  We consider quantum circuits consisting of randomly chosen two-local gates
and study the number of gates needed for the distribution over measurement
outcomes for typical circuit instances to be anti-concentrated, roughly meaning
that the probability mass is not too concentrated on a small number of
measurement outcomes. Understanding the conditions for anti-concentration is
important for determining which quantum circuits are difficult to simulate
classically, as anti-concentration has been in some cases an ingredient of
mathematical arguments that simulation is hard and in other cases a necessary
condition for easy simulation. Our definition of anti-concentration is that the
expected collision probability, that is, the probability that two independently
drawn outcomes will agree, is only a constant factor larger than if the
distribution were uniform. We show that when the 2-local gates are each drawn
from the Haar measure (or any two-design), at least $\Omega(n \log(n))$ gates
(and thus $\Omega(\log(n))$ circuit depth) are needed for this condition to be
met on an $n$ qudit circuit. In both the case where the gates are
nearest-neighbor on a 1D ring and the case where gates are long-range, we show
$O(n \log(n))$ gates are also sufficient, and we precisely compute the optimal
constant prefactor for the $n \log(n)$. The technique we employ relies upon a
mapping from the expected collision probability to the partition function of an
Ising-like classical statistical mechanical model, which we manage to bound
using stochastic and combinatorial techniques.
