When expressing computations in terms of a quantum circuit, one makes use of gates, that is, (typically) unitary evolutions.
In some sense, these are rather mysterious objects, in that they perform "magic" discrete operations on the states. They are essentially black boxes, whose inner workings are not often dealt with while studying quantum algorithms. However, that is not how quantum mechanics works: states evolve in a continuous fashion following Schrödinger's equation.
In other words, when talking about quantum gates and operations, one neglects the dynamic (that is, the Hamiltonian) realising said evolution, which is how the gates are actually implemented in experimental architectures.
One method is to decompose the gate in terms of elementary (in a given experimental architecture) ones. Is this the only way? What about such "elementary" gates? How are the dynamics implementing those typically found?