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.
Highlights
Mathematics Subject Classifications: 05C50, 81P68 where H is some Hermitian matrix associated with X
We develop the theory of fractional revival in the quantum walk on a graph using its Laplacian matrix as the Hamiltonian
We say a graph admits fractional revival at time τ from vertex a to vertex b if, for some complex numbers α and β, U (τ )ea = αea + βeb. Such a state has physical significance: if neither α nor β is zero, the two qubits at a and b are entangled at time τ ; if β = 0, the system recovers its original state at qubit a after time τ ; and if α = 0, the original state of qubit a is completely transferred to qubit b at time τ
Summary
The Laplacian matrix of X, denoted L, is given by. It has 0 as an eigenvalue with eigenvector 1. [2] Let X be a connected graph on n vertices whose largest Laplacian eigenvalue is μ. Given an orientation of X, the signed incidence matrix, denoted B, is the matrix whose rows are indexed by the vertices and columns by the edges with. The following lemma shows a connection between the Laplacian matrix and the signed incidence matrices of a graph. Let X be a graph with Laplacian matrix L. Let B be the signed incidence matrix of any orientation of X. Let X be a graph on n vertices with Laplacian matrix L. Nq equals the product of non-zero eigenvalues of L
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