  We examine an unexplored quantum phenomenon we call oscillatory localization,
where a discrete-time quantum walk with Grover's diffusion coin jumps back and
forth between two vertices. We then connect it to the power dissipation of a
related electric network. Namely, we show that there are only two kinds of
oscillating states, called uniform states and flip states, and that the
projection of an arbitrary state onto a flip state is bounded by the power
dissipation of an electric circuit. By applying this framework to states along
a single edge of a graph, we show that low effective resistance implies
oscillatory localization of the quantum walk. This reveals that oscillatory
localization occurs on a large variety of regular graphs, including
edge-transitive, expander, and high degree graphs. As a corollary, high
edge-connectivity also implies localization of these states, since it is
closely related to electric resistance.
