$\mathsf{QAC}^0$ Contains $\mathsf{TC}^0$ (with Many Copies of the Input)

TL;DR

This paper demonstrates that quantum constant-depth circuits ($ extsf{QAC}^0$) are more powerful than classical ones, capable of computing complex functions like $ extsf{TC}^0$ when multiple input copies are available, and establishes key separations.

Contribution

It proves an unconditional separation between $ extsf{QAC}^0$ and classical $ extsf{AC}^0[p]$, and shows $ extsf{QAC}^0$ can compute $ extsf{TC}^0$ functions with multiple input copies.

Findings

$ extsf{QAC}^0$ is not contained in $ extsf{AC}^0[p]$

$ extsf{TC}^0$ functions are computable in $ extsf{QAC}^0$ with multiple copies

Introduces an amplitude amplification technique for exact constructions

Abstract

$\mathsf{QAC}^0$ is the class of constant-depth polynomial-size quantum circuits constructed from arbitrary single-qubit gates and generalized Toffoli gates. It is arguably the smallest natural class of constant-depth quantum computation which has not been shown useful for computing any non-trivial Boolean function. Despite this, many attempts to port classical $\mathsf{AC}^0$ lower bounds to $\mathsf{QAC}^0$ have failed. We give one possible explanation of this: $\mathsf{QAC}^0$ circuits are significantly more powerful than their classical counterparts. We show the unconditional separation $\mathsf{QAC}^0\not\subset\mathsf{AC}^0[p]$ for decision problems, which also resolves for the first time whether $\mathsf{AC}^0$ could be more powerful than $\mathsf{QAC}^0$. Moreover, we prove that $\mathsf{QAC}^0$ circuits can compute a wide range of Boolean functions if given multiple copies of the input: $\mathsf{TC}^0 \subseteq \mathsf{QAC}^0 \circ \mathsf{NC}^0$. Along the way, we introduce an amplitude amplification technique that makes several approximate constant-depth constructions exact.