My task is to simulate quantum evolution. To do that I need to perform this operation
$$w = e^{-itH}v$$
where $H$ is a sparse matrix and $v$ is the initial column vector. I am wondering if there is a way to calculate vector $w$ without evaluating the MatrixExp? What I mean is: if $t$ is really small (time step) is there some kind of an algorithm (stable and accurate) which uses only matrix-vector multiplication?
For example:
$$w(t) = \left(e^{-i\frac{t}{n}H}\right)^{n}v\\ \frac{t}{n} = dt$$ Algorithm would go like this: $$w(0) = v \\ w(dt) = e^{-idtH}w(0)\\ w(2dt) = e^{-idtH}w(dt)\\ \ldots$$
and I don't know, replace $$e^{-idtH}\approx 1 -idtH$$
One thing should remain unchanged: norm of the vector $||w(t)||=1$