We study the limitations and fast-forwarding of quantum algorithms for linear ordinary differential equation (ODE) systems with a particular focus on non-quantum dynamics, where the coefficient matrix in the ODE is not anti-Hermitian or the ODE is inhomogeneous. On the one hand, for generic homogeneous linear ODEs, by proving worst-case lower bounds, we show that quantum algorithms suffer from computational overheads due to two types of ``non-quantumness'': real part gap and non-normality of the coefficient matrix. We then show that homogeneous ODEs in the absence of both types of ``non-quantumness'' are equivalent to quantum dynamics, and reach the conclusion that quantum algorithms for quantum dynamics work best. We generalize our results to the inhomogeneous case and find that existing generic quantum ODE solvers cannot be substantially improved. To obtain these lower bounds, we propose a general framework for proving lower bounds on quantum algorithms that are amplifiers, meaning that they amplify the difference between a pair of input quantum states. On the other hand, we show how to fast-forward quantum algorithms for solving special classes of ODEs which leads to improved efficiency. More specifically, we obtain quadratic improvements in the evolution time $T$ for inhomogeneous ODEs with a negative semi-definite coefficient matrix, and exponential improvements in both $T$ and the spectral norm of the coefficient matrix for inhomogeneous ODEs with efficiently implementable eigensystems, including various spatially discretized linear evolutionary partial differential equations. We give fast-forwarding algorithms that are conceptually different from existing ones in the sense that they neither require time discretization nor solving high-dimensional linear systems.

翻译：我们研究线性常微分方程（ODE）系统量子算法的局限性及加速现象，重点关注非量子动力学情形，即ODE系数矩阵非反厄米或方程非齐次。一方面，针对一般齐次线性ODE，通过证明最坏情况下的下界，我们展示了量子算法因两种“非量子特性”——系数矩阵的实部间隙与非正规性——而产生计算开销。进而证明，当齐次ODE同时排除这两种“非量子特性”时，其等价于量子动力学，并得出量子动力学算法最优的结论。我们将结论推广至非齐次情形，发现现有通用量子ODE求解器难以实现本质改进。为获得这些下界，我们提出了一套通用框架，用于证明作为放大器的量子算法的下界——即通过放大输入量子态对的差异实现运算。另一方面，我们展示了如何针对特殊类ODE加速量子算法从而提升效率：对于系数矩阵半负定的非齐次ODE，在演化时间$T$上获得二次加速；对于具有可高效实现本征系统的非齐次ODE（包括多种空间离散化线性演化偏微分方程），在$T$和系数矩阵谱范数上同时获得指数级加速。我们提出的加速算法在概念上区别于现有方法：既无需时间离散化，也无需求解高维线性系统。