  We present a quantum algorithm for systems of (possibly inhomogeneous) linear
ordinary differential equations with constant coefficients. The algorithm
produces a quantum state that is proportional to the solution at a desired
final time. The complexity of the algorithm is polynomial in the logarithm of
the inverse error, an exponential improvement over previous quantum algorithms
for this problem. Our result builds upon recent advances in quantum linear
systems algorithms by encoding the simulation into a sparse, well-conditioned
linear system that approximates evolution according to the propagator using a
Taylor series. Unlike with finite difference methods, our approach does not
require additional hypotheses to ensure numerical stability.
