Universality of single qudit gates

Abstract: We consider the problem of deciding if a set of quantum one-qudit gates $\mathcal{S}={g_1,\ldots,g_n}\subset G$ is universal, i.e if the closure $\overline{<\mathcal{S}>}$ is equal to $G$, where $G$ is either the special unitary or the special orthogonal group. To every gate $g$ in $\mathcal{S}$ we asign its image under the adjoint representation $\mathrm{Ad}g$, where $\mathrm{Ad}:G\rightarrow SO(\mathfrak{g})$ and $\mathfrak{g}$ is the Lie algebra of $G$. The necessary condition for the universality of $\mathcal{S}$ is that the only matrices that commute with all $\mathrm{Ad}{g_i}$'s are proportional to the identity. If in addition there is an element in $<\mathcal{S}>$ whose Hilbert-Schmidt distance from the centre of $G$ belongs to $]0,\frac{1}{\sqrt{2}}]$, then $\mathcal{S}$ is universal. Using these we provide a simple algorithm that allows deciding the universality of any set of $d$-dimensional gates in a finite number of steps and formulate the general classification theorem.