  We construct a Pauli stabilizer model for every two-dimensional Abelian
topological order that admits a gapped boundary. Our primary example is a Pauli
stabilizer model on four-dimensional qudits that belongs to the double semion
(DS) phase of matter. The DS stabilizer Hamiltonian is constructed by
condensing an emergent boson in a $\mathbb{Z}_4$ toric code, where the
condensation is implemented by making certain two-body measurements. We
rigorously verify the topological order of the DS stabilizer model by
identifying an explicit finite-depth quantum circuit (with ancillary qubits)
that maps its ground state subspace to that of a DS string-net model. We show
that the construction of the DS stabilizer Hamiltonian generalizes to all
twisted quantum doubles (TQDs) with Abelian anyons. This yields a Pauli
stabilizer code on composite-dimensional qudits for each such TQD, implying
that the classification of topological Pauli stabilizer codes extends well
beyond stacks of toric codes - in fact, exhausting all Abelian anyon theories
that admit a gapped boundary. We also demonstrate that symmetry-protected
topological phases of matter characterized by type I and type II cocycles can
be modeled by Pauli stabilizer Hamiltonians by gauging certain 1-form
symmetries of the TQD stabilizer models.
