  We propose a family of explicit geometrically local circuits realizing any
abelian non-chiral topological phase as an actively error-corrected
fault-tolerant memory. These circuits are constructed from measuring 1-form
symmetries in discrete fixed-point path integrals, which we express through
cellular cohomology and higher-order cup products. The specific path integral
we use is the abelian Dijkgraaf-Witten state sum on a 3-dimensional
cellulation, which is a spacetime representation of the twisted quantum double
model. The resulting circuits are based on a syndrome extraction circuit of the
(qudit) stabilizer toric code, into which we insert non-Clifford phase gates
that implement the ``twist''. The overhead compared to the toric code is
moderate, in contrast to known constructions for twisted abelian phases. We
also show that other architectures for the (qudit) toric code phase, like
measurement-based topological quantum computation or Floquet codes, can be
enriched with phase gates to implement twisted quantum doubles instead of their
untwisted versions. As a further result, we prove fault tolerance under
arbitrary local (including non-Pauli) noise for a very general class of
topological circuits that we call 1-form symmetric fixed-point circuits. This
notion unifies the circuits in this paper as well as the stabilizer toric code,
subsystem toric code, measurement-based topological quantum computation, or the
(CSS) honeycomb Floquet code. We also demonstrate how our method can be adapted
to construct fault-tolerant circuits for specific non-Abelian phases. In the
appendix we present an explicit combinatorial procedure to define formulas for
higher cup products on arbitrary cellulations, which might be interesting in
its own right to the TQFT and topological-phases community.
