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<ε≤δ≤ 1, we define BQNC $^{0}_{\epsilon ,\delta}$ 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 $^{0}_{\epsilon ,\delta} \subseteq$ 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 & DiVincenzo [1] to show that, for any family $\mathcal{F}$ of quantum gates including Hadamard and CNOT gates, computing the acceptance probabilities of depth-five circuits over $\mathcal{F}$ is just as hard as computing these probabilities for arbitrary quantum circuits over $\mathcal{F}$. 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 ⊆ TC ⁰ [2].

This work was supported in part by the National Security Agency (NSA) and Advanced Research and Development Agency (ARDA) under Army Research Office (ARO) contract numbers DAAD 19-02-1-0058 (for S. Homer, and F. Green) and DAAD 19-02-1-0048 (for S. Fenner and Y. Zhang).