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：我们提出了相较于先前工作在非齐次线性和非线性常微分方程（ODE）方面大幅推广和改进的量子算法。具体而言，我们展示了矩阵指数范数如何表征线性ODE量子算法的运行时间，从而为将其应用于更广泛的线性和非线性ODE类别打开了大门。在Berry等人（2017）的工作中，针对特定类别的线性ODE给出了量子算法，其中所涉及的矩阵需可对角化。本文提出的线性ODE量子算法扩展至许多类别的不可对角化矩阵。对于某些类别的可对角化矩阵，该算法在运行时间上比Berry等人（2017）推导的界限呈指数级加速。随后，我们将线性ODE算法通过Carleman线性化（我们近期在Liu等人（2021）中采用的方法）应用于非线性微分方程。相较于该结果的改进体现在两个方面。首先，我们获得了对误差的指数级更好依赖关系。这种对误差的对数依赖关系也在Xue等人（2021）中实现，但仅适用于齐次非线性方程。其次，当前算法可处理任何稀疏、可逆的矩阵（用于建模耗散），前提是其具有负对数范数（包括不可对角化矩阵），而Liu等人（2021）和Xue等人（2021）还额外要求正规性。