*Citation:* Bravyi, Sergey, Matthew B. Hastings, and Spyridon Michalakis. "Topological quantum order: stability under local perturbations." Journal of mathematical physics 51.9 (2010).

*Web:* https://arxiv.org/abs/1001.0344

*Tags:* Error-correcting-codes, Foundational, Toric-code

This is the first paper to prove general stability results about (2+1)-dimensional topological phases of matter. That is, showing that certain Hamiltonian implementations of certain phases remain gapped under local perturbations. Before this there had been some partial results about the spin-1 Heisenberg chain near a critical point,

> Yarotsky, D. A. "Ground states in relatively bounded quantum perturbations of classical lattice systems." Communications in mathematical physics 261.3 (2006): 799-819.

and some partial results about the toric code

> Trebst, Simon, et al. "Breakdown of a topological phase: Quantum phase transition in a loop gas model with tension." Physical review letters 98.7 (2007): 070602.

> Klich, Israel. "On the stability of topological phases on a lattice." Annals of Physics 325.10 (2010): 2120-2131.

This new paper does fantastically more. Not only does it show that the standard toric code Hamiltonian is stable under generic small perturbations, but the argument also applies to all local commuting projector models which satisfy certain topological order conditions. These topological order conditions on their own are a very interesting feature of the paper, and suggest that this could be the route towards defining what a topological phase should be. The two axioms are as follows, re-worded to my liking:

- TQO-1: The ground subspace is an error correcting code whose code distance scales as some power of the system size
- TQO-2: Local ground subspaces are consistent under reduction.

The axiom TQO-1 doesn't need much of an introduction. The axiom TQO-2 is a bit more unexpected. It says that, given any contractible regions $A\subeteq B$, the ground states of the stabilizers in $A$ should be the same as the ground states of the stabilizers in $B$ when restricted to a region sufficiently far from the boundaries of $A$ and $B$. This axiom can be viewed as saying that the topological protection can't be stored more in some spots that others - the stabilizers around every lattice site have to all be the same in some sense.

A fantastic feature of this article is the inclusion of a counterexample of phase stability when only TQO-1 is assumed. This example is very useful to keep in mind, since it can be used as a counterexample to many other conjectures as well.

Both of the TQO axioms are satisfied by the standard Kitaev quantum double model as well as the Levin-Wen model.