  Motivated by the existence of exact many-body quantum scars in the AKLT
chain, we explore the connection between Matrix Product State (MPS)
wavefunctions and many-body quantum scarred Hamiltonians. We provide a method
to systematically search for and construct parent Hamiltonians with towers of
exact eigenstates composed of quasiparticles on top of an MPS wavefunction.
These exact eigenstates have low entanglement in spite of being in the middle
of the spectrum, thus violating the strong Eigenstate Thermalization Hypothesis
(ETH). Using our approach, we recover the AKLT chain starting from the MPS of
its ground state, and we derive the most general nearest-neighbor Hamiltonian
that shares the AKLT quasiparticle tower of exact eigenstates. We further apply
this formalism to other simple MPS wavefunctions, and derive new families of
Hamiltonians that exhibit AKLT-like quantum scars. As a consequence, we also
construct a scar-preserving deformation that connects the AKLT chain to the
integrable spin-1 pure biquadratic model. Finally, we also derive other
families of Hamiltonians that exhibit new types of exact quantum scars,
including a $U(1)$-invariant perturbed Potts model.
