Quantum walks and the size of the graph

TL;DR

This paper establishes lower bounds on the size of graphs necessary for quantum phenomena like periodicity and perfect state transfer in continuous-time quantum walks, linking graph structure to quantum behavior.

Contribution

It provides the first lower bounds on graph size related to quantum phenomena, specifically relating edges and eccentricity in the adjacency matrix model.

Findings

Number of edges bounded below by a cubic function of eccentricity

Insights into graph shapes that admit periodicity or perfect state transfer

Raises extremal questions for future research

Abstract

A continuous-time quantum walk is modelled using a graph. In this short paper, we provide lower bounds on the size of a graph that would allow for some quantum phenomena to occur. Among other things, we show that, in the adjacency matrix quantum walk model, the number of edges is bounded below by a cubic function on the eccentricity of a periodic vertex. This gives some idea on the shape of a graph that would admit periodicity or perfect state transfer. We also raise some extremal type of questions in the end that could lead to future research.