Quantum walks have been shown to have impressive transport properties compared to classical random walks. However, imperfections in the quantum-walk algorithm can destroy any quantum-mechanical speedup due to Anderson localization. We numerically study the effect of static disorder on a quantum walk on the glued trees graph. For small disorder, we find that the dominant effect is a type of quantum decay, and not quantum localization. For intermediate disorder, there is a crossover to diffusive transport, while a localization transition is observed at large disorder, in agreement with Anderson localization on the Cayley tree.Quantum walks have been shown to have impressive transport properties compared to classical random walks. However, imperfections in the quantum-walk algorithm can destroy any quantum-mechanical speedup due to Anderson localization. We numerically study the effect of static disorder on a quantum walk on the glued trees graph. For small disorder, we find that the dominant effect is a type of quantum decay, and not quantum localization. For intermediate disorder, there is a crossover to diffusive transport, while a localization transition is observed at large disorder, in agreement with Anderson localization on the Cayley tree.

arXiv:1206.3178Physical Review AAug 27, 201228 citations

Tree-Line graphs and their quantum walks

Musung Kang

For a simple graph $Γ$ , a (bipartite)tree-line graph and a tree-graph of $Γ$ can be defined. With a (bipartite)tree-line graph constructed by the function $(b)\ell$ , we study the continuous quantum walk on $(b)\ell ^n Γ$ . An equitable partition of a bipartite tree-line graph is obtained by its corresponding derived tree graph. This paper also examines quantum walks on derived graphs, whose vertices represent their basis state.For a simple graph $Γ$, a (bipartite)tree-line graph and a tree-graph of $Γ$ can be defined. With a (bipartite)tree-line graph constructed by the function $(b)\ell$, we study the continuous quantum walk on $(b)\ell ^n Γ$. An equitable partition of a bipartite tree-line graph is obtained by its corresponding derived tree graph. This paper also examines quantum walks on derived graphs, whose vertices represent their basis state.

arXiv:2504.13000Apr 17, 20250 citations

For a simple graph

Γ

, a (bipartite)tree-line graph and a tree-graph of

Γ

can be defined. With a (bipartite)tree-line graph constructed by the function

(b)\ell

, we study the continuous quantum walk on

(b)\ell ^n Γ

. An equitable partition of a bipartite tree-line graph is obtained by its corresponding derived tree graph. This paper also examines quantum walks on derived graphs, whose vertices represent their basis state.