# Percolated quantum walks with a general shift operator: From trapping to transport

## TL;DR

This paper introduces a generalized framework for discrete-time quantum walks on arbitrary graphs, analyzing their asymptotic behavior and transport efficiency, with applications to percolated quantum transport and localization phenomena.

## Contribution

It develops a broad, flexible definition of quantum walks incorporating various shift operators and graph geometries, enabling detailed analysis of percolation effects and localization.

## Key findings

- Edge-3-coloring enables non-stationary asymptotic behavior.
- Localized attractors influence quantum transport efficiency.
- Percolated cube demonstrates transport dynamics on complex graphs.

## Abstract

We present a generalized definition of discrete-time quantum walks convenient for capturing a rather broad spectrum of walker's behavior on arbitrary graphs. It includes and covers both: the geometry of possible walker's positions with interconnecting links and the prescribed rule in which directions the walker will move at each vertex. While the former allows for the analysis of inhomogeneous quantum walks on graphs with vertices of varying degree, the latter offers us to choose, investigate, and compare quantum walks with different shift operators. The synthesis of both key ingredients constitutes a well-suited playground for analyzing percolated quantum walks on a quite general class of graphs. Analytical treatment of the asymptotic behavior of percolated quantum walks is presented and worked out in details for the Grover walk on graphs with maximal degree 3. We find, that for these walks with cyclic shift operators the existence of an edge-3-coloring of the graph allows for non-stationary asymptotic behavior of the walk. For different shift operators the general structure of localized attractors is investigated, which determines the overall efficiency of a source-to-sink quantum transport across a dynamically changing medium. As a simple nontrivial example of the theory we treat a single excitation transport on a percolated cube.