arXiv:1605.05260 · The Electronic Journal of Combinatorics

Quantum State Transfer in Coronas

Ethan Ackelsberg, Zachary Brehm, Ada Chan, Joshua Mundinger, Christino Tamon

Published May 19, 2017• 22 citations

Abstract

We study state transfer in quantum walks on graphs relative to the adjacency matrix. Our motivation is to understand how the addition of pendant subgraphs affect state transfer. For two graphs $G$ and $H$, the Frucht-Harary corona product $G \circ H$ is obtained by taking $|G|$ copies of the cone $K_{1} + H$ and by identifying the conical vertices according to $G$. Our work explores conditions under which the corona $G \circ H$ exhibits state transfer. We also describe new families of graphs with state transfer based on the corona product. Some of these constructions provide a generalization of related known results.

DOI: 10.37236/6145

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Perfect state transfer in neighborhood coronas

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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.

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arXiv:1605.05260 · The Electronic Journal of Combinatorics

Quantum State Transfer in Coronas

Ethan Ackelsberg, Zachary Brehm, Ada Chan, Joshua Mundinger, Christino Tamon

Published May 19, 2017• 22 citations

arXiv HTML PDF

Abstract

We study state transfer in quantum walks on graphs relative to the adjacency matrix. Our motivation is to understand how the addition of pendant subgraphs affect state transfer. For two graphs

G

and

H

, the Frucht-Harary corona product

G \circ H

is obtained by taking

|G|

copies of the cone

K_{1} + H

and by identifying the conical vertices according to

G

. Our work explores conditions under which the corona

G \circ H

exhibits state transfer. We also describe new families of graphs with state transfer based on the corona product. Some of these constructions provide a generalization of related known results.

DOI: 10.37236/6145

Related papers

Perfect state transfer in neighborhood coronas

Xing-Kun Song

The neighborhood corona $G \star H$ is the graph obtained by taking one copy $G$ and $|G|$ copies of $H$ , and joining each vertex of the $j$ th copy of $H$ to all neighbors of $v_{j}$ in $G$ . In this paper, we study the state transfer of neighborhood coronas related to the adjacency matrix. Concretely, we provide some necessary conditions under which the neighborhood corona $G \star H$ admits perfect state transfer, and obtain a new family of graphs with pretty good state transfer based on neighborhood coronas.The neighborhood corona $G \star H$ is the graph obtained by taking one copy $G$ and $|G|$ copies of $H$, and joining each vertex of the $j$th copy of $H$ to all neighbors of $v_{j}$ in $G$. In this paper, we study the state transfer of neighborhood coronas related to the adjacency matrix. Concretely, we provide some necessary conditions under which the neighborhood corona $G \star H$ admits perfect state transfer, and obtain a new family of graphs with pretty good state transfer based on neighborhood coronas.

arXiv:2204.05034Apr 11, 20220 citations

Quantum state transfer on edge coronae

Gao-Le Dai, Shu‐Yu Cui, Gui‐Xian Tian +1 more

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.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.

RAIRO - Operations ResearchSep 29, 20260 citations

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ABSTRACT The efficient implementation of quantum information processing relies on the reliable transfer of quantum states between specified vertices. This paper investigates the time required to achieve perfect state transfer (PST) on corona product graphs . To leverage the high symmetry of such graph structures, we employ a compression method to decompose their state space into several disjoint strata. By designing the coupling strengths between vertices, we analyze the time required for PST from a fixed vertex to vertices within each stratum. Furthermore, while keeping the number of vertices in the base graph fixed, we examine how the time required for PST on the corona product graph varies as the number of vertices in the other graph changes. Finally, we extend the obtained results to the general case where the base graph has an arbitrary number of vertices, thereby providing a complete characterization of the conditions for achieving PST.ABSTRACT The efficient implementation of quantum information processing relies on the reliable transfer of quantum states between specified vertices. This paper investigates the time required to achieve perfect state transfer (PST) on corona product graphs . To leverage the high symmetry of such graph structures, we employ a compression method to decompose their state space into several disjoint strata. By designing the coupling strengths between vertices, we analyze the time required for PST from a fixed vertex to vertices within each stratum. Furthermore, while keeping the number of vertices in the base graph fixed, we examine how the time required for PST on the corona product graph varies as the number of vertices in the other graph changes. Finally, we extend the obtained results to the general case where the base graph has an arbitrary number of vertices, thereby providing a complete characterization of the conditions for achieving PST.

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Quantum walks and graph operations

Joy Cooper, Homer De Vera, Hermie Monterde +3 more

Let $U_X(t)$ be the transition matrix of a quantum walk on a graph $X$ relative to its adjacency matrix $A$ or the Laplacian matrix $L$ . This paper investigates the behavior of quantum walks under Cartesian products, joins, and graph complements. We have two main goals. First, we characterize the conditions such that peak state transfer and pretty good state transfer are preserved under these operations, allowing us to construct new families of graphs admitting these properties. Our second goal is to analyze the relationship between the quantum walks on a graph and its complement. We provide bounds for $f_{u,v}(t)=\big|U_{X^c}(t)_{u,v}-e^{itδ}U_X(-t)_{u,v}\big|$ and $g_{u,v}(t)=\big||U_X(t)_{u,v}|-|U_{X^c}(t)_{u,v}|\big|$ , where $δ=-1$ when dealing with $A$ and $δ=n$ otherwise. Note that $f_{u,v}(t)$ and $g_{u,v}(t)$ both measure the difference between the behavior of quantum state transfer between vertices $u$ and $v$ in a graph and its complement. If $X$ is regular or $M=L$ , then $f_{u,v}(t)$ is bounded above by $\frac{2}{|V(X)|}$ . If $X$ is non-regular and $M=A$ , then we utilize the main eigenvalues of a graph to obtain an upper bound for $f_{u,v}(t)$ which depends only on $A$ . We also use the bounding matrix of the graph to give bounds for the Nordhaus-Gaddum type relations $|U_X(t)_{u,v}|+|U_{X^c}(t)_{u,v}|$ and $|U_X(t)_{u,v}|\cdot |U_{X^c}(t)_{u,v}|$ . Finally, we demonstrate that most of our bounds are sharp for certain families of graphs.Let $U_X(t)$ be the transition matrix of a quantum walk on a graph $X$ relative to its adjacency matrix $A$ or the Laplacian matrix $L$. This paper investigates the behavior of quantum walks under Cartesian products, joins, and graph complements. We have two main goals. First, we characterize the conditions such that peak state transfer and pretty good state transfer are preserved under these operations, allowing us to construct new families of graphs admitting these properties. Our second goal is to analyze the relationship between the quantum walks on a graph and its complement. We provide bounds for $f_{u,v}(t)=\big|U_{X^c}(t)_{u,v}-e^{itδ}U_X(-t)_{u,v}\big|$ and $g_{u,v}(t)=\big||U_X(t)_{u,v}|-|U_{X^c}(t)_{u,v}|\big|$, where $δ=-1$ when dealing with $A$ and $δ=n$ otherwise. Note that $f_{u,v}(t)$ and $g_{u,v}(t)$ both measure the difference between the behavior of quantum state transfer between vertices $u$ and $v$ in a graph and its complement. If $X$ is regular or $M=L$, then $f_{u,v}(t)$ is bounded above by $\frac{2}{|V(X)|}$. If $X$ is non-regular and $M=A$, then we utilize the main eigenvalues of a graph to obtain an upper bound for $f_{u,v}(t)$ which depends only on $A$. We also use the bounding matrix of the graph to give bounds for the Nordhaus-Gaddum type relations $|U_X(t)_{u,v}|+|U_{X^c}(t)_{u,v}|$ and $|U_X(t)_{u,v}|\cdot |U_{X^c}(t)_{u,v}|$. Finally, we demonstrate that most of our bounds are sharp for certain families of graphs.

arXiv:2610.07673Oct 6, 20260 citations

Signless Laplacian State Transfer on Vertex Complemented Coronae

Zhu, Ke-Yu, Gui‐Xian Tian, Shu‐Yu Cui

Given a graph $G$ with vertex set $V(G)=\{v_1,v_2,\ldots,v_{n_1}\}$ and a graph $H$ of order $n_2$ , the vertex complemented corona, denoted by $G\tilde{\circ}{H}$ , is the graph produced by copying $H$ $n_1$ times, with the $i$ -th copy of $H$ corresponding to the vertex $v_i$ , and then adding edges between any vertex in $V(G)\setminus\{v_{i}\}$ and any vertex of the $i$ -th copy of $H$ . The present article deals with quantum state transfer of vertex complemented coronae concerning signless Laplacian matrix. Our research investigates conditions in which signless Laplacian perfect state transfer exists or not on vertex complemented coronae. Additionally, we also provide some mild conditions for the class of graphs under consideration that allow signless Laplacian pretty good state transfer.Given a graph $G$ with vertex set $V(G)=\{v_1,v_2,\ldots,v_{n_1}\}$ and a graph $H$ of order $n_2$, the vertex complemented corona, denoted by $G\tilde{\circ}{H}$, is the graph produced by copying $H$ $n_1$ times, with the $i$-th copy of $H$ corresponding to the vertex $v_i$, and then adding edges between any vertex in $V(G)\setminus\{v_{i}\}$ and any vertex of the $i$-th copy of $H$. The present article deals with quantum state transfer of vertex complemented coronae concerning signless Laplacian matrix. Our research investigates conditions in which signless Laplacian perfect state transfer exists or not on vertex complemented coronae. Additionally, we also provide some mild conditions for the class of graphs under consideration that allow signless Laplacian pretty good state transfer.

arXiv:2502.14460Feb 20, 20250 citations

Perfect state transfer in products and covers of graphs

Gabriel Coutinho, Chris D. Godsil

A continuous-time quantum walk on a graph is represented by the complex matrix , where is the adjacency matrix of and is a non-negative time. If the graph models a network of interacting qubits, transfer of state among such qubits throughout time can be formalized as the action of the continuous-time quantum walk operator in the characteristic vectors of the vertices. Here, we are concerned with the problem of determining which graphs admit a perfect transfer of state. More specifically, we will study graphs whose adjacency matrix is a sum of tensor products of -matrices, focusing on the case where a graph is the tensor product of two other graphs. As a result, we will construct many new examples of perfect state transfer.A continuous-time quantum walk on a graph is represented by the complex matrix , where is the adjacency matrix of and is a non-negative time. If the graph models a network of interacting qubits, transfer of state among such qubits throughout time can be formalized as the action of the continuous-time quantum walk operator in the characteristic vectors of the vertices. Here, we are concerned with the problem of determining which graphs admit a perfect transfer of state. More specifically, we will study graphs whose adjacency matrix is a sum of tensor products of -matrices, focusing on the case where a graph is the tensor product of two other graphs. As a result, we will construct many new examples of perfect state transfer.

arXiv:1501.04396Linear and Multilinear AlgebraApr 22, 201527 citations

We study state transfer in quantum walks on graphs relative to the adjacency matrix. Our motivation is to understand how the addition of pendant subgraphs affect state transfer. For two graphs

G

and

H

, the Frucht-Harary corona product

G \circ H

is obtained by taking

|G|

copies of the cone

K_{1} + H

and by identifying the conical vertices according to

G

. Our work explores conditions under which the corona

G \circ H

exhibits state transfer. We also describe new families of graphs with state transfer based on the corona product. Some of these constructions provide a generalization of related known results.

The neighborhood corona

G \star H

is the graph obtained by taking one copy

G

and

|G|

copies of

H

, and joining each vertex of the

j

th copy of

H

to all neighbors of

v_{j}

G

. In this paper, we study the state transfer of neighborhood coronas related to the adjacency matrix. Concretely, we provide some necessary conditions under which the neighborhood corona

G \star H

admits perfect state transfer, and obtain a new family of graphs with pretty good state transfer based on neighborhood coronas.

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.

Let

U_X(t)

be the transition matrix of a quantum walk on a graph

X

relative to its adjacency matrix

A

or the Laplacian matrix

L

. This paper investigates the behavior of quantum walks under Cartesian products, joins, and graph complements. We have two main goals. First, we characterize the conditions such that peak state transfer and pretty good state transfer are preserved under these operations, allowing us to construct new families of graphs admitting these properties. Our second goal is to analyze the relationship between the quantum walks on a graph and its complement. We provide bounds for

f_{u,v}(t)=\big|U_{X^c}(t)_{u,v}-e^{itδ}U_X(-t)_{u,v}\big|

and

g_{u,v}(t)=\big||U_X(t)_{u,v}|-|U_{X^c}(t)_{u,v}|\big|

, where

δ=-1

when dealing with

A

and

δ=n

otherwise. Note that

f_{u,v}(t)

and

g_{u,v}(t)

both measure the difference between the behavior of quantum state transfer between vertices

u

and

v

in a graph and its complement. If

X

is regular or

M=L

, then

f_{u,v}(t)

is bounded above by

\frac{2}{|V(X)|}

. If

X

is non-regular and

M=A

, then we utilize the main eigenvalues of a graph to obtain an upper bound for

f_{u,v}(t)

which depends only on

A

. We also use the bounding matrix of the graph to give bounds for the Nordhaus-Gaddum type relations

|U_X(t)_{u,v}|+|U_{X^c}(t)_{u,v}|

and

|U_X(t)_{u,v}|\cdot |U_{X^c}(t)_{u,v}|

. Finally, we demonstrate that most of our bounds are sharp for certain families of graphs.

Given a graph

G

with vertex set

V(G)=\{v_1,v_2,\ldots,v_{n_1}\}

and a graph

H

of order

n_2

, the vertex complemented corona, denoted by

G\tilde{\circ}{H}

, is the graph produced by copying

H

n_1

times, with the

i

-th copy of

H

corresponding to the vertex

v_i

, and then adding edges between any vertex in

V(G)\setminus\{v_{i}\}

and any vertex of the

i

-th copy of

H

. The present article deals with quantum state transfer of vertex complemented coronae concerning signless Laplacian matrix. Our research investigates conditions in which signless Laplacian perfect state transfer exists or not on vertex complemented coronae. Additionally, we also provide some mild conditions for the class of graphs under consideration that allow signless Laplacian pretty good state transfer.