  We study the properties of output distributions of noisy, random circuits. We
obtain upper and lower bounds on the expected distance of the output
distribution from the "useless" uniform distribution. These bounds are tight
with respect to the dependence on circuit depth. Our proof techniques also
allow us to make statements about the presence or absence of anticoncentration
for both noisy and noiseless circuits. We uncover a number of interesting
consequences for hardness proofs of sampling schemes that aim to show a quantum
computational advantage over classical computation. Specifically, we discuss
recent barrier results for depth-agnostic and/or noise-agnostic proof
techniques. We show that in certain depth regimes, noise-agnostic proof
techniques might still work in order to prove an often-conjectured claim in the
literature on quantum computational advantage, contrary to what was thought
prior to this work.
