Convergence Rates for Arbitrary Statistical Moments of Random Quantum Circuits

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 mapping the superoperator that describes t order moments on n qubits to a multilevel SU(4(t)) Lipkin-Meshkov-Glick Hamiltonian. We show that, for arbitrary fixed t, the ground-state manifold is exactly spanned by factorized eigenstates and, under the assumption that a mean-field ansatz accurately describes the low-lying excitations, 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.