Graphs of Unitary Matrices and Positive Semidefinite Zero Forcing

Abstract

The possible zero/nonzero patterns of unitary matrices are of interest in quantum evolution and the study of quantum systems on graphs. In particular, a quantum random walk can be defined on a directed graph if and only if that graph is associated with a unitary matrix pattern. We propose positive semidefinite zero forcing as a way to determine whether there exists a unitary matrix with a given zero/nonzero pattern. We show that zero forcing is a better criterion than strong quadrangularity of the pattern and prove that an n -by-n pattern supports a unitary matrix if and only if its positive semidefinite zero forcing number equals n for n ≤ 5.