  We calculate the frame potential for Brownian clusters of $N$ spins or
fermions with time-dependent all-to-all interactions. In both cases the problem
can be mapped to an effective statistical mechanics problem which we study
using a path integral approach. We argue that the $k$th frame potential comes
within $\epsilon$ of the Haar value after a time of order $t \sim k N + k \log
k + \log \epsilon^{-1}$. Using a bound on the diamond norm, this implies that
such circuits are capable of coming very close to a unitary $k$-design after a
time of order $t \sim k N$. We also consider the same question for systems with
a time-independent Hamiltonian and argue that a small amount of time-dependent
randomness is sufficient to generate a $k$-design in linear time provided the
underlying Hamiltonian is quantum chaotic. These models provide explicit
examples of linear complexity growth that are also analytically tractable.
