Symmetry Protection of Measurement-based Teleportation in Ising Graphs

Symmetry protected topological (SPT) phases can be identified from degeneracies in their entanglement spectrum and show stability to certain unitary perturbations thanks to string symmetries. Graph state chains have been shown, incidentally, to possess such SPT order which, in turn, protects measurement-based teleportation along the chain. We discuss a new class of SPT states which we call Ising graphs. We define a hierarchy to construct stabilizers that establish multiple string symmetries in Ising graphs beyond just chain geometries. Using our approach, protection against Ising-type unitary perturbations can be engineered within the Ising graph. We find that Ising graphs can be constructed to host redundancy of the degeneracy in the entanglement spectrum which then protects measurement-based teleportation by allowing information copies to move along various paths within the graph. We perform experiments on a real quantum device to demonstrate such multi-path stability. Our analysis of the entanglement spectrum of certain Ising graphs provides insight into encoding SPT order into higher dimensional quantum states.