How well can quantum computers simulate classical dynamical systems? There is increasing effort in developing quantum algorithms to efficiently simulate dynamics beyond Hamiltonian simulation, but so far exact resource estimates are not known. In this work, we provide two significant contributions. First, we give the first non-asymptotic computation of the cost of encoding the solution to general linear ordinary differential equations into quantum states – either the solution at a final time, or an encoding of the whole history within a time interval. Second, we show that the stability properties of a large class of classical dynamics allow their fast-forwarding, making their quantum simulation much more time-efficient. From this point of view, quantum Hamiltonian dynamics is a boundary case that does not allow this form of stability-induced fast-forwarding. In particular, we find that the history state can always be output with complexity $O(T^{1/2})$ for any stable linear system. We present a range of asymptotic improvements over state-of-the-art in various regimes. We illustrate our results with a family of dynamics including linearized collisional plasma problems, coupled, damped, forced harmonic oscillators and dissipative nonlinear problems. In this case the scaling is quadratically improved, and leads to significant reductions in the query counts after inclusion of all relevant constant prefactors.

中文翻译：

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在量子计算机上求解线性微分方程的成本：快进到显式资源计数


量子计算机模拟经典动力学系统的程度如何？人们越来越努力开发量子算法来有效地模拟哈密顿模拟之外的动力学，但到目前为止，确切的资源估计尚不清楚。在这项工作中，我们提供了两个重要的贡献。首先，我们给出了将一般线性常微分方程的解编码为量子态的成本的第一次非渐近计算——要么是最终时间的解，要么是某个时间间隔内整个历史的编码。其次，我们表明，一大类经典动力学的稳定性特性允许它们快进，从而使它们的量子模拟更加省时。从这个角度来看，量子哈密顿动力学是一个边界情况，不允许这种形式的稳定性诱导的快进。特别是，我们发现对于任何稳定的线性系统，历史状态总是可以以复杂度 $O（T^{1/2}）$ 输出。我们提出了一系列在各种制度下对最先进的技术的渐近改进。我们用一系列动力学来说明我们的结果，包括线性碰撞等离子体问题、耦合、阻尼、强制谐波振荡器和耗散非线性问题。在这种情况下，缩放得到了二次方改进，并在包含所有相关的常量前提因子后导致查询计数显著减少。