Irrational Quantum Walks
Gabriel Coutinho, Pedro Ferreira Baptista, Chris D. Godsil, Thomás Jung Spier, Reinhard F. Werner · SIAM Journal on Applied Algebra and Geometry · 2023
Abstract. The adjacency matrix of a graph [Formula: see text] is the Hamiltonian for a continuous-time quantum walk on the vertices of [Formula: see text]. Although the entries of the adjacency matrix are integers, its eigenvalues are generally irrational and, because of this, the behavior of the walk is typically not periodic. In this paper, we develop a theory to exactly study any quantum walk generated by an integral Hamiltonian, and we put emphasis on those with irrational eigenvalues—what we call irrational quantum walks. As a result, we provide exact methods to compute the average of the mixing matrices, and to decide whether pretty good (or almost perfect) state transfer occurs in a given graph. We also use our methods to study geometric properties of beautiful curves arising from entries of the quantum walk matrix and discuss possible applications of these results. Throughout the paper, we emphasize the interplay between different fields of mathematics applied to the study of quantum walks.