Sedentary quantum walks

Abstract

Let X be a graph with adjacency matrix A. The continuous quantum walk on X is determined by the unitary matrices U(t)=exp⁡(itA) (for t∈R). If X is the complete graph Kn and a∈V(X), then1−|U(t)a,a|≤2/n. Roughly speaking, this means that a quantum walk on a complete graph stays home with high probability. We say that a family of graphs is sedentary if there is a constant c such that 1−|U(t)a,a|≤c/|V(X)| for all t. In this paper we investigate this condition, and produce further examples of sedentary graphs.A cone over a graph X is the graph we get by adjoining a new vertex and making it adjacent to each vertex of X. We prove that if X is the cone over an ℓ-regular graph on n vertices, then |U(t)a,a|≤ℓ2/(ℓ2+4n). It follows that if we choose ℓ and n such that n/ℓ2→0, then a continuous quantum walk starting on the “conical” vertex will remain there with probability close to 1. On the other hand, if ℓ≤2, we show there is a time t such that all entries in the a-column of U(t)ea have absolute value 1/n. We show that there are large classes of strongly regular graphs such that 1−|U(t)a,a|≤c/V(X) for some constant c. On the other hand, for Paley graphs on n vertices, we prove that if t=π/n, then |U(t)a,a|≤1/n.