Recently, Okuyama and Ohzek [1] derived a speed limit for the imaginary-time Schrödinger equation, which is inspired by the prior work of Kieu, who had shown a new class of time–energy uncertainty relations suitable for actually evaluating the speed limit of quantum dynamics. In this paper, we apply the result of Okuyama and Ohzek to obtain a generalized speed limit for Grover’s search in imaginary-time quantum annealing. An estimate of the lower bound on the computational time is shown, from which the role of the coefficient function corresponding to the final Hamiltonian played in the quantum dynamics for the problem is sticking out. However, when trying to apply the speed limit to the analogue of Grover’s problem, we find that not only the coefficient of the target Hamiltonian is related to the time complexity of the algorithm, but also the coefficient of the initial Hamiltonian is crucial for determining the time complexity. This is new and generalizes one of the results in our previous work.

中文翻译：

---

虚时间薛定谔方程的速度极限及其在量子搜索中的应用

最近，Okuyama 和 Ohzek [1] 推导出了虚时间薛定谔方程的速度限制，这是受到 Kieu 先前工作的启发，他展示了一种新的时间-能量不确定性关系，适用于实际评估速度限制量子动力学。在本文中，我们应用 Okuyama 和 Ohzek 的结果来获得 Grover 在虚时间量子退火中搜索的广义速度限制。显示了对计算时间下限的估计，从中可以看出对应于最终哈密顿量的系数函数在该问题的量子动力学中所起的作用。然而，当试图将速度限制应用于 Grover 问题的类比时，我们发现不仅目标哈密顿量的系数与算法的时间复杂度有关，而且初始哈密顿量的系数对于确定时间复杂度至关重要。这是新的，并且概括了我们之前工作中的一个结果。