k阶常微分方程初值问题的随机量子解
Randomized and Quantum Solution of Initial-Value Problems for Ordinary Differential Equations of Order k
摘要：
我们研究了随机和量子计算在k阶常微分方程标量初值问题中相对于确定性计算的可能优势。对于一阶方程组，这个问题已经在\cite{Kacewicz05}中解决了一些细节。\cite{Kacewicz05}中所示的确定性计算的加速与相对于右侧函数的解的增加的正则性有关。对于一个k阶标量方程（可以转化为一阶特殊系统），解的正则性增加了k个数量级。这导致在确定性设置中依赖于线性信息的k的改进的复杂度界限，参见\cite{Szczesny05}。这可能表明，在随机化和量子设置中，也可以根据k来实现加速。在本文中，我们表明，依赖于k的加速是不可能的随机和量子设置。我们建立了较低的复杂性界限，表明随机和量子复杂性保持在一定的水平，为一阶系统，无论k是多大。因此，从\cite{Kacewicz05}的算法保持（几乎）最优，即使我们限制自己的一个子类的系统所产生的标量方程的阶k。
英文摘要：
We study possible advantages of randomized and quantum computing over deterministic computing for scalar initial-value problems for ordinary differential equations of order k. For systems of equations of the first order this question has been settled modulo some details in \cite{Kacewicz05}. A speed-up over deterministic computing shown in \cite{Kacewicz05} is related to the increased regularity of the solution with respect to that of the right-hand side function. For a scalar equation of order k (which can be transformed into a special system of the first order), the regularity of the solution is increased by k orders of magnitude. This leads to improved complexity bounds depending on k for linear information in the deterministic setting, see \cite{Szczesny05}. This may suggest that in the randomized and quantum settings a speed-up can also be achieved depending on k. We show in this paper that a speed-up dependent on k is not possible in the randomized and quantum settings. We establish lower complexity bounds, showing that the randomized and quantum complexities remain at the some level as for systems of the first order, no matter how large k is. Thus, the algorithms from \cite{Kacewicz05} remain (almost) optimal, even if we restrict ourselves to a subclass of systems arising from scalar equations of order k.
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