A Criterion for Post-Selected Quantum Advantage

TL;DR

This paper establishes a criterion based on group density properties to identify when certain quantum circuits cannot be efficiently simulated classically, demonstrating quantum advantage for various circuit classes under the assumption of an infinite polynomial hierarchy.

Contribution

It introduces a new criterion for quantum advantage based on subgroup density, providing simplified proofs and settling open questions about specific circuit classes' classical simulability.

Findings

IQP and conjugated Clifford circuits have quantum advantage.

Commuting Clifford circuits also have quantum advantage.

Circuits over (U†⊗U†)CZ(U⊗U) are classically hard to simulate for almost all U.

Abstract

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