arXiv:2410.03310

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

Rachana Soni, Neelam Choudhary, Singh, Navneet Pratap

Published Oct 4, 2024• 0 citations

Abstract

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.

DOI: 10.48550/arxiv.2410.03310

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arXiv:2410.03310

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

Rachana Soni, Neelam Choudhary, Singh, Navneet Pratap

Published Oct 4, 2024• 0 citations

arXiv HTML PDF

Abstract

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.

DOI: 10.48550/arxiv.2410.03310

Related papers

Quantum Fractional Revival and Entanglement Entropy in Unitary Cayley Graphs

Duaa Abdullah

This paper extends the theory of quantum fractional revival (QFR) on unitary Cayley graphs $X=(V(\mathbb{Z}_n),E(S))$ in several directions that remained unresolved in previous work. First, we investigate QFR with respect to the Laplacian matrix Hamiltonian in addition to the adjacency matrix Hamiltonian. In particular, we prove that for regular graphs the two models differ only by a global phase factor, and we determine the conditions under which the Laplacian framework independently admits QFR. Second, for unitary Cayley graphs of order $n=2p$ , where $p$ is an odd prime, we derive an explicit closed-form expression for the minimum revival time, $t^{*}=\frac{2π}{p},$ and show that the associated revival amplitudes are given by $α=\cos\!\left(\frac{2π}{p}\right), \qquad β=-i\sin\!\left(\frac{2π}{p}\right).$ Third, we provide a complete characterization of strongly cospectral vertex pairs in $X=(V(\mathbb{Z}_n),E(S))$ through the arithmetic structure of $\mathbb{Z}_n$ , establishing that strong cospectrality is equivalent to antipodality whenever $n$ is twice a prime. Finally, we compute the von Neumann entanglement entropy generated by QFR for all admissible graphs, thereby obtaining a collection of quantum information measures and proving that the entropy depends solely on the revival amplitudes $|α|$ and $|β|$ .This paper extends the theory of quantum fractional revival (QFR) on unitary Cayley graphs $X=(V(\mathbb{Z}_n),E(S))$ in several directions that remained unresolved in previous work. First, we investigate QFR with respect to the Laplacian matrix Hamiltonian in addition to the adjacency matrix Hamiltonian. In particular, we prove that for regular graphs the two models differ only by a global phase factor, and we determine the conditions under which the Laplacian framework independently admits QFR. Second, for unitary Cayley graphs of order $n=2p$, where $p$ is an odd prime, we derive an explicit closed-form expression for the minimum revival time, $t^{*}=\frac{2π}{p},$ and show that the associated revival amplitudes are given by \[ α=\cos\!\left(\frac{2π}{p}\right), \qquad β=-i\sin\!\left(\frac{2π}{p}\right). \] Third, we provide a complete characterization of strongly cospectral vertex pairs in $X=(V(\mathbb{Z}_n),E(S))$ through the arithmetic structure of $\mathbb{Z}_n$, establishing that strong cospectrality is equivalent to antipodality whenever $n$ is twice a prime. Finally, we compute the von Neumann entanglement entropy generated by QFR for all admissible graphs, thereby obtaining a collection of quantum information measures and proving that the entropy depends solely on the revival amplitudes $|α|$ and $|β|$.

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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.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.

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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

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

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

This paper extends the theory of quantum fractional revival (QFR) on unitary Cayley graphs

X=(V(\mathbb{Z}_n),E(S))

in several directions that remained unresolved in previous work. First, we investigate QFR with respect to the Laplacian matrix Hamiltonian in addition to the adjacency matrix Hamiltonian. In particular, we prove that for regular graphs the two models differ only by a global phase factor, and we determine the conditions under which the Laplacian framework independently admits QFR. Second, for unitary Cayley graphs of order

n=2p

, where

p

is an odd prime, we derive an explicit closed-form expression for the minimum revival time,

t^{*}=\frac{2π}{p},

and show that the associated revival amplitudes are given by

α=\cos\!\left(\frac{2π}{p}\right), \qquad β=-i\sin\!\left(\frac{2π}{p}\right).

Third, we provide a complete characterization of strongly cospectral vertex pairs in

X=(V(\mathbb{Z}_n),E(S))

through the arithmetic structure of

\mathbb{Z}_n

, establishing that strong cospectrality is equivalent to antipodality whenever

n

is twice a prime. Finally, we compute the von Neumann entanglement entropy generated by QFR for all admissible graphs, thereby obtaining a collection of quantum information measures and proving that the entropy depends solely on the revival amplitudes

|α|

and

|β|

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.

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