  We present simple lattice realizations of symmetry-protected topological
(SPT) phases with $q$-form global symmetries where charged excitations have $q$
spatial dimensions. Specifically, we construct $d$ space-dimensional models
supported on a $(d+1)$-colorable graph by using a family of unitary phase
gates, known as multi-qubit control-$Z$ gates in quantum information community.
In our construction, charged excitations of different dimensionality may
coexist and form a short-range entangled state which is protected by symmetry
operators of different dimensionality. Non-triviality of proposed models, in a
sense of quantum circuit complexity, is confirmed by studying protected
boundary modes, gauged models and corresponding gapped domain walls. We also
comment on applications of our construction to quantum error-correcting codes,
and discuss corresponding fault-tolerant logical gates.
