Akimasa Miyake

We extend the connection between degenerate entanglement spectra present in symmetry-protected topological orders (SPTO's) of 1D spin chains and their use in measurement-based quantum computation (MQC) to the setting of 2D systems. We find surprisingly that the 2D cluster state, an archetypal resource state for MQC, is in a trivial 2D SPTO phase, and show, by a more fine-grained classification, that it does have nontrivial SPTO, but of the same nature as 1D spin chains. In contrast, we introduce a new ground state which possesses nontrivial SPTO entirely of a 2D nature, and show that it is universal for MQC. By utilizing genuine higher-dimensional SPTO, our results open up a research avenue to directly harness its greater quantum-gate complexity within the so-called Clifford hierarchy for the first time in MQC.