  Recently, Bravyi, Gosset, and K\"{o}nig (Science, 2018) exhibited a search
problem called the 2D Hidden Linear Function (2D HLF) problem that can be
solved exactly by a constant-depth quantum circuit using bounded fan-in gates
(or QNC^0 circuits), but cannot be solved by any constant-depth classical
circuit using bounded fan-in AND, OR, and NOT gates (or NC^0 circuits). In
other words, they exhibited a search problem in QNC^0 that is not in NC^0.
  We strengthen their result by proving that the 2D HLF problem is not
contained in AC^0, the class of classical, polynomial-size, constant-depth
circuits over the gate set of unbounded fan-in AND and OR gates, and NOT gates.
We also supplement this worst-case lower bound with an average-case result:
There exists a simple distribution under which any AC^0 circuit (even of nearly
exponential size) has exponentially small correlation with the 2D HLF problem.
Our results are shown by constructing a new problem in QNC^0, which we call the
Relaxed Parity Halving Problem, which is easier to work with. We prove our AC^0
lower bounds for this problem, and then show that it reduces to the 2D HLF
problem.
  As a step towards even stronger lower bounds, we present a search problem
that we call the Parity Bending Problem, which is in QNC^0/qpoly (QNC^0
circuits that are allowed to start with a quantum state of their choice that is
independent of the input), but is not even in AC^0[2] (the class AC^0 with
unbounded fan-in XOR gates).
  All the quantum circuits in our paper are simple, and the main difficulty
lies in proving the classical lower bounds. For this we employ a host of
techniques, including a refinement of H{\aa}stad's switching lemmas for
multi-output circuits that may be of independent interest, the
Razborov-Smolensky AC^0[2] lower bound, Vazirani's XOR lemma, and lower bounds
for non-local games.
