  Time-dependent linear differential equations are a common type of problem
that needs to be solved in classical physics. Here we provide a quantum
algorithm for solving time-dependent linear differential equations with
logarithmic dependence of the complexity on the error and derivative. As usual,
there is an exponential improvement over classical approaches in the scaling of
the complexity with the dimension, with the caveat that the solution is encoded
in the amplitudes of a quantum state. Our method is to encode the Dyson series
in a system of linear equations, then solve via the optimal quantum linear
equation solver. Our method also provides a simplified approach in the case of
time-independent differential equations.
