  Universal quantum computation can be achieved by simply performing
single-spin measurements on a highly entangled resource state, such as cluster
states. The family of Affleck-Kennedy-Lieb-Tasaki (AKLT) states has recently
been explored; for example, the spin-1 AKLT chain can be used to simulate
single-qubit gate operations on a single qubit, and the spin-3/2
two-dimensional AKLT state on the honeycomb lattice can be used as a universal
resource. However, it is unclear whether such universality is a coincidence for
the specific state or a shared feature in all two-dimensional AKLT states. Here
we consider the family of spin-3/2 AKLT states on various trivalent Archimedean
lattices and show that in addition to the honeycomb lattice, the spin-3/2 AKLT
states on the square octagon $(4,8^2)$ and the `cross' $(4,6,12)$ lattices are
also universal resource, whereas the AKLT state on the `star' $(3,12^2)$
lattice is likely not due to geometric frustration.

Quantum Physics;Statistical Mechanics