We study the existence of quantum state transfer in $\mathcal{Q}$ -graphs in this paper. The $\mathcal{Q}$ -graph of a graph $G$ , denoted by $\mathcal{Q}(G)$ , is the graph derived from $G$ by plugging a new vertex to each edge of $G$ and joining two new vertices which lie on adjacent edges of $G$ by an edge. We show that, if all eigenvalues of a regular graph $G$ are integers, then its $\mathcal{Q}$ -graph $\mathcal{Q}(G)$ has no perfect state transfer. In contrast, we also prove that the $\mathcal{Q}$ -graph of a regular graph has pretty good state transfer under some mild conditions. Finally, applying the obtained results, we also exhibit many new families of $\mathcal{Q}$ -graphs having no perfect state transfer, but admitting pretty good state transfer.We study the existence of quantum state transfer in $\mathcal{Q}$-graphs in this paper. The $\mathcal{Q}$-graph of a graph $G$, denoted by $\mathcal{Q}(G)$, is the graph derived from $G$ by plugging a new vertex to each edge of $G$ and joining two new vertices which lie on adjacent edges of $G$ by an edge. We show that, if all eigenvalues of a regular graph $G$ are integers, then its $\mathcal{Q}$-graph $\mathcal{Q}(G)$ has no perfect state transfer. In contrast, we also prove that the $\mathcal{Q}$-graph of a regular graph has pretty good state transfer under some mild conditions. Finally, applying the obtained results, we also exhibit many new families of $\mathcal{Q}$-graphs having no perfect state transfer, but admitting pretty good state transfer.

arXiv:2108.07590Aug 17, 20210 citations

Strongly regular and strongly walk-regular graphs that admit perfect state transfer

Sho Kubota, Hiroto Sekido, Harunobu Yata +1 more

We study perfect state transfer in Grover walks on two important classes of graphs: strongly regular graphs and strongly walk-regular graphs. The latter class is a generalization of the former. We first give a complete classification of strongly regular graphs that admit perfect state transfer. The only such graphs are the complete bipartite graph $K_{2,2}$ and the complete tripartite graph $K_{2,2,2}$ . We then show that, if a connected strongly walk-regular graph that is not a strongly regular graph admits perfect state transfer, then its spectrum must be of the form $\{[k]^1, [\frac{k}{2}]^α, [0]^β, [-\frac{k}{2}]^γ\}$ , and we enumerate all feasible spectra of this form up to $k=20$ with the help of a computer. These results are obtained using techniques from algebraic number theory and spectral graph theory, particularly through the analysis of eigenvalues and eigenprojections of a normalized adjacency matrix. While the setting is in quantum walks, the core discussion is developed entirely within the framework of spectral graph theory.We study perfect state transfer in Grover walks on two important classes of graphs: strongly regular graphs and strongly walk-regular graphs. The latter class is a generalization of the former. We first give a complete classification of strongly regular graphs that admit perfect state transfer. The only such graphs are the complete bipartite graph $K_{2,2}$ and the complete tripartite graph $K_{2,2,2}$. We then show that, if a connected strongly walk-regular graph that is not a strongly regular graph admits perfect state transfer, then its spectrum must be of the form $\{[k]^1, [\frac{k}{2}]^α, [0]^β, [-\frac{k}{2}]^γ\}$, and we enumerate all feasible spectra of this form up to $k=20$ with the help of a computer. These results are obtained using techniques from algebraic number theory and spectral graph theory, particularly through the analysis of eigenvalues and eigenprojections of a normalized adjacency matrix. While the setting is in quantum walks, the core discussion is developed entirely within the framework of spectral graph theory.

arXiv:2506.02530Jun 3, 20250 citations

Perfect state transfer in cubelike graphs

Wang-Chi Cheung, Chris D. Godsil

arXiv:1010.4721Linear Algebra and its ApplicationsMay 20, 201179 citations

We study the existence of quantum state transfer in

\mathcal{Q}

-graphs in this paper. The

\mathcal{Q}

-graph of a graph

G

, denoted by

\mathcal{Q}(G)

, is the graph derived from

G

by plugging a new vertex to each edge of

G

and joining two new vertices which lie on adjacent edges of

G

by an edge. We show that, if all eigenvalues of a regular graph

G

are integers, then its

\mathcal{Q}

-graph

\mathcal{Q}(G)

has no perfect state transfer. In contrast, we also prove that the

\mathcal{Q}

-graph of a regular graph has pretty good state transfer under some mild conditions. Finally, applying the obtained results, we also exhibit many new families of

\mathcal{Q}

-graphs having no perfect state transfer, but admitting pretty good state transfer.

K_{2,2}

and the complete tripartite graph

K_{2,2,2}

. We then show that, if a connected strongly walk-regular graph that is not a strongly regular graph admits perfect state transfer, then its spectrum must be of the form

\{[k]^1, [\frac{k}{2}]^α, [0]^β, [-\frac{k}{2}]^γ\}

, and we enumerate all feasible spectra of this form up to

k=20

with the help of a computer. These results are obtained using techniques from algebraic number theory and spectral graph theory, particularly through the analysis of eigenvalues and eigenprojections of a normalized adjacency matrix. While the setting is in quantum walks, the core discussion is developed entirely within the framework of spectral graph theory.