We present substantially generalized and improved quantum algorithms over prior work for inhomogeneous linear and nonlinear ordinary differential equations (ODE). Specifically, we show how the norm of the matrix exponential characterizes the run time of quantum algorithms for linear ODEs opening the door to an application to a wider class of linear and nonlinear ODEs. In Berry et al., (2017), a quantum algorithm for a certain class of linear ODEs is given, where the matrix involved needs to be diagonalizable. The quantum algorithm for linear ODEs presented here extends to many classes of non-diagonalizable matrices. The algorithm here is also exponentially faster than the bounds derived in Berry et al., (2017) for certain classes of diagonalizable matrices. Our linear ODE algorithm is then applied to nonlinear differential equations using Carleman linearization (an approach taken recently by us in Liu et al., (2021)). The improvement over that result is two-fold. First, we obtain an exponentially better dependence on error. This kind of logarithmic dependence on error has also been achieved by Xue et al., (2021), but only for homogeneous nonlinear equations. Second, the present algorithm can handle any sparse, invertible matrix (that models dissipation) if it has a negative log-norm (including non-diagonalizable matrices), whereas Liu et al., (2021) and Xue et al., (2021) additionally require normality.

翻译：在Berry等人(2017年)中,我们提出了与先前的不相容的线性和非线性普通差异方程式(ODE)相比的大幅度和改良量子算法。具体地说,我们展示了矩阵指数值的规范如何成为线性数数运算的运行时间的特征,从而打开了应用范围更广的线性非线性非线性ODE的大门。在Berry等人(2017年)中,我们给出了某类线性ODE的量子算法,其中涉及的矩阵需要进行分解。在这里提出的线性ODE的量算法扩展至许多非可对等矩阵的类别。这里的算法也比Berry等人(2017年)中某些可对可对线性数运算的基质算法的运行速度要快得多。我们的线性Orental 算法适用于非线性方程(我们最近在Li等人等人(2021年)中采用的算法是二倍的改进。首先,我们获得了对错误的指数性更高的依赖度(2040) 和对可辨性基质性矩阵的逻辑依赖性(如果是Scialalal-al-al-al-al-al-al-al-lial-al-al-al-al-al-al-al-li-al-al-al-al-al-al-al-al-al-al-al-al-al-al-al-al-al-al-al-al-algal-al-algal-al-al-al-al-algalgal-al-al-al-al-al-al-al-al-al-al-al-al-al-al-al-al-al-al-al-al-al-al-al-al-al-al-al-al-al-al-al-al-al-al-al-al-al-al-al-al-al-al-al-al-al-al-al-al-al-al-al-al-al-al-al-al-al-al-al-al-al-al-al-al-al-al-al-al-al-al-al-al-al-al-al-al-l