Oscillatory Localization of Quantum Walks Analyzed by Classical Electric Circuits

TL;DR

This paper introduces the concept of oscillatory localization in quantum walks, linking it to electric circuit dissipation, and demonstrates its occurrence on various regular graphs through a novel classical-quantum analogy.

Contribution

It establishes a new framework connecting quantum walk localization to electric circuit properties, specifically power dissipation and effective resistance.

Findings

Oscillatory localization occurs on regular graphs with low effective resistance.

Two types of oscillating states identified: uniform and flip states.

High edge-connectivity implies localization due to low electric resistance.

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.