The relevance of shallow-depth quantum circuits has recently increased, mainly due to their applicability to near-term devices. In this context, one of the main goals of quantum circuit complexity is to find problems that can be solved by quantum shallow circuits but require more computational resources classically. Our first contribution in this work is to prove new separations between classical and quantum constant-depth circuits. Firstly, we show a separation between constant-depth quantum circuits with quantum advice $\mathsf{QNC}^0/\mathsf{qpoly}$, and $\mathsf{AC}^0[p]$, which is the class of classical constant-depth circuits with unbounded-fan in and $\mathsf{MOD}_{p}$ gates. Additionally, we show a separation between $\mathsf{QAC}^0$, the circuit class containing Toffoli gates with unbounded control, and $\mathsf{AC}^0[p]$, when $\mathsf{QAC}^0$ is augmented with additional mid-circuit measurements and classical fanout. This establishes the first such separation for a shallow-depth quantum class that does not involve quantum fanout gates, while relying solely on finite quantum gate sets. Equivalently, this yields a separation between $\mathsf{AC}^0[p]$ and $[\mathsf{QNC}^0, \mathsf{AC}^0]^2$, i.e., shallow quantum circuits interleaved with simple classical computation. Secondly, we consider $\mathsf{QNC}^0$ circuits with infinite-size gate sets. We show that these circuits, along with quantum prime modular gates or classical prime modular gates in combination with classical fanout, can implement threshold gates, showing that $\mathsf{QNC}^0[p]=\mathsf{QTC}^0$. Finally, we also show that in the infinite-size gate set case, these quantum circuit classes for higher-dimensional Hilbert spaces do not offer any advantage to standard qubit implementations.

翻译：浅层量子电路的相关性近来日益凸显，这主要得益于其在近期设备上的适用性。在此背景下，量子电路复杂度的核心目标之一是寻找能被量子浅层电路解决、但经典计算需要更多资源的问题。本工作的首要贡献是证明了经典与量子常数深度电路之间的新分离结果。首先，我们展示了具有量子建议的常数深度量子电路类$\mathsf{QNC}^0/\mathsf{qpoly}$与经典常数深度电路类$\mathsf{AC}^0[p]$（包含无界扇入与$\mathsf{MOD}_{p}$门）之间的分离。此外，我们证明了包含无界控制Toffoli门的电路类$\mathsf{QAC}^0$在与额外的中电路测量及经典扇出结合时，与$\mathsf{AC}^0[p]$之间存在分离。这首次为不涉及量子扇出门、仅依赖有限量子门集的浅层量子类建立了此类分离。等价地，这得出了$\mathsf{AC}^0[p]$与$[\mathsf{QNC}^0, \mathsf{AC}^0]^2$（即与简单经典计算交织的浅层量子电路）之间的分离。其次，我们研究了具有无限规模门集的$\mathsf{QNC}^0$电路。我们证明此类电路结合量子素数模门或经典素数模门与经典扇出，可实现阈值门，从而表明$\mathsf{QNC}^0[p]=\mathsf{QTC}^0$。最后，我们还证明了在无限规模门集情形下，针对高维希尔伯特空间的这些量子电路类相较于标准量子比特实现并未提供任何优势。