Seminarium "The Trans-Carpathian Seminar on Geometry & Physics"

I will consider the problem of deciding if a finite set of quantum one-qudit gates is universal, i.e if the generated group is either the special unitary or the special orthogonal group. To every gate I will assign its image under the adjoint representation. The necessary condition for the universality is that the only matrices that commute with all the adjoint representation matrices are proportional to the identity. If in addition there is an element in the considered group whose Hilbert-Schmidt distance from the centre is smaller than 1/\sqrt{2}, then the set of gates is universal. Using these I will present a simple algorithm that allows deciding the universality of any set of d-dimensional gates in a finite number of steps. Moreover, I will formulate the general classification theorem.This is a joint work with Katarzyna Karnas.