  What kinds of symmetry-protected topologically ordered (SPTO) ground states
can be used for universal measurement-based quantum computation in a similar
fashion to the 2D cluster state? 2D SPTO states are classified not only by
global on-site symmetries but also by subsystem symmetries, which are
fine-grained symmetries dependent on the lattice geometry. Recently, all states
within so-called SPTO cluster phases on the square and hexagonal lattices have
been shown to be universal, based on the presence of subsystem symmetries and
associated structures of quantum cellular automata. Motivated by this
observation, we analyze the computational capability of SPTO cluster phases on
all vertex-translative 2D Archimedean lattices. There are four subsystem
symmetries here called ribbon, cone, fractal, and 1-form symmetries, and the
former three are fundamentally in one-to-one correspondence with three classes
of Clifford quantum cellular automata. We conclude that nine out of the eleven
Archimedean lattices support universal cluster phases protected by one of the
former three symmetries, while the remaining lattices possess 1-form symmetries
and have a different capability related to error correction.
