  How well can quantum computers simulate classical dynamical systems? There is
increasing effort in developing quantum algorithms to efficiently simulate
dynamics beyond Hamiltonian simulation, but so far exact resource estimates are
not known. In this work, we provide two significant contributions. First, we
give the first non-asymptotic computation of the cost of encoding the solution
to general linear ordinary differential equations into quantum states -- either
the solution at a final time, or an encoding of the whole history within a time
interval. Second, we show that the stability properties of a large class of
classical dynamics allow their fast-forwarding, making their quantum simulation
much more time-efficient. From this point of view, quantum Hamiltonian dynamics
is a boundary case that does not allow this form of stability-induced
fast-forwarding. In particular, we find that the history state can always be
output with complexity $O(T^{1/2})$ for any stable linear system. We present a
range of asymptotic improvements over state-of-the-art in various regimes. We
illustrate our results with a family of dynamics including linearized
collisional plasma problems, coupled, damped, forced harmonic oscillators and
dissipative nonlinear problems. In this case the scaling is quadratically
improved, and leads to significant reductions in the query counts after
inclusion of all relevant constant prefactors.
