Bounds on the power of constant-depth quantum circuits

• Authors: Stephen Fenner;Frederic Green;Steven Homer;Yong Zhang
• Venue: FCT'05 Proceedings of the 15th international conference on Fundamentals of Computation Theory
• Year: 2005

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 0ε≤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 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].