  We prove that $poly(t) \cdot n^{1/D}$-depth local random quantum circuits
with two qudit nearest-neighbor gates on a $D$-dimensional lattice with n
qudits are approximate $t$-designs in various measures. These include the
"monomial" measure, meaning that the monomials of a random circuit from this
family have expectation close to the value that would result from the Haar
measure. Previously, the best bound was $poly(t)\cdot n$ due to
Brandao-Harrow-Horodecki (BHH) for $D=1$. We also improve the "scrambling" and
"decoupling" bounds for spatially local random circuits due to Brown and Fawzi.
  One consequence of our result is that assuming the polynomial hierarchy (PH)
is infinite and that certain counting problems are $\#P$-hard on average,
sampling within total variation distance from these circuits is hard for
classical computers. Previously, exact sampling from the outputs of even
constant-depth quantum circuits was known to be hard for classical computers
under the assumption that PH is infinite. However, to show the hardness of
approximate sampling using this strategy requires that the quantum circuits
have a property called "anti-concentration", meaning roughly that the output
has near-maximal entropy. Unitary 2-designs have the desired anti-concentration
property. Thus our result improves the required depth for this level of
anti-concentration from linear depth to a sub-linear value, depending on the
geometry of the interactions. This is relevant to a recent proposal by the
Google Quantum AI group to perform such a sampling task with 49 qubits on a
two-dimensional lattice and confirms their conjecture that $O(\sqrt n)$ depth
suffices for anti-concentration. We also prove that anti-concentration is
possible in depth O(log(n) loglog(n)) using a different model.
