Evolutions of local Hamiltonians in short times are expected to remain local and thus limited. In this paper, we validate this intuition by proving some limitations on short-time evolutions of local time-dependent Hamiltonians. We show that the distribution of the measurement output of short-time (at most logarithmic) evolutions of local Hamiltonians are $concentrated$ and satisfy an $\textit{isoperimetric inequality}$. To showcase explicit applications of our results, we study the $M$$\small{AX}$$C$$\small{UT}$ problem and conclude that quantum annealing needs at least a run-time that scales logarithmically in the problem size to beat classical algorithms on $M$$\small{AX}$$C$$\small{UT}$. To establish our results, we also prove a Lieb-Robinson bound that works for time-dependent Hamiltonians which might be of independent interest.

中文翻译：

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局部哈密顿量短时演化的极限

局部哈密顿量在短时间内的演化预计将保持局部，因此是有限的。在本文中，我们通过证明对局部时间相关哈密顿量的短时间演化的一些限制来验证这种直觉。我们证明了局部哈密顿量的短时间（最多对数）演化的测量输出分布是集中的并且满足等周不等式。为了展示我们的结果的明确应用，我们研究了 $M$$\small{AX}$$C$$\small{UT}$ 问题并得出结论，量子退火至少需要一个在问题中以对数方式扩展的运行时间大小以击败 $M$$\small{AX}$$C$$\small{UT}$ 上的经典算法。为了确定我们的结果，我们还证明了一个 Lieb-Robinson 界，该界适用于可能具有独立兴趣的时间相关哈密顿量。