A Criterion for Quantum Advantage

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 $U \in \mathrm{U}(2)$.