arXiv:2208.05107

Fractional revival on abelian Cayley graphs

Xiwang Cao, Gaojun Luo

Published Aug 10, 2022• 0 citations

Abstract

Fractional revival, known as a quantum transport phenomenon, is essential for entanglement generation in quantum spin networks. The concept of fractional revival is a generalization of perfect state transfer and periodicity on graphs. In this paper, we propose a sufficient and necessary condition for abelian Cayley graphs having fractional revival between any two distinct vertices. With this characterization, two general constructions of abelian Cayley graphs having fractional revival is presented. Meanwhile, we establish several new families of abelian Cayley graphs admitting fractional revival.

DOI: 10.48550/arxiv.2208.05107

Related papers

Fractional revival on quasi-abelian Cayley graphs

Yi Fang, Xueyi Huang, Xiaogang Liu +1 more

arXiv:2502.14330Feb 20, 20250 citations

Fractional revival on oriented Cayley and semi-Cayley graphs over abelian groups

Ming Jiang, Xiaogang Liu, Jing Wang

arXiv:2608.19961Aug 20, 20260 citations

Quantum fractional revival governed by adjacency matrix Hamiltonian in unitary Cayley graphs

Rachana Soni, Neelam Choudhary, Singh, Navneet Pratap

arXiv:2410.03310Oct 4, 20240 citations

Perfect state transfer and Cayley presentations

Arnbjörg Soffía Árnadóttir, Krystal Guo

math.COCross-listedarXiv:2608.20476Aug 24, 20260 citations

Continuous-time Quantum Walks on Cayley Graphs of Extraspecial Groups

Peter Sin, Julien Sorci

We study continuous-time quantum walks on normal Cayley graphs of certain non-abelian groups called extraspecial groups. By applying general results for graphs in association schemes we determine the precise conditions for perfect state transfer and fractional revival. Using partial spreads, we construct Cayley graphs on extraspecial 2 -groups that admit these phenomena. We also show that there is no normal Cayley graph of an extraspecial group that admits instantaneous uniform mixing.

arXiv:2011.07566Algebraic CombinatoricsSep 8, 20228 citations

Generalized fractional revival on graphs

Whitney Drazen

Jan 1, 20210 citations

arXiv:2208.05107

Fractional revival on abelian Cayley graphs

Xiwang Cao, Gaojun Luo

Published Aug 10, 2022• 0 citations

arXiv HTML PDF

Abstract

DOI: 10.48550/arxiv.2208.05107

Related papers

Fractional revival on quasi-abelian Cayley graphs

Yi Fang, Xueyi Huang, Xiaogang Liu +1 more

Fractional revival, a quantum transport phenomenon critical to entanglement generation in quantum spin networks, generalizes the notion of perfect state transfer on graphs. A Cayley graph $\mathrm{Cay}(G,S)$ is called quasi-abelian if its connection set $S$ is a union of conjugacy classes of the group $G$ . In this paper, we establish a necessary and sufficient condition for quasi-abelian Cayley graphs to have fractional revival. This extends a result of Cao and Luo (2022) on the existence of fractional revival in Cayley graphs over abelian groups.Fractional revival, a quantum transport phenomenon critical to entanglement generation in quantum spin networks, generalizes the notion of perfect state transfer on graphs. A Cayley graph $\mathrm{Cay}(G,S)$ is called quasi-abelian if its connection set $S$ is a union of conjugacy classes of the group $G$. In this paper, we establish a necessary and sufficient condition for quasi-abelian Cayley graphs to have fractional revival. This extends a result of Cao and Luo (2022) on the existence of fractional revival in Cayley graphs over abelian groups.

arXiv:2502.14330Feb 20, 20250 citations

Fractional revival on oriented Cayley and semi-Cayley graphs over abelian groups

Ming Jiang, Xiaogang Liu, Jing Wang

Fractional revival (FR), a generalization of perfect state transfer (PST), is a significant phenomenon in quantum state transfer that allows quantum information to be transmitted between two qubits with a certain probability. The existence of FR has been extensively studied on many classes of graphs. However, oriented graphs have not yet been investigated. In this paper, we investigate the existence of proper FR on oriented graphs. We first establish necessary and sufficient conditions for oriented graphs to admit proper FR between strongly cospectral vertices. Furthermore, we prove that oriented Cayley graphs over abelian groups do not admit proper FR, and we subsequently characterize the conditions under which oriented semi-Cayley graphs over abelian groups admit proper FR.Fractional revival (FR), a generalization of perfect state transfer (PST), is a significant phenomenon in quantum state transfer that allows quantum information to be transmitted between two qubits with a certain probability. The existence of FR has been extensively studied on many classes of graphs. However, oriented graphs have not yet been investigated. In this paper, we investigate the existence of proper FR on oriented graphs. We first establish necessary and sufficient conditions for oriented graphs to admit proper FR between strongly cospectral vertices. Furthermore, we prove that oriented Cayley graphs over abelian groups do not admit proper FR, and we subsequently characterize the conditions under which oriented semi-Cayley graphs over abelian groups admit proper FR.

arXiv:2608.19961Aug 20, 20260 citations

Quantum fractional revival governed by adjacency matrix Hamiltonian in unitary Cayley graphs

Rachana Soni, Neelam Choudhary, Singh, Navneet Pratap

In this article, we give characterization for existence of quantum fractional revival in unitary Cayley graph utilizing adjacency matrix Hamiltonian. Unitary Cayley graph $X=( Z_n, S)$ is a special graph as connection set $S \subseteq Z_n$ is the collection of coprimes to $n$ . Unitary Cayley graph is an integral graph and its adjacency matrix is a circulant one. We prove that quantum fractional revival in unitary Cayley graphs exists only when the number of vertices is even. Number-theoretic and spectral characterizations are given for unitary Cayley graph admitting quantum fractional revival. Quantum fractional revival is analogous to quantum entanglement. It is one of qubit state transfer phenomena useful in communication of quantum information.In this article, we give characterization for existence of quantum fractional revival in unitary Cayley graph utilizing adjacency matrix Hamiltonian. Unitary Cayley graph $X=( Z_n, S)$ is a special graph as connection set $S \subseteq Z_n$ is the collection of coprimes to $n$. Unitary Cayley graph is an integral graph and its adjacency matrix is a circulant one. We prove that quantum fractional revival in unitary Cayley graphs exists only when the number of vertices is even. Number-theoretic and spectral characterizations are given for unitary Cayley graph admitting quantum fractional revival. Quantum fractional revival is analogous to quantum entanglement. It is one of qubit state transfer phenomena useful in communication of quantum information.

arXiv:2410.03310Oct 4, 20240 citations

Perfect state transfer and Cayley presentations

Arnbjörg Soffía Árnadóttir, Krystal Guo

We study perfect state transfer on Cayley graphs from the point of view that state transfer is a property of a graph and not of a group. This paper is a bridge between the classical question about isomorphic Cayley graphs of non-isomorphic groups and quantum walks on graphs. We show that a Cayley graph of a group with an abelian subgroup of index two is a Cayley graph of an abelian group under any one of three hypotheses, two drawn from the theory of isomorphic Cayley graphs. A statement of the same kind holds for extraspecial groups: every Cayley graph of an extraspecial $p$ -group of order $p^{2n+1}$ with a conjugacy-closed connection set is a Cayley graph of $Z_p^{2n+1}$ . From these results we deduce that every explicit construction of perfect state transfer in the six papers we survey, on dihedral, dicyclic, generalized dihedral, $V_{8n}$ and extraspecial $2$ -groups, is a non-abelian presentation of an abelian Cayley graph. Moreover, we show that a non-abelian group with an abelian subgroup of index two admits a connected Cayley graph with perfect state transfer if and only if its order is divisible by four. Genuinely non-abelian examples do exist. We prove that, for every odd prime power $q\ge 5$ , the $SL(2,q)$ graph of Pantangi and Sin, which they showed to admit perfect state transfer, is a Cayley graph of no abelian group; to our knowledge, this is the first infinite family of Cayley graphs with perfect state transfer provably admitting no abelian Cayley presentation. We also construct an infinite family of Cayley graphs with peak state transfer and determine all regular subgroups of the automorphism group of every member. An appendix records a census of the connected vertex-transitive graphs with perfect state transfer on at most $30$ vertices.We study perfect state transfer on Cayley graphs from the point of view that state transfer is a property of a graph and not of a group. This paper is a bridge between the classical question about isomorphic Cayley graphs of non-isomorphic groups and quantum walks on graphs. We show that a Cayley graph of a group with an abelian subgroup of index two is a Cayley graph of an abelian group under any one of three hypotheses, two drawn from the theory of isomorphic Cayley graphs. A statement of the same kind holds for extraspecial groups: every Cayley graph of an extraspecial $p$-group of order $p^{2n+1}$ with a conjugacy-closed connection set is a Cayley graph of $Z_p^{2n+1}$. From these results we deduce that every explicit construction of perfect state transfer in the six papers we survey, on dihedral, dicyclic, generalized dihedral, $V_{8n}$ and extraspecial $2$-groups, is a non-abelian presentation of an abelian Cayley graph. Moreover, we show that a non-abelian group with an abelian subgroup of index two admits a connected Cayley graph with perfect state transfer if and only if its order is divisible by four. Genuinely non-abelian examples do exist. We prove that, for every odd prime power $q\ge 5$, the $SL(2,q)$ graph of Pantangi and Sin, which they showed to admit perfect state transfer, is a Cayley graph of no abelian group; to our knowledge, this is the first infinite family of Cayley graphs with perfect state transfer provably admitting no abelian Cayley presentation. We also construct an infinite family of Cayley graphs with peak state transfer and determine all regular subgroups of the automorphism group of every member. An appendix records a census of the connected vertex-transitive graphs with perfect state transfer on at most $30$ vertices.

math.COCross-listedarXiv:2608.20476Aug 24, 20260 citations

Continuous-time Quantum Walks on Cayley Graphs of Extraspecial Groups

Peter Sin, Julien Sorci

arXiv:2011.07566Algebraic CombinatoricsSep 8, 20228 citations

Generalized fractional revival on graphs

Whitney Drazen

A continuous-time quantum walk is a process on a network of quantum particles that is governed by the transition matrix $U(t) = e^{-itA}$ , where is $A$ is the adjacency matrix of the graph representing the network. Certain instantaneous phenomena occuring in quantum walks can be utilized in quantum computing algorithms. The two-vertex phenomenon fractional revival occurs between vertices $u$ and $v$ at time $t$ if the columns of $U(t)$ corresponding to $u$ and $v$ are only supported on the rows indexed by those same two vertices. This means that given any initial state supported only on vertices $u$ and $v$ , the state of the system at time $t$ will again be supported only on those same two vertices. Fractional revival can be used to generate entanglement between two quantum particles. We will discuss $K$ -fractional revival, the generalization of fractional revival to $K$ , a larger subset of the vertices in our graph. We will also explore the related concept approximate $K$ -fractional revival, which denotes the existence of a sequence of times for which $U(t)$ comes arbitrarily close to achieving $K$ -fractional revival. Next, we will examine some spectral characterizations of these phenomena, which will allow us to determine if a given graph exhibits either $K$ -fractional revival or approximate $K$ -fractional revival. Lastly, we will detail families of examples of approximate $K$ -fractional revival on path graphs.--Author's abstractA continuous-time quantum walk is a process on a network of quantum particles that is governed by the transition matrix $U(t) = e^{-itA}$, where is $A$ is the adjacency matrix of the graph representing the network. Certain instantaneous phenomena occuring in quantum walks can be utilized in quantum computing algorithms. The two-vertex phenomenon fractional revival occurs between vertices $u$ and $v$ at time $t$ if the columns of $U(t)$ corresponding to $u$ and $v$ are only supported on the rows indexed by those same two vertices. This means that given any initial state supported only on vertices $u$ and $v$, the state of the system at time $t$ will again be supported only on those same two vertices. Fractional revival can be used to generate entanglement between two quantum particles. We will discuss $K$-fractional revival, the generalization of fractional revival to $K$, a larger subset of the vertices in our graph. We will also explore the related concept approximate $K$-fractional revival, which denotes the existence of a sequence of times for which $U(t)$ comes arbitrarily close to achieving $K$-fractional revival. Next, we will examine some spectral characterizations of these phenomena, which will allow us to determine if a given graph exhibits either $K$-fractional revival or approximate $K$-fractional revival. Lastly, we will detail families of examples of approximate $K$-fractional revival on path graphs.--Author's abstract

Jan 1, 20210 citations

Fractional revival, a quantum transport phenomenon critical to entanglement generation in quantum spin networks, generalizes the notion of perfect state transfer on graphs. A Cayley graph

\mathrm{Cay}(G,S)

is called quasi-abelian if its connection set

S

is a union of conjugacy classes of the group

G

. In this paper, we establish a necessary and sufficient condition for quasi-abelian Cayley graphs to have fractional revival. This extends a result of Cao and Luo (2022) on the existence of fractional revival in Cayley graphs over abelian groups.

In this article, we give characterization for existence of quantum fractional revival in unitary Cayley graph utilizing adjacency matrix Hamiltonian. Unitary Cayley graph

X=( Z_n, S)

is a special graph as connection set

S \subseteq Z_n

is the collection of coprimes to

n

. Unitary Cayley graph is an integral graph and its adjacency matrix is a circulant one. We prove that quantum fractional revival in unitary Cayley graphs exists only when the number of vertices is even. Number-theoretic and spectral characterizations are given for unitary Cayley graph admitting quantum fractional revival. Quantum fractional revival is analogous to quantum entanglement. It is one of qubit state transfer phenomena useful in communication of quantum information.

p

-group of order

p^{2n+1}

with a conjugacy-closed connection set is a Cayley graph of

Z_p^{2n+1}

. From these results we deduce that every explicit construction of perfect state transfer in the six papers we survey, on dihedral, dicyclic, generalized dihedral,

V_{8n}

and extraspecial

2

-groups, is a non-abelian presentation of an abelian Cayley graph. Moreover, we show that a non-abelian group with an abelian subgroup of index two admits a connected Cayley graph with perfect state transfer if and only if its order is divisible by four. Genuinely non-abelian examples do exist. We prove that, for every odd prime power

q\ge 5

, the

SL(2,q)

graph of Pantangi and Sin, which they showed to admit perfect state transfer, is a Cayley graph of no abelian group; to our knowledge, this is the first infinite family of Cayley graphs with perfect state transfer provably admitting no abelian Cayley presentation. We also construct an infinite family of Cayley graphs with peak state transfer and determine all regular subgroups of the automorphism group of every member. An appendix records a census of the connected vertex-transitive graphs with perfect state transfer on at most

30

vertices.

A continuous-time quantum walk is a process on a network of quantum particles that is governed by the transition matrix

U(t) = e^{-itA}

, where is

A

is the adjacency matrix of the graph representing the network. Certain instantaneous phenomena occuring in quantum walks can be utilized in quantum computing algorithms. The two-vertex phenomenon fractional revival occurs between vertices

u

and

v

at time

t

if the columns of

U(t)

corresponding to

u

and

v

are only supported on the rows indexed by those same two vertices. This means that given any initial state supported only on vertices

u

and

v

, the state of the system at time

t

will again be supported only on those same two vertices. Fractional revival can be used to generate entanglement between two quantum particles. We will discuss

K

-fractional revival, the generalization of fractional revival to

K

, a larger subset of the vertices in our graph. We will also explore the related concept approximate

K

-fractional revival, which denotes the existence of a sequence of times for which

U(t)

comes arbitrarily close to achieving

K

-fractional revival. Next, we will examine some spectral characterizations of these phenomena, which will allow us to determine if a given graph exhibits either

K

-fractional revival or approximate

K

-fractional revival. Lastly, we will detail families of examples of approximate

K

-fractional revival on path graphs.--Author's abstract