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We study the limitations and fast-forwarding of quantum algorithms for linearordinary differential equation (ODE) systems with a particular focus onnon-quantum dynamics, where the coefficient matrix in the ODE is notanti-Hermitian or the ODE is inhomogeneous. On the one hand, for generichomogeneous linear ODEs, by proving worst-case lower bounds, we show thatquantum 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 theconclusion that quantum algorithms for quantum dynamics work best. Wegeneralize our results to the inhomogeneous case and find that existing genericquantum ODE solvers cannot be substantially improved. To obtain these lowerbounds, we propose a general framework for proving lower bounds on quantumalgorithms that are amplifiers, meaning that they amplify the differencebetween a pair of input quantum states. On the other hand, we show how tofast-forward quantum algorithms for solving special classes of ODEs which leadsto improved efficiency. More specifically, we obtain quadratic improvements inthe evolution time $T$ for inhomogeneous ODEs with a negative semi-definitecoefficient matrix, and exponential improvements in both $T$ and the spectralnorm of the coefficient matrix for inhomogeneous ODEs with efficientlyimplementable eigensystems, including various spatially discretized linearevolutionary partial differential equations. We give fast-forwarding algorithmsthat are conceptually different from existing ones in the sense that theyneither require time discretization nor solving high-dimensional linearsystems.

A theory of quantum differential equation solvers: limitations and fast-forwarding