  We investigate entanglement naturally present in the 1D topologically ordered
phase protected with the on-site symmetry group of an octahedron as a potential
resource for teleportation-based quantum computation. We show that, as long as
certain characteristic lengths are finite, all its ground states have the
capability to implement any unit-fidelity one-qubit gate operation
asymptotically as a key computational building block. This feature is intrinsic
to the entire phase, in that perfect gate fidelity coincides with perfect
string order parameters under a state-insensitive renormalization procedure.
Our approach may pave the way toward a novel program to classify quantum
many-body systems based on their operational use for quantum information
processing.
