Improved separation between quantum and classical computers for sampling and functional tasks

TL;DR

This paper demonstrates that certain quantum sampling experiments, if classically simulable, would cause a collapse of the polynomial hierarchy, strengthening the evidence that quantum computers outperform classical ones in specific tasks.

Contribution

It provides new complexity-theoretic results linking quantum sampling problems to the collapse of the polynomial hierarchy, improving upon previous work by considering approximate sampling and postselection.

Findings

Quantum sampling experiments can be approximately simulated classically under certain assumptions.

Classical simulation of these experiments implies a collapse of the polynomial hierarchy to its second level.

A new complexity result relates exact and approximate counting oracles to hierarchy collapse.

Abstract

This paper furthers existing evidence that quantum computers are capable of computations beyond classical computers. Specifically, we strengthen the collapse of the polynomial hierarchy to the second level if: (i) Quantum computers with postselection are as powerful as classical computers with postselection ($\mathsf{PostBQP=PostBPP}$), (ii) any one of several quantum sampling experiments ($\mathsf{BosonSampling}$, $\mathsf{IQP}$, $\mathsf{DQC1}$) can be approximately performed by a classical computer (contingent on existing assumptions). This last result implies that if any of these experiment's hardness conjectures hold, then quantum computers can implement functions classical computers cannot ($\mathsf{FBQP\neq FBPP}$) unless the polynomial hierarchy collapses to its 2nd level. These results are an improvement over previous work which either achieved a collapse to the third level or were concerned with exact sampling, a physically impractical case. The workhorse of these results is a new technical complexity-theoretic result which we believe could have value beyond quantum computation. In particular, we prove that if there exists an equivalence between problems solvable with an exact counting oracle and problems solvable with an approximate counting oracle, then the polynomial hierarchy collapses to its second level, indeed to $\mathsf{ZPP^{NP}}$.