Generalized fractional revival on graphs

Abstract

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