论文标题
微分方程分析的量子优势
Quantum advantage for differential equation analysis
论文作者
论文摘要
用于差异方程式和机器学习的量子算法可能在所有已知的经典算法上都具有指数加速。但是，在有用的问题实例中获得这种潜在的加速也存在障碍。量子微分方程求解的基本障碍是输出有用的信息可能需要困难的后处理，而量子机器学习的基本障碍是，输入训练集只是一项艰巨的任务。在本文中，我们证明了这些困难彼此解决。我们展示了量子微分方程求解的输出如何用作量子机学习的输入，从而可以通过主组件，功率谱和小波的分解来动态分析。为了说明这一点，我们考虑了马尔可夫在流行病学和社交网络上的连续时间。这些量子算法比现有的经典蒙特卡洛方法提供了指数优势。
Quantum algorithms for both differential equation solving and for machine learning potentially offer an exponential speedup over all known classical algorithms. However, there also exist obstacles to obtaining this potential speedup in useful problem instances. The essential obstacle for quantum differential equation solving is that outputting useful information may require difficult post-processing, and the essential obstacle for quantum machine learning is that inputting the training set is a difficult task just by itself. In this paper, we demonstrate, when combined, these difficulties solve one another. We show how the output of quantum differential equation solving can serve as the input for quantum machine learning, allowing dynamical analysis in terms of principal components, power spectra, and wavelet decompositions. To illustrate this, we consider continuous time Markov processes on epidemiological and social networks. These quantum algorithms provide an exponential advantage over existing classical Monte Carlo methods.
