Classical simulation of commuting quantum computations implies collapse of the polynomial hierarchy
Michael J. Bremner, Richard Jozsa, Dan J. Shepherd
TL;DR
The paper investigates commuting quantum computations (IQP) and argues that sampling their output distributions is unlikely to be classically efficient. It introduces post-IQP and proves post-IQP = PP, connecting IQP with the well-studied post-selected quantum class. Moreover, it shows that efficient classical sampling of IQP outputs within multiplicative error less than sqrt(2) would cause the polynomial hierarchy to collapse to Δ_3, a highly implausible consequence. The work also proves that exact classical sampling is feasible when restricting to O(log n) output lines, and discusses extensions to other restricted circuit classes, underscoring a robust boundary between quantum advantage and classical simulability in structured quantum models.
Abstract
We consider quantum computations comprising only commuting gates, known as IQP computations, and provide compelling evidence that the task of sampling their output probability distributions is unlikely to be achievable by any efficient classical means. More specifically we introduce the class post-IQP of languages decided with bounded error by uniform families of IQP circuits with post-selection, and prove first that post-IQP equals the classical class PP. Using this result we show that if the output distributions of uniform IQP circuit families could be classically efficiently sampled, even up to 41% multiplicative error in the probabilities, then the infinite tower of classical complexity classes known as the polynomial hierarchy, would collapse to its third level. We mention some further results on the classical simulation properties of IQP circuit families, in particular showing that if the output distribution results from measurements on only O(log n) lines then it may in fact be classically efficiently sampled.