Although quantum computing is expected to outperform universal classical computing, an unconditional proof of this assertion seems to be hard because an unconditional separation between ${\sf BQP}$ and ${\sf BPP}$ implies ${\sf P}\neq{\sf PSPACE}$. Because of this, the quantum-computational-supremacy approach has been actively studied; it shows that if the output probability distributions from a family of quantum circuits can be efficiently simulated in classical polynomial time, then the polynomial hierarchy collapses to its second or third level. Since it is widely believed that the polynomial hierarchy does not collapse, this approach shows one kind of quantum advantage under a plausible assumption. On the other hand, the limitations of universal quantum computing are also actively studied. For example, it is believed to be impossible to generate ground states of any local Hamiltonians in quantum polynomial time. In this paper, we give evidence for this impossibility by applying an argument used in the quantum-computational-supremacy approach. More precisely, we show that if ground states of any $3$-local Hamiltonians can be approximately generated in quantum polynomial time with postselection, then the counting hierarchy collapses to its first level. Our evidence is superior to the existing findings in the sense that we reduce the impossibility to an unlikely relation between classical complexity classes. Furthermore, our argument can be used to give evidence that at least one $3$-local Hamiltonian exists such that its ground state cannot be represented by a polynomial number of bits, which may be related to a gap between ${\sf QMA}$ and ${\sf QCMA}$.

中文翻译：

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后选量子计算很难有效地生成基态

尽管预计量子计算将优于通用经典计算，但这一断言的无条件证明似乎很难，因为 ${\sf BQP}$ 和 ${\sf BPP}$ 之间的无条件分离意味着 ${\sf P}\neq {\sf PSPACE}$。正因为如此，量子计算至上的方法得到了积极的研究。它表明，如果可以在经典多项式时间内有效地模拟来自一系列量子电路的输出概率分布，那么多项式层次结构就会崩溃到它的第二或第三级。由于人们普遍认为多项式层次结构不会崩溃，因此这种方法在合理假设下显示出一种量子优势。另一方面，通用量子计算的局限性也得到了积极研究。例如，人们认为在量子多项式时间内不可能产生任何局部哈密顿量的基态。在本文中，我们通过应用量子计算至上方法中使用的论证来证明这种不可能性。更准确地说，我们表明，如果任何 $3$-局部哈密顿量的基态可以在量子多项式时间内通过后选择近似生成，那么计数层次结构就会崩溃到它的第一级。我们的证据优于现有的发现，因为我们将经典复杂性类别之间的可能性降低到不太可能的关系。此外，我们的论点可用于证明至少存在一个 $3$-局部哈密顿量，其基态不能用多项式位数表示，