  Projected entangled pair states (PEPS) provide a natural ansatz for the
ground states of gapped, local Hamiltonians in which global characteristics of
a quantum state are encoded in properties of local tensors. We develop a
framework to describe on-site symmetries, as occurring in systems exhibiting
symmetry-protected topological (SPT) quantum order, in terms of virtual
symmetries of the local tensors expressed as a set of matrix product operators
(MPOs) labeled by distinct group elements. These MPOs describe the possibly
anomalous symmetry of the edge theory, whose local degrees of freedom are
concretely identified in a PEPS. A classification of SPT phases is obtained by
studying the obstructions to continuously deforming one set of MPOs into
another, recovering the results derived for fixed-point models [X. Chen et al.,
Phys. Rev. B 87, 155114 (2013)]. Our formalism accommodates perturbations away
from fixed point models, opening the possibility of studying phase transitions
between different SPT phases. We also demonstrate that applying the recently
developed quantum state gauging procedure to a SPT PEPS yields a PEPS with
topological order determined by the initial symmetry MPOs. The MPO framework
thus unifies the different approaches to classifying SPT phases, via
fixed-points models, boundary anomalies, or gauging the symmetry, into the
single problem of classifying inequivalent sets of matrix product operator
symmetries that are defined purely in terms of a PEPS.
