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 shallow quantum 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期显著提升，主要源于其在近期设备上的适用性。在此背景下，量子电路复杂性的核心目标之一是寻找可由浅层量子电路解决、但经典计算需要更多资源的问题。本文的首个贡献是证明了经典与量子常量深度电路之间的新分离性。首先，我们展示了带量子建议的常量深度量子电路类 $\mathsf{QNC}^0/\mathsf{qpoly}$ 与经典常量深度电路类 $\mathsf{AC}^0[p]$（包含无界扇入和 $\mathsf{MOD}_{p}$ 门）之间的分离性。其次，我们证明了包含无界控制托佛利门的电路类 $\mathsf{QAC}^0$ 与 $\mathsf{AC}^0[p]$ 之间的分离性——当 $\mathsf{QAC}^0$ 附加中间电路测量和经典扇出时。这建立了首个不涉及量子扇出门、仅依赖有限量子门集的浅层量子类别的此类分离性。等价地，这揭示了 $\mathsf{AC}^0[p]$ 与 $[\mathsf{QNC}^0, \mathsf{AC}^0]^2$（即浅层量子电路与简单经典计算的交错结构）之间的分离性。其次，我们研究了含无限大小门集的 $\mathsf{QNC}^0$ 电路。结果表明，此类电路结合量子素数模门或带经典扇出的经典素数模门，可实现阈值门功能，从而证明 $\mathsf{QNC}^0[p]=\mathsf{QTC}^0$。最后，我们还指出在无限大小门集情形下，这些针对高维希尔伯特空间的量子电路类相较于标准量子比特实现并无优势。