  Matrix exponentiation (ME) is widely used in various fields of science and
engineering. For example, the unitary dynamics of quantum systems is described
by exponentiation of Hamiltonian operators. However, despite a significant
attention, the numerical evaluation of ME remains computationally expensive,
particularly for large dimensions. Often this process becomes a bottleneck in
algorithms requiring iterative evaluation of ME. Here we propose a method for
approximating ME of a single operator with a bounded coefficient into a product
of certain discrete operators. This approach, which we refer to as Rapid
Exponentiation using Discrete Operators (REDO), is particularly efficient for
iterating ME over large numbers. We describe REDO in the context of a quantum
system with a constant as well as a time-dependent Hamiltonian, although in
principle, it can be adapted in a more general setting. As concrete examples,
we choose two applications. First, we incorporate REDO in optimal quantum
control algorithms and report a speed-up of several folds over a wide range of
system size. Secondly, we propose REDO for numerical simulations of quantum
dynamics. In particular, we study exotic quantum freezing with noisy drive
fields.
