Time evolution of continuous-time quantum walks on dynamical percolation graphs

TL;DR

This paper analyzes how continuous-time quantum walks evolve on graphs that change randomly over time, revealing a universal time rescaling effect caused by percolation in the rapid-change limit.

Contribution

It derives explicit formulas for quantum walk evolution on rapidly changing graphs and proves that percolation induces a universal time rescaling effect.

Findings

Percolation causes a universal time rescaling in quantum walks.

Explicit formulas for the evolution in the rapid-change limit are derived.

Numerical simulations confirm the theoretical predictions.

Abstract

We study the time evolution of continuous-time quantum walks on randomly changing graphs. At certain moments edges of the graph appear or disappear with a given probability. We focus on the case when the time interval between subsequent changes of the graph tends to zero. We derive explicit formulae for the general evolution in this limit. We find that the percolation in this limit causes an effective time rescaling. Independently of the graph and the initial state of the walk, the time is rescaled by the probability of keeping and edge. Both the individual trajectories for a single system and average properties with a superoperator formalism are discussed. We give an analytical proof for our theorem and we also present results from numerical simulations of the phenomena for different graphs.