Qubit resource states for measurement-based quantum teleportation beyond the Haldane phase

All currently known resource states for measurement-based quantum computation possess symmetry protected topological order (STPO), known as Haldane-phase states, but it is not clear if this is a necessary or sufficient condition. This work investigates the ability of one-dimensional qubit states to perform universal and deterministic measurement-based quantum gate teleportation (MBQT) in correlation space, within the framework of matrix product states with bond dimension four, without making any assumptions about SPTO. Several families of resource states are identified and expressed analytically that permit MBQT of both single-qubit and two-qubit gates. We find examples of states without SPTO that are able to perform MBQT, and examples of Haldane-phase states that are unable to do so, even via infinitesimal gates. The results reveal a complex relationship between SPTO and MBQT for qubits.