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 $\pmod{p}$ gates. In addition, we show a separation between $\mathsf{QAC}^0$, which additionally has Toffoli gates with unbounded control, and $\mathsf{AC}^0[p]$. This establishes the first such separation for a shallow-depth quantum class that does not involve quantum fan-out gates. Secondly, we consider $\mathsf{QNC}^0$ circuits with infinite-size gate sets. We show that these circuits, along with (classical or quantum) prime modular gates, can implement threshold gates, showing that $\mathsf{QNC}^0[p]=\mathsf{QTC}^0$. Finally, we also show that in the infinite-size gateset case, these quantum circuit classes for higher-dimensional Hilbert spaces do not offer any advantage to standard qubit implementations.

翻译：近年来，浅层量子电路的相关性显著提升，主要得益于其在近中期设备上的应用潜力。在此背景下，量子电路复杂性的核心目标之一是寻找那些可由量子浅层电路解决、但在经典计算中需要更多资源的问题。本文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，我们展示了带量子建议的常数深度量子电路 $\mathsf{QNC}^0/\mathsf{qpoly}$ 与经典常数深度电路类 $\mathsf{AC}^0[p]$（允许无扇入限制和 $\pmod{p}$ 门）之间的分离。其次，我们进一步证明了 $\mathsf{QAC}^0$（额外包含无界控制Toffoli门）与 $\mathsf{AC}^0[p]$ 之间的分离，这是首次针对不涉及量子扇出门的浅层量子类建立此类分离。接着，我们考虑使用无限大门集的 $\mathsf{QNC}^0$ 电路，表明这类电路结合（经典或量子）素数模门可实现阈值门，从而证明 $\mathsf{QNC}^0[p]=\mathsf{QTC}^0$。最后，我们证明在无限大门集情形下，面向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未提供任何优势。