Assuming the polynomial hierarchy is infinite, we prove a sufficient condition for determining if uniform and polynomial size quantum circuits over a non-universal gate set are not efficiently classically simulable in the weak multiplicative sense. Our criterion exploits the fact that subgroups of $\mathrm{SL}(2;\mathbb{C})$ are essentially either discrete or dense in $\mathrm{SL}(2;\mathbb{C})$. Using our criterion, we give a new proof that both instantaneous quantum polynomial (IQP) circuits and conjugated Clifford circuits (CCCs) afford a quantum advantage. We also prove that both commuting CCCs and CCCs over various fragments of the Clifford group afford a quantum advantage, which settles two questions of Bouland, Fitzsimons, and Koh. Our results imply that circuits over just $(U^\dagger \otimes U^\dagger) \mathrm{CZ} (U \otimes U)$ afford a quantum advantage for almost all $U \in \mathrm{U}(2)$.

翻译：假设多项式层级是无限的，我们证明了一个充分条件，用于判定基于非通用门集的均匀多项式规模量子电路是否在弱乘法意义上无法被经典计算机高效模拟。我们的准则利用了$\mathrm{SL}(2;\mathbb{C})$的子群本质上要么是离散的、要么在$\mathrm{SL}(2;\mathbb{C})$中稠密这一事实。应用该准则，我们给出了瞬时量子多项式（IQP）电路与共轭克利福德电路（CCC）均能实现量子优势的新证明。我们还证明了交换CCC以及基于克利福德群不同子结构的CCC均能实现量子优势，这解决了Bouland、Fitzsimons和Koh提出的两个问题。我们的结果表明，对于几乎所有$U \in \mathrm{U}(2)$，仅由$(U^\dagger \otimes U^\dagger) \mathrm{CZ} (U \otimes U)$构成的电路即可实现量子优势。