Comfortability of quantum walks on embedded graphs on surfaces

TL;DR

This paper introduces a quantum walk model on embedded graphs that incorporates surface topology, analyzes scattering to determine surface orientability, and finds that quantum walkers prefer surfaces with lower genus.

Contribution

It presents a novel quantum walk model reflecting surface embedding and uses scattering matrices to detect surface orientability and comfortability of quantum walkers.

Findings

Quantum walkers are more comfortable on surfaces with small genus.

Scattering matrices reveal the orientability of the embedding.

The model links quantum walk dynamics with surface topology.

Abstract

The time evolutions of discrete-time quantum walks on graphs are determined by the local adjacency relations of the graphs. In this paper, first, we construct a discrete-time quantum walk model that reflects the embedding on the surface so that an underlying global geometric information is reflected. Second, we consider the scattering problem of this quantum walk model. We obtain the scattering matrix characterized by the faces on the surface and detect the orientablility of the embedding using scattering information. For the stationary state in the scattering problem, the comfortability is defined as the square norm of the stationary state restricted to the internal. This indicates how a quantum walker is stored in the internal under the embedding. Then we find that a quantum walker feels more comfortable on a surface with small genus in some natural setting. We illustrate our results with some interesting examples.