Google's landmark result last year was to compute a task with a quantum computer that a classical computer could not compute, and they chose random circuit sampling. Part of their justification was complexity-theoretic reasons that, if one can efficiently compute this classically, it collapses the polynomial hierarchy (they cite 1,2,3 for this). Paper 2 in that list says that the hardness result comes from a reduction to computing the permanent of a random matrix.
Based on a quick search, computing an approximation to the permanent seems to be easy for many classes of random matrices. So is it possible there is some classical algorithm that could efficiently approximate the random circuit sampling problem?
Second, is the quantum computer solving this exactly or approximately? I'm not quite sure what that means to approximately sample (since sampling is inherently noisy anyway). That is: if I had a quantum computer that was (up to some noise) sampling from random circuits, and a classical computer efficiently approximating the same task, could you tell the difference?