Oscillatory Localization of Quantum Walks by Classical Electric Circuits
Andris Ambainis, Krišjānis Prūsis, Jevgēnijs Vihrovs, Thomas G. Wong · arXiv (Cornell University) · 2016
Power dissipation and effective resistance are ubiquitous quantities in electric circuits. We connect these classical notions with a new quantum phenomenon, proving that a discrete-time quantum walk oscillates between two states if the power dissipation on a related electric network is low. By applying this framework to starting states along a single edge of the graph, we show that low effective resistance implies oscillatory localization of the quantum walk. This includes a large variety of regular graphs, such as edge-transitive, expander, and high degree graphs. As a corollary, it follows that high edge-connectivity also implies localization of these states, since it is closely related to electric resistance.