  The projected ensemble is based on the study of the quantum state of a
subsystem $A$ conditioned on projective measurements in its complement. Recent
studies have observed that a more refined measure of the thermalization of a
chaotic quantum system can be defined on the basis of convergence of the
projected ensemble to a quantum state design, i.e. a system thermalizes when it
becomes indistinguishable, up to the $k$-th moment, from a Haar ensemble of
uniformly distributed pure states. Here we consider a random unitary circuit
with the brick-wall geometry and analyze its convergence to the Haar ensemble
through the frame potential and its mapping to a statistical mechanical
problem. This approach allows us to highlight a geometric interpretation of the
frame potential based on the existence of a fluctuating membrane, similar to
those appearing in the study of entanglement entropies. At large local Hilbert
space dimension $q$, we find that all moments converge simultaneously with a
time scaling linearly in the size of region $A$, a feature previously observed
in dual unitary models. However, based on the geometric interpretation, we
argue that the scaling at finite $q$ on the basis of rare membrane
fluctuations, finding the logarithmic scaling of design times $t_k = O(\log
k)$. Our results are supported with numerical simulations performed at $q=2$.
