  Topological quantum computing is believed to be inherently fault-tolerant.
One mathematical justification would be to prove that ground subspaces or
ground state manifolds of topological phases of matter behave as error
correction codes with macroscopic distance. While this is widely assumed and
used as a definition of topological phases of matter in the physics literature,
besides the doubled abelian anyon models in Kitaev's seminal paper, no
non-abelian models are proven to be so mathematically until recently. Cui et al
extended the theorem from doubled abelian anyon models to all Kitaev models
based on any finite group. Those proofs are very explicit using detailed
knowledge of the Hamiltonians, so it seems to be hard to further extend the
proof to cover other models such as the Levin-Wen. We pursue a totally
different approach based on topological quantum field theories (TQFTs), and
prove that a lattice implementation of the disk axiom and annulus axiom in
TQFTs as essentially the equivalence of TQO1 and TQO2 conditions. We confirm
the error correcting properties of ground subspaces for topological lattice
Hamiltonian schemas of the Levin-Wen model and Dijkgraaf-Witten TQFTs by
providing a lattice version of the disk axiom and annulus of the underlying
TQFTs. The error correcting property of ground subspaces is also shared by
gapped fracton models such as the Haah codes. We propose to characterize
topological phases of matter via error correcting properties, and refer to
gapped fracton models as lax-topological.
