  An emerging insight is that ground states of symmetry-protected topological
orders (SPTO's) possess latent computational complexity in terms of their
many-body entanglement. By introducing a fractional symmetry of SPTO, which
requires the invariance under 3-colorable symmetries of a lattice, we prove
that every renormalization fixed-point state of 2D $(\mathbb{Z}_2)^m$ SPTO with
fractional symmetry can be utilized for universal quantum computation using
only Pauli measurements, as long as it belongs to a nontrivial 2D SPTO phase.
Our infinite family of fixed-point states may serve as a base model to
demonstrate the idea of a "quantum computational phase" of matter, whose states
share universal computational complexity ubiquitously.
