  Fully-augmented Matrix Product States (FAMPS) was proposed recently (Chin.
Phys. Lett. 40, 057102 (2023)) as an accurate numerical tool to study
two-dimensional quantum many-body systems. It is constructed by including a
disentangler layer upon MPS. The cost of simulating quantum models with FAMPS
is similar as DMRG (with small overhead), but FAMPS can support area-law
entanglement entropy for two-dimensional systems. These properties make FAMPS
an effective and efficient tool. In this work, we demonstrate that for each
FAMPS we can construct a two-dimensional Hamiltonian with the FAMPS being its
ground state. We show how to construct the parent Hamiltonian for given FAMPS.
We also perform numerical simulation to show that the algorithm proposed in
Chin. Phys. Lett. 40, 057102 (2023) can find the exact FAMPS for the parent
Hamiltonian. FAMPS and the corresponding parent Hamiltonian provides a useful
framework for the future study of two-dimensional quantum many-body systems
