Measurement-based quantum gate teleportation and the degeneracy of the entanglement spectrum: beyond symmetry protected topological order

Quantum states with symmetry protected topological order (SPTO), such as the gapped degenerate ground states of the AKLT spin chain for a suitable choice of parameters, have a key common attribute: the Schmidt coefficients for the decomposition of any bipartite density operator are degenerate. This attribute is also shared by cluster states, the first resource to be identified for measurement-based quantum computation (MBQC), which obeys a global on-site symmetry. While the existence of SPTO is sufficient to ensure (noisy) MBQC, it is not clear what role the degeneracy of the entanglement spectrum itself plays; in principle it is possible that accidental degeneracy could suffice for MBQC, even in the absence of SPTO. We explore the implications of this possibility for quantum gate teleportation in one-dimensional spin chains; while the teleported gates are offered no protection from errors in this scheme, it would greatly expand the resource states able to support MBQC.