  We show that if a language is recognized within certain error bounds by
constant-depth quantum circuits over a finite family of gates, then it is
computable in (classical) polynomial time. In particular, our results imply
EQNC^0 is contained in P, where EQNC^0 is the constant-depth analog of the
class EQP. On the other hand, we adapt and extend ideas of Terhal and
DiVincenzo (quant-ph/0205133) to show that, for any family F of quantum gates
including Hadamard and CNOT gates, computing the acceptance probabilities of
depth-five circuits over F is just as hard as computing these probabilities for
circuits over F. In particular, this implies that NQNC^0 = NQACC = NQP = coC=P
where NQNC^0 is the constant-depth analog of the class NQP. This essentially
refutes a conjecture of Green et al. that NQACC is contained in TC^0
(quant-ph/0106017).
