Universal measurement-based quantum computation in two-dimensional SPT phases

Recent progress in characterization for gapped quantum phases has also triggered the search of universal resource for quantum computation in symmetric gapped phases. Prior works in one dimension suggest that it is a feature more common than previously thought that nontrivial 1D symmetry-protected topological (SPT) phases provide quantum computational power characterized by the algebraic structure defining these phases. Progress in two and higher dimensions so far has been limited to special fixed points in SPT phases. Here we provide two families of 2D $Z_2$ symmetric wave functions such that there exists a finite region of the parameter in the SPT phases that supports universal quantum computation. The quantum computational power loses its universality at the boundary between the SPT and symmetry-breaking phases.