  Quantifying the flatness of the objective-function landscape associated with
unstructured parameterized quantum circuits is important for understanding the
performance of variational algorithms utilizing a "hardware-efficient ansatz",
particularly for ensuring that a prohibitively flat landscape -- a so-called
"barren plateau" -- is avoided. For a model of such ans\"{a}tze, we relate the
typical landscape flatness to a certain family of random walks, enabling us to
derive a Monte Carlo algorithm for efficiently, classically estimating the
landscape flatness for any architecture. The statistical picture additionally
allows us to prove new analytic bounds on the barren plateau phenomenon, and
more generally provides novel insights into the phenomenon's dependence on the
ansatz depth, architecture, qudit dimension, and Hamiltonian combinatorial and
spatial locality. Our analysis utilizes techniques originally developed by
Dalzell et al. to study anti-concentration in random circuits.
