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

Oscillatory Localization of Quantum Walks Analyzed by Classical Electric Circuits