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

Graph states as ground states of two-body frustration-free Hamiltonians