RAIRO - Operations Research

Quantum state transfer on 𝒬-graphs

Xiao-Qin Zhang, Shu‐Yu Cui, Gui‐Xian Tian

Published Apr 11, 2025• 2 citations

Abstract

Given a graph G, the graph 𝒬(G) is obtained by inserting a new vertex into every edge of G, then connecting those inserted vertices on the neighbouring edges in G. We usually refer to 𝒬(G) as the 𝒬-graph of G. This article mainly focuses on the effects of 𝒬-graph operation on quantum state transfer. It is shown that if G is an integral r-regular graph for r ≥ 2, then 𝒬(G) always avoids perfect state transfer. In contrast, we also show that, this 𝒬-graph exists pretty good state transfer under some special circumstances. Subsequently, by applying the results obtained above, we also display many new classes of 𝒬-graphs that avoid perfect state transfer but exist pretty good state transfer.

DOI: 10.1051/ro/2025046

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RAIRO - Operations Research

Quantum state transfer on 𝒬-graphs

Xiao-Qin Zhang, Shu‐Yu Cui, Gui‐Xian Tian

Published Apr 11, 2025• 2 citations

Publisher

Abstract

DOI: 10.1051/ro/2025046

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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.

The edge corona product of a graph

G

with size

m

and another graph

H

, usually expressed as

G\diamond H

, is formed by taking one copy of

G

and

m

copies of

H

, and then connecting each endpoint of the

i

-th edge of

G

to each vertex of the

i

-th copy of

H

. This paper investigates perfect state transfer (PST) and pretty good state transfer (PGST) in continuous-time quantum walks on edge corona product of graphs. We establish some sufficient conditions under which this product has no periodic vertices; consequently, PST does not occur. We also exhibit an infinite family of periodic edge corona graphs that do not admit PST, and further give sufficient conditions for the occurrence of PGST on edge corona product of graphs.