The 3-dimensional cube is the only periodic, connected cubic graph with perfect state transfer

TL;DR

This paper proves that among connected cubic graphs, only the 3-dimensional cube exhibits perfect state transfer and periodicity, highlighting its unique role in quantum information transfer.

Contribution

The authors demonstrate that the 3-dimensional cube is uniquely the only connected cubic graph with perfect state transfer and periodicity, and conjecture this uniqueness extends to all connected cubic graphs.

Findings

3D cube is the only such graph with these properties

The result emphasizes the special role of the 3D cube in quantum networks

Conjecture that no other connected cubic graphs have perfect state transfer

Abstract

There is perfect state transfer between two vertices of a graph, if a single excitation can travel with fidelity one between the corresponding sites of a spin system modeled by the graph. When the excitation is back at the initial site, for all sites at the same time, the graph is said to be periodic. A graph is cubic if each of its vertices has a neighbourhood of size exactly three. We prove that the 3-dimensional cube is the only periodic, connected cubic graph with perfect state transfer. We conjecture that this is also the only connected cubic graph with perfect state transfer.