Oscillatory Localization of Quantum Walks by Classical Electric Circuits
We introduce a new quantum phenomenon of oscillatory localization in which a discrete time quantum walk oscillates between two states. We then connect it to the power dissipation of a related electric network. Namely, the exact form of quantum oscillatory localization occurs for only two kinds of states, called uniform states and flip states, and 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.