  We propose definitions of $\QAC^0$, the quantum analog of the classical class
$\AC^0$ of constant-depth circuits with AND and OR gates of arbitrary fan-in,
and $\QACC[q]$, the analog of the class $\ACC[q]$ where $\Mod_q$ gates are also
allowed. We prove that parity or fanout allows us to construct quantum $\MOD_q$
gates in constant depth for any $q$, so $\QACC[2] = \QACC$. More generally, we
show that for any $q,p > 1$, $\MOD_q$ is equivalent to $\MOD_p$ (up to constant
depth). This implies that $\QAC^0$ with unbounded fanout gates, denoted
$\QACwf^0$, is the same as $\QACC[q]$ and $\QACC$ for all $q$. Since $\ACC[p]
\ne \ACC[q]$ whenever $p$ and $q$ are distinct primes, $\QACC[q]$ is strictly
more powerful than its classical counterpart, as is $\QAC^0$ when fanout is
allowed. This adds to the growing list of quantum complexity classes which are
provably more powerful than their classical counterparts.
  We also develop techniques for proving upper bounds for $\QACC^0$ in terms of
related language classes. We define classes of languages $\EQACC$, $\NQACC$ and
$\BQACC_{\rats}$. We define a notion of $\log$-planar $\QACC$ operators and
show the appropriately restricted versions of $\EQACC$ and $\NQACC$ are
contained in $\P/\poly$. We also define a notion of $\log$-gate restricted
$\QACC$ operators and show the appropriately restricted versions of $\EQACC$
and $\NQACC$ are contained in $\TC^0$.
