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arXiv:2010.10413 · The Electronic Journal of Combinatorics

Laplacian Fractional Revival on Graphs

Ada Chan, Bobae Johnson, Mengzhen Liu, Malena Schmidt, Zhanghan Yin, Hanmeng Zhan

Published Jul 16, 2021• 8 citations
arXivHTMLPDF

Abstract

We develop the theory of fractional revival in the quantum walk on a graph using its Laplacian matrix as the Hamiltonian. We first give a spectral characterization of Laplacian fractional revival, which leads to a polynomial time algorithm to check this phenomenon and find the earliest time when it occurs. We then apply the characterization theorem to special families of graphs. In particular, we show that no tree admits Laplacian fractional revival except for the paths on two and three vertices, and the only graphs on a prime number of vertices that admit Laplacian fractional revival are double cones. Finally, we construct, through Cartesian products and joins, several infinite families of graphs that admit Laplacian fractional revival; some of these graphs exhibit polygamous fractional revival.

DOI
10.37236/10146

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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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We develop the theory of fractional revival in the quantum walk on a graph using its Laplacian matrix as the Hamiltonian. We first give a spectral characterization of Laplacian fractional revival, which leads to a polynomial time algorithm to check this phenomenon and find the earliest time when it occurs. We then apply the characterization theorem to special families of graphs. In particular, we show that no tree admits Laplacian fractional revival except for the paths on two and three vertices, and the only graphs on a prime number of vertices that admit Laplacian fractional revival are double cones. Finally, we construct, through Cartesian products and joins, several infinite families of graphs that admit Laplacian fractional revival; some of these graphs exhibit polygamous fractional revival.
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
This paper extends the theory of quantum fractional revival (QFR) on unitary Cayley graphs X=(V(Zn),E(S))X=(V(\mathbb{Z}_n),E(S))X=(V(Zn​),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=2pn=2pn=2p, where ppp is an odd prime, we derive an explicit closed-form expression for the minimum revival time, t∗=2πp,t^{*}=\frac{2π}{p},t∗=p2π​, and show that the associated revival amplitudes are given by α=cos⁡ ⁣(2πp),β=−isin⁡ ⁣(2πp). α=\cos\!\left(\frac{2π}{p}\right), \qquad β=-i\sin\!\left(\frac{2π}{p}\right). α=cos(p2π​),β=−isin(p2π​). Third, we provide a complete characterization of strongly cospectral vertex pairs in X=(V(Zn),E(S))X=(V(\mathbb{Z}_n),E(S))X=(V(Zn​),E(S)) through the arithmetic structure of Zn\mathbb{Z}_nZn​, establishing that strong cospectrality is equivalent to antipodality whenever nnn 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∣|α|∣α∣ and ∣β∣|β|∣β∣.
In this article, we give characterization for existence of quantum fractional revival in unitary Cayley graph utilizing adjacency matrix Hamiltonian. Unitary Cayley graph X=(Zn,S)X=( Z_n, S)X=(Zn​,S) is a special graph as connection set S⊆ZnS \subseteq Z_nS⊆Zn​ is the collection of coprimes to nnn. 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.
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.
We initiate the study of approximate quantum fractional revival in graphs, a generalization of pretty good quantum state transfer in graphs. We give a complete characterization of approximate fractional revival in a graph in terms of the eigenvalues and eigenvectors of the adjacency matrix of a graph. This characterization follows from a lemma due to Kronecker on Diophantine approximation, and is similar to the spectral characterization of pretty good state transfer in graphs. Using this, we give a complete characterizations of when approximate fractional revival can occur in paths and in cycles.
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.