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 $\mathsf{ZPP^{NP}}$.

翻译：本文进一步证实了量子计算机能够执行超越经典计算机的计算能力。具体而言，我们强化了多项式层级塌缩至第二层的条件：若（i）具备后选择的量子计算机与具备后选择的经典计算机具有同等计算能力（$\mathsf{PostBQP=PostBPP}$），或（ii）若干量子采样实验（$\mathsf{BosonSampling}$、$\mathsf{IQP}$、$\mathsf{DQC1}$）中的任意一项可由经典计算机近似实现（基于现有假设）。最后一项结果表明，若这些实验中的任一硬度猜想成立，则量子计算机能够实现经典计算机无法执行的功能（$\mathsf{FBQP\neq FBPP}$），除非多项式层级塌缩至第二层。这些结果相较于先前研究有所改进——既往工作要么仅实现向第三层的塌缩，要么仅关注物理上不切实际的精确采样情形。本研究的核心贡献是一项新的技术性复杂性理论结果，我们认为其价值可能超越量子计算领域。具体而言，我们证明：若存在精确计数预言机可解问题与近似计数预言机可解问题之间的等价性，则多项式层级将塌缩至第二层，确切而言是塌缩至$\mathsf{ZPP^{NP}}$。