Bounds on the Power of Constant-Depth Quantum Circuits

We show that if a language is recognized within certain error bounds by constantdepth quantum circuits over a nite family of gates, then it is computable in (classical) polynomial time. In particular, our results imply EQNC P; where EQNC is the constant-depth analogue of the class EQP. On the other hand, we adapt and extend ideas of DiVincenzo & Terhal [?] to show that, for any family F of quantum gates including Hadamard and CNOT gates, computing the acceptance probabilities of depthve circuits over F is just as hard as computing these probabilities for arbitrary quantum circuits over F . In particular, this implies that NQNC = NQACC = NQP = coC=P; where NQNC is the constant-depth analogue of the class NQP. This essentially refutes a conjecture of Green et al. that NQACC TC [?].