  There are several known techniques to construct a Hamiltonian with an
expected value that is minimized uniquely by a given quantum state. Common
approaches include the parent Hamiltonian construction from matrix product
states, building approximate ground state projectors, and, in a common case,
developing penalty functions from the generalized Ising model. Here we consider
the framework that enables one to engineer exact parent Hamiltonians from
commuting polynomials. We derive elementary classification results of quadratic
Ising parent Hamiltonians and to generally derive a non-injective parent
Hamiltonian construction. We also consider that any $n$-qubit stabilizer state
has a commutative parent Hamiltonian with $n+1$ terms and we develop an
approach that allows the derivation of parent Hamiltonians by composition of
network elements that embed the truth tables of discrete functions into a
kernel space. This work presents a unifying framework that captures components
of what is known about exact parent Hamiltonians and bridges a few techniques
across the domains that are concerned with such constructions.
