  The time-marching strategy, which propagates the solution from one time step
to the next, is a natural strategy for solving time-dependent differential
equations on classical computers, as well as for solving the Hamiltonian
simulation problem on quantum computers. For more general linear differential
equations, a time-marching based quantum solver can suffer from exponentially
vanishing success probability with respect to the number of time steps and is
thus considered impractical. We solve this problem by repeatedly invoking a
technique called the uniform singular value amplification, and the overall
success probability can be lower bounded by a quantity that is independent of
the number of time steps. The success probability can be further improved using
a compression gadget lemma. This provides a path of designing quantum
differential equation solvers that is alternative to those based on quantum
linear systems algorithms (QLSA). We demonstrate the performance of the
time-marching strategy with a high-order integrator based on the truncated
Dyson series. The complexity of the algorithm depends linearly on the
amplification ratio, which quantifies the deviation from a unitary dynamics. We
prove that the linear dependence on the amplification ratio attains the query
complexity lower bound and thus cannot be improved in the worst case. This
algorithm also surpasses existing QLSA based solvers in three aspects: (1) the
coefficient matrix $A(t)$ does not need to be diagonalizable. (2) $A(t)$ can be
non-smooth, and is only of bounded variation. (3) It can use fewer queries to
the initial state. Finally, we demonstrate the time-marching strategy with a
first-order truncated Magnus series, while retaining the aforementioned
benefits. Our analysis also raises some open questions concerning the
differences between time-marching and QLSA based methods for solving
differential equations.
