  We construct a tensor network representation of the 3d toric code ground
state that is stable to all local tensor perturbations, including those that do
not map to local operators on the physical Hilbert space. The stability is
established by mapping the phase diagram of the perturbed tensor network to
that of the 3d Ising gauge theory, which has a non-zero finite temperature
transition. More generally, we find that the stability of a topological tensor
network state is determined by the form of its virtual symmetries and the
topological excitations created by virtual operators that break those
symmetries. In particular, a dual representation of the 3d toric code ground
state, as well as representations of the X-cube and cubic code ground states,
for which point-like excitations are created by such operators, are found to be
unstable.
