Random Quantum Circuits Anticoncentrate in Log Depth

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 anticoncentrated, roughly meaning that the probability mass is not too concentrated on a small number of measurement outcomes. An understanding of the conditions for anticoncentration is important for determining which quantum circuits are difficult to simulate classically, as anticoncentration 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 anticoncentration is that the expected collision probability of the distribution—that is, the probability that two independently drawn outcomes will agree—is only a constant factor larger than the collision probability for the uniform distribution. We show that when the two-local gates are each drawn from the Haar measure (or any 2-design), at least $\mathrm{\Omega }(n\log (n))$ gates (and thus $\mathrm{\Omega }(\log (n))$ 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 one-dimensional ring and the case where gates are long range, we show that $O(n\log (n))$ 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.