  In this short letter, I present a powerful application in dimensionality
reduction of the lesser-used Jacobi-Davidson algorithm for the generalized
eigenvalue decomposition. When combined with matrix-free implementations of
relevant operators, this technique allows for the computation of the dynamics
of an arbitrary quantum state to be done in $\mathcal{O}(n)$ time, where $n$ is
the size of the original Hilbert space.
