Classical simulation of commuting quantum computations implies collapse of the polynomial hierarchy

TL;DR

This paper demonstrates that efficiently simulating the output of commuting quantum computations (IQP) classically would cause the polynomial hierarchy to collapse, indicating such simulations are likely infeasible.

Contribution

It introduces the class post-IQP, proves its equivalence to PP, and links classical simulation difficulty of IQP to the collapse of the polynomial hierarchy.

Findings

Post-IQP equals PP, establishing a link between quantum and classical complexity.

Efficient classical sampling of IQP outputs would collapse the polynomial hierarchy.

Sampling from measurements on O(log n) lines is classically feasible.

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.