  It has long been known that long-ranged entangled topological phases can be
exploited to protect quantum information against unwanted local errors. Indeed,
conditions for intrinsic topological order are reminiscent of criteria for
faithful quantum error correction. At the same time, the promise of using
general topological orders for practical error correction remains largely
unfulfilled to date. In this work, we significantly contribute to establishing
such a connection by showing that Abelian twisted quantum double models can be
used for quantum error correction. By exploiting the group cohomological data
sitting at the heart of these lattice models, we transmute the terms of these
Hamiltonians into full-rank, pairwise commuting operators, defining commuting
stabilizers. The resulting codes are defined by non-Pauli commuting
stabilizers, with local systems that can either be qubits or higher dimensional
quantum systems. Thus, this work establishes a new connection between condensed
matter physics and quantum information theory, and constructs tools to
systematically devise new topological quantum error correcting codes beyond
toric or surface code models.
