  We numerically investigate the statement that local random quantum circuits
acting on n qubits composed of polynomially many nearest neighbour two-qubit
gates form an approximate unitary poly(n)-design [F.G.S.L. Brandao et al.,
arXiv:1208.0692]. Using a group theory formalism, spectral gaps that give a
ratio of convergence to a given t-design are evaluated for a different number
of qubits n (up to 20) and degrees t (t=2,3,4 and 5), improving previously
known results for n=2 in the case of t=2 and 3. Their values lead to a
conclusion that the previously used lower bound that bounds spectral gaps
values may give very little information about the real situation and in most
cases, only tells that a gap is closed. We compare our results to the another
lower bounding technique, again showing that its results may not be tight.
