  We explore methods to generate quantum coherence through unitary evolutions,
by introducing and studying the coherence generating capacity of Hamiltonians.
This quantity is defined as the maximum derivative of coherence that can be
achieved by a Hamiltonian. By adopting the relative entropy of coherence as our
figure of merit, we evaluate the maximal coherence generating capacity with the
constraint of a bounded Hilbert-Schmidt norm for the Hamiltonian. Our
investigation yields closed-form expressions for both Hamiltonians and quantum
states that induce the maximal derivative of coherence under these conditions.
Specifically, for qubit systems, we solve this problem comprehensively for any
given Hamiltonian, identifying the quantum states that lead to the largest
coherence derivative induced by the Hamiltonian. Our investigation enables a
precise identification of conditions under which quantum coherence is optimally
enhanced, offering valuable insights for the manipulation and control of
quantum coherence in quantum systems.
