  The framework of measurement-based quantum computation (MBQC) allows us to
view the ground states of local Hamiltonians as potential resources for
universal quantum computation. A central goal in this field is to find models
with ground states that are universal for MBQC and that are also natural in the
sense that they involve only two-body interactions and have a small local
Hilbert space dimension. Graph states are the original resource states for
MBQC, and while it is not possible to obtain graph states as exact ground
states of two-body Hamiltonians here we construct two-body frustration-free
Hamiltonians that have arbitrarily good approximations of graph states as
unique ground states. The construction involves taking a two-body
frustration-free model that has a ground state convertible to a graph state
with stochastic local operations, then deforming the model such that its ground
state is close to a graph state. Each graph state qubit resides in a subspace
of a higher dimensional particle. This deformation can be applied to two-body
frustration-free Affleck-Kennedy-Lieb-Tasaki (AKLT) models, yielding
Hamiltonians that are exactly solvable with exact tensor network expressions
for ground states. For the star-lattice AKLT model, the ground state of which
is not expected to be a universal resource for MBQC, applying such a
deformation appears to enhance the computational power of the ground state,
promoting it to a universal resource for MBQC. Transitions in computational
power, similar to percolation phase transitions, can be observed when
Hamiltonians are deformed in this way. Improving the fidelity of the ground
state comes at the cost of a shrinking gap. While analytically proving gap
properties for these types of models is difficult in general, we provide a
detailed analysis of the deformation of a spin-1 AKLT state to a linear graph
state.
