  Mesoscopic quantum systems exhibit complex many-body quantum phenomena, where
interactions between spins and charges give rise to collective modes and
topological states. Even simple, non-interacting theories display a rich
landscape of energy states --- distinct many-particle configurations connected
by spin- and energy-dependent transition rates. The collective energy landscape
is difficult to characterize or predict, especially in regimes of frustration
where many-body effects create a multiply degenerate landscape. Here we use
network science to characterize the complex interconnection patterns of these
energy-state transitions. Using an experimentally verified computational model
of electronic transport through quantum antidots, we construct networks where
nodes represent accessible energy states and edges represent allowed
transitions. We then explore how physical changes in currents and voltages are
reflected in the network topology. We find that the networks exhibit Rentian
scaling, which is characteristic of efficient transportation systems in
computer circuitry, neural circuitry, and human mobility, and can be used to
measure the interconnection complexity of a network. Remarkably, networks
corresponding to points of frustration in quantum transport (due, for example,
to spin-blockade effects) exhibit an enhanced topological complexity relative
to networks not experiencing frustration. Our results demonstrate that network
characterizations of the abstract topological structure of energy landscapes
can capture salient properties of quantum transport. More broadly, our approach
motivates future efforts to use network science in understanding the dynamics
and control of complex quantum systems.
