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^0[q], the analog of the class
\ACC^0[q] where \Mod_q gates are also allowed. We show that it is
possible to make a `cat' state on n qubits in constant depth if and
only if we can construct a parity or \Mod_2 gate in constant depth;
therefore, any circuit class that can fan out a qubit to n copies in
constant depth also includes \QACC^0[2]. In addition, we prove the
somewhat surprising result that parity or fanout allows us to
construct \Mod_q gates in constant depth for any q, so \QACC^0[2] = \QACC^0. Since \ACC^0[p] \ne \ACC^0[q] whenever p and q are
distinct primes, \QACC^0[2] is strictly more powerful than its
classical counterpart, as is \QAC^0 when fanout is allowed.