Bounds on the Power of Constant-Depth Quantum Circuits

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 < ≤ δ ≤ 1, we define BQNC ,δ to be the class of languages recognized by constant depth, polynomial-size quantum circuits with acceptance probability either < (for rejection) or ≥ δ (for acceptance). We show that BQNC 0 ,δ ⊆ P, provided that 1− δ ≤ 2(1− ), where d is the circuit depth. On the other hand, we adapt and extend ideas of Terhal & DiVincenzo [TD04] 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 is the constant-depth analog of the class NQP. This essentially refutes a conjecture of Green et al. that NQACC ⊆ TC [GHMP02].