Criteria for universality of quantum gates

Abstract: We consider the problem of deciding if a set of quantum one-qudit gates $\mathcal{S}={U_1,\ldots,U_n}$ is universal. We provide the compact form criteria leading to a simple algorithm that allows deciding universality of any given set of gates in a finite number of steps. Moreover, for a non-universal $\mathcal{S}$ our criteria indicate what type of gates can be added to $\mathcal{S}$ to turn it into a universal set.