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arXiv · 1606.02136

Oscillatory Localization of Quantum Walks Analyzed by Classical Electric Circuits

Andris Ambainis·Krišjānis Prūsis·Jevgēnijs Vihrovs·Thomas G. Wong
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Abstract

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.

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Andris Ambainis, Krišjānis Prūsis, Jevgēnijs Vihrovs, Thomas G. Wong. 2016-11-16. Oscillatory Localization of Quantum Walks Analyzed by Classical Electric Circuits. https://doi.org/10.1103/physreva.94.062324

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