  We show how to perform universal adiabatic quantum computation using a
Hamiltonian which describes a set of particles with local interactions on a
two-dimensional grid. A single parameter in the Hamiltonian is adiabatically
changed as a function of time to simulate the quantum circuit. We bound the
eigenvalue gap above the unique groundstate by mapping our model onto the
ferromagnetic XXZ chain with kink boundary conditions; the gap of this spin
chain was computed exactly by Koma and Nachtergaele using its $q$-deformed
version of SU(2) symmetry. We also discuss a related time-independent
Hamiltonian which was shown by Janzing to be capable of universal computation.
We observe that in the limit of large system size, the time evolution is
equivalent to the exactly solvable quantum walk on Young's lattice.
