  We propose an approach to factorize the time-evolution operator of a quantum
system through a (finite) sequence of elementary operations that are
time-ordered. Our proposal borrows from previous approaches based on Lie
algebra techniques and other factorization procedures, and requires a set of
optimization operations that provide the final result. Concretely, the
algorithm produces at each step three optimal quantities, namely the optimal
duration of the desired unitary operation, the optimal functional dependence of
the driving function on the optimal time, and the optimal elementary Hermitian
operation that induces the additional unitary operation to be implemented. The
resulting sequence of unitary operations that is obtained this way is
sequential with time. We compare our proposal with existing approaches, and
highlight which key assumptions can be relaxed for practical implementations.
