  Given a generic time-dependent many-body quantum state, we determine the
associated parent Hamiltonian. This procedure may require, in general,
interactions of any sort. Enforcing the requirement of a fixed set of
engineerable Hamiltonians, we find the optimal Hamiltonian once a set of
realistic elementary interactions is defined. We provide three examples of this
approach. We first apply the optimization protocol to the ground states of the
one-dimensional Ising model and a ferromagnetic $p$-spin model but with
time-dependent coefficients. We also consider a time-dependent state that
interpolates between a product state and the ground state of a $p$-spin model.
We determine the time-dependent optimal parent Hamiltonian for these states and
analyze the capability of this Hamiltonian of generating the state evolution.
Finally, we discuss the connections of our approach to shortcuts to
adiabaticity.

Quantum Physics