant concepts related to lines in complex
space, including SIC-POVMs, MUBs and geometries such as the Coxeter-Todd
lattice, and many examples obtained as induced subgraphs by employing a
technique parallel to the dismantling of association schemes. Finally, we touch
on an innovative application of simulated annealing to find examples by
computer.
