Bounds on the power of constant-depth quantum circuits

Abstract

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∈,δ0 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∈,δ0, ⊆ P, provided that 1 - δ ≤ 2-2d(1 - ∈), where d is the circuit depth. On the other hand, we adapt and extend ideas of Terhal & DiVincenzo [1] 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 NQNC0 = NQACC = NQP = coC=P, where NQNC0 is the constant-depth analog of the class NQP. This essentially refutes a conjecture of Green et al. that NQACC ∈ TC0 [2]. © Springer-Verlag Berlin Heidelberg 2005.