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更为普遍化的改进型量子算法，用于求解非齐次线性和非线性常微分方程。具体而言，我们揭示了矩阵指数范数如何决定线性常微分方程量子算法的运行时间，从而为更广泛的线性和非线性方程开辟了应用途径。Berry等人（2017）针对特定类别的线性常微分方程提出了量子算法，但要求所涉矩阵可对角化。本文提出的线性常微分方程量子算法可扩展至多种不可对角化矩阵类别，且对于某些可对角化矩阵，其运行时间比Berry等人（2017）推导的界呈指数级加速。随后，我们将线性常微分方程算法通过Carleman线性化（我们近期在Liu等人（2021）工作中采用的方案）应用于非线性微分方程。相较于该结果，本研究具有双重改进：其一，我们获得了对误差的指数级更优依赖性——这种对数级误差依赖虽已由Xue等人（2021）实现，但仅针对齐次非线性方程；其二，当前算法可处理任意具有负对数范数（包括不可对角化矩阵）的稀疏可逆矩阵（用于模拟耗散过程），而Liu等人（2021）和Xue等人（2021）的工作额外要求矩阵具有正规性。