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arXiv:2605.00518

Quantum fractional revival on zero-divisor graphs over $\mathbb{Z}_n$

Bui Phuoc Minh, Songpon Sriwongsa

Published May 1, 2026• 0 citations
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Abstract

In this paper, we characterize the existence of perfect state transfer (PST) and fractional revival in continuous-time quantum walks on the zero-divisor graph $Γ(\mathbb{Z}_n)$. By using the canonical equitable partition of $Γ(\mathbb{Z}_n)$ induced by the proper divisors of $n$, we derive a sufficient condition on $n$ for PST to occur between a pair of vertices. We show that fractional revival is restricted to cells of size $2$ within the equitable partition. Furthermore, assuming $-1$ is not an eigenvalue of the quotient spectrum, we establish that two vertices in $Γ(\mathbb{Z}_n)$ are strongly cospectral if and only if they form a cell of size $2$ within the equitable partition that is either a set of false twins or true twins. Finally, we provide a characterization of fractional revival on bipartite $Γ(\mathbb{Z}_n)$ and prove the non-existence of fractional revival on $Γ(\mathbb{Z}_{p^2q})$.

DOI
10.48550/arxiv.2605.00518

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Quantum fractional revival on zero-divisor graphs over Zn\mathbb{Z}_nZn​
In this paper, we characterize the existence of perfect state transfer (PST) and fractional revival in continuous-time quantum walks on the zero-divisor graph Γ(Zn)Γ(\mathbb{Z}_n)Γ(Zn​). By using the canonical equitable partition of Γ(Zn)Γ(\mathbb{Z}_n)Γ(Zn​) induced by the proper divisors of nnn, we derive a sufficient condition on nnn for PST to occur between a pair of vertices. We show that fractional revival is restricted to cells of size 222 within the equitable partition. Furthermore, assuming −1-1−1 is not an eigenvalue of the quotient spectrum, we establish that two vertices in Γ(Zn)Γ(\mathbb{Z}_n)Γ(Zn​) are strongly cospectral if and only if they form a cell of size 222 within the equitable partition that is either a set of false twins or true twins. Finally, we provide a characterization of fractional revival on bipartite Γ(Zn)Γ(\mathbb{Z}_n)Γ(Zn​) and prove the non-existence of fractional revival on Γ(Zp2q)Γ(\mathbb{Z}_{p^2q})Γ(Zp2q​).
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.
This paper investigates perfect state transfer (PST) in discrete-time quantum walks constructed via the shunt decomposition method. The walks are defined on a graph GGG and its associated quotient graph G/πG/πG/π, induced by an equitable partition πππ. Through the shunt decomposition of GGG, we derive an explicit relation between the shift operator of the parent graph GGG and that of its quotient graph G/πG/πG/π. We construct a reflection operator based on the characteristic matrix, which establishes a connection between the transition operator of the parent graph and that of its lower-dimensional quotient graph. We then prove that PST occurs on GGG if and only if it occurs on G/πG/πG/π. Furthermore, we express the unitary evolution operator of the quotient graph in terms of Chebyshev polynomials of the first kind, from which we derive explicit criteria for PST. As an application, we establish PST on the cycle graph CnC_{n}Cn​ at time k=n/2k = n/2k=n/2, and lift the result to the parent graph C2nC_{2n}C2n​ via the equitable partition πππ. We further show that if an equitable partition πππ of GGG induces a quotient isomorphic to Kn↺K_n^{\circlearrowleft}Kn↺​, the complete digraph on nnn vertices with a loop at every vertex, then PST occurs at step k=nk = nk=n, and the walk is periodic at k=2nk = 2nk=2n. This framework is applied to two families of graphs, which are the complete bipartite digraph Kn,n⇌K_{n,n}^{\rightleftharpoons}Kn,n⇌​ and the circulant graph Circ⁡(2n,S)\operatorname{Circ}(2n, S)Circ(2n,S), where SSS consists of all odd residues modulo 2n2n2n and n=2sn = 2^sn=2s for some s≥1s \geq 1s≥1, establishing PST in their respective line digraphs. Collectively, these results also answer the question posed by Godsil and Zhan concerning which shunt decompositions or embeddings of a graph admit PST.
We prove new results on perfect state transfer of quantum walks on quotient graphs. Since a graph G has perfect state transfer if and only if its quotient G/\pi, under any equitable partition \pi, has perfect state transfer, we exhibit graphs with perfect state transfer between two vertices but which lack automorphism swapping them. This answers a question of Godsil (Discrete Mathematics 312(1):129-147, 2011). We also show that the Cartesian product of quotient graphs \Box_{k} G_{k}/\pi_{k} is isomorphic to the quotient graph \Box_{k} G_{k}/\pi, for some equitable partition \pi. This provides an algebraic description of a construction due to Feder (Physical Review Letters 97, 180502, 2006) which is based on many-boson quantum walk.
A continuous-time quantum walk is a process on a network of quantum particles that is governed by the transition matrix U(t)=e−itAU(t) = e^{-itA}U(t)=e−itA, where is AAA 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 uuu and vvv at time ttt if the columns of U(t)U(t)U(t) corresponding to uuu and vvv are only supported on the rows indexed by those same two vertices. This means that given any initial state supported only on vertices uuu and vvv, the state of the system at time ttt 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 KKK-fractional revival, the generalization of fractional revival to KKK, a larger subset of the vertices in our graph. We will also explore the related concept approximate KKK-fractional revival, which denotes the existence of a sequence of times for which U(t)U(t)U(t) comes arbitrarily close to achieving KKK-fractional revival. Next, we will examine some spectral characterizations of these phenomena, which will allow us to determine if a given graph exhibits either KKK-fractional revival or approximate KKK-fractional revival. Lastly, we will detail families of examples of approximate KKK-fractional revival on path graphs.--Author's 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.