  Universal quantum computation can be achieved by simply performing
single-qubit measurements on a highly entangled resource state. Resource states
can arise from ground states of carefully designed two-body interacting
Hamiltonians. This opens up an appealing possibility of creating them by
cooling. The family of Affleck-Kennedy-Lieb-Tasaki (AKLT) states are the ground
states of particularly simple Hamiltonians with high symmetry, and their
potential use in quantum computation gives rise to a new research direction.
Expanding on our prior work [T.-C. Wei, I. Affleck, and R. Raussendorf, Phys.
Rev. Lett. 106, 070501 (2011)], we give detailed analysis to explain why the
spin-3/2 AKLT state on a two-dimensional honeycomb lattice is a universal
resource for measurement-based quantum computation. Along the way, we also
provide an alternative proof that the 1D spin-1 AKLT state can be used to
simulate arbitrary one-qubit unitary gates. Moreover, we connect the quantum
computational universality of 2D random graph states to their percolation
property and show that these states whose graphs are in the supercritical (i.e.
percolated) phase are also universal resources for measurement-based quantum
computation.
