In this paper, we present a detailed mathematical description of the error correction process for Kitaev's model for finite Abelian groups. The number of errors Kitaev's model can correct depends on the lattice and its topology. Although there is a theoretical maximum number of errors that can be corrected, we prove that correcting this number of errors, in general, is an NP-complete problem. Consequently, we introduce a polynomial-time correction algorithm that corrects a number of errors below the theoretical maximum.
Calderbank-Shor-Steane (CSS) codes are a versatile quantum error-correcting family built out of commuting
X- and
Z-type checks. We introduce CSS-like codes on
G-valued qudits for any finite group
G that reduce to qubit CSS codes for
G=Z2 yet generalize the Kitaev quantum double model for general groups. The
X-checks of our group-CSS codes correspond to left and/or right multiplication by group elements, while
Z-checks project onto solutions to group word equations. We describe quantum-double models on oriented two-dimensional CW complexes (which need not cellulate a manifold) and prove that, when
G is non-Abelian and simple, every
G-covariant group-CSS code with suitably upper-bounded
Z-check weight and lower-bounded
Z-distance reduces to a CW quantum double. We describe the codespace and logical operators of CW quantum doubles via the same intuition used to obtain logical structure of surface codes. We obtain distance bounds for codes on non-Abelian simple groups from the graph underlying the CW complex, and construct intrinsically non-Abelian code families with asymptotically optimal rate and distances. Adding "ghost vertices" to the CW complex generalizes quantum double models with defects and rough boundary conditions whose logical structure can be understood without reference to non-Abelian anyons or defects. Several non-invertible symmetry-protected topological states, both with ordinary and higher-form symmetries, are the unique codewords of simply-connected CW quantum doubles with a single ghost vertex.
Kitaev's quantum double models in 2D provide some of the most commonly studied examples of topological quantum order. In particular, the ground space is thought to yield a quantum error-correcting code. We offer an explicit proof that this is the case for arbitrary finite groups. Actually a stronger claim is shown: any two states with zero energy density in some contractible region must have the same reduced state in that region. Alternatively, the local properties of a gauge-invariant state are fully determined by specifying that its holonomies in the region are trivial. We contrast this result with the fact that local properties of gauge-invariant states are not generally determined by specifying all of their non-Abelian fluxes --- that is, the Wilson loops of lattice gauge theory do not form a complete commuting set of observables. We also note that the methods developed by P. Naaijkens (PhD thesis, 2012) under a different context can be adapted to provide another proof of the error correcting property of Kitaev's model. Finally, we compute the topological entanglement entropy in Kitaev's model, and show, contrary to previous claims in the literature, that it does not depend on whether the ``log dim R'' term is included in the definition of entanglement entropy.