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, for 0 < ε ≤ δ ≤ 1, we define BQNC_(ε,δ)~0 to be the class of languages recognized by constant depth, polynomial-size quantum circuits with acceptance probability either < ε (for rejection) or ≥ δ (for acceptance). We show that BQNC_(ε,δ)~0 is contained in P, provided that 1 - δ ≤ 2~(-2d)(1 - ε), where d is the circuit depth. On the other hand, we adapt and extend ideas of Terhal & DiVin-cenzo 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 arbitrary quantum 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.
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