The Robustness of QAC0

Abstract: In this work we study the robustness of $\mathsf{QAC}<sup>0$ with respect to error tolerance and modifications to its gate-set. First, we investigate whether the non-zero error typically allowed for $\mathsf{QAC}<sup>0$ circuits computing Boolean functions is truly necessary. We show that the error inherent in the parallel $W$-test of \cite{grier_morris_wu} can be eliminated entirely via a novel application of exact amplitude amplification in the many-copies context. Consequently, we find that $\mathsf{QAC}<sup>0$ can \textit{exactly} simulate $\mathsf{TC}<sup>0$ with polynomially many copies of the classical input and that for every fixed prime $p$ exact $\mathsf{QAC}<sup>0$, $\mathsf{EQAC}<sup>0$, can compute total Boolean functions outside of $\mathsf{AC}<sup>0[p]$. Second, we ask to what extent the computational power of $\mathsf{QAC}<sup>0$ follows from the fact that arbitrary single-qubit gates may be used at any point in the circuit. We find that $\mathsf{QAC}<sup>0$ is in fact robust to restrictions on which single-qubit gates are permitted: every $\mathsf{QAC}<sup>0$ circuit can be approximately implemented by a $\mathsf{QAC}<sup>0$ circuit consisting of just generalized Toffoli, $S$, and Hadamard gates. Moreover, this approximating circuit can be constructed efficiently from a classical description of the original circuit.