  We consider a class of random quantum circuits where at each step a gate from
a universal set is applied to a random pair of qubits, and determine how
quickly averages of arbitrary finite-degree polynomials in the matrix elements
of the resulting unitary converge to Haar measure averages. This is
accomplished by establishing an exact mapping between the superoperator that
describes t-order moments on n qubits and a multilevel SU(4^t)
Lipkin-Meshkov-Glick Hamiltonian. For arbitrary fixed t, we find that the
spectral gap scales as 1/n in the thermodynamic limit. Our results imply that
random quantum circuits yield an efficient implementation of
\epsilon-approximate unitary t-designs.
