Kitaev's quantum double model as an error correcting code

Abstract

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.

Quasi-Locality Bounds for Quantum Lattice Systems. Part II. Perturbations of Frustration-Free Spin Models with Gapped Ground States

“…In this case, the indistinguishability radius, which does not depend on x, is essentially the code distance, meaning, the number of bits one has to modify to make an unrecoverable error. The case of general quantum double models was worked out in [32]. iv.…”

a×b=c in 2+1D TQFT

“…In fact, most generally, we might expect automorphisms of the fusion rules that are not even symmetries of the modular data (e.g., as studied recently in[36]). 33 By the results of[21], these theories cannot have such fusions involving lines that carry magnetic flux 34. It would be interesting to know if our results here have any connection with moonshine phenomena observed involving M 24 as in[39][40][41].Accepted in Quantum 2021-06-01, click title to verify.…”

“…The smallest group that has this feature has order 2 7[33]. See[34] for an application of groups that have at least some class-preserving outer automorphisms to quantum doubles.Accepted in Quantum 2021-06-01, click title to verify. Published under CC-BY 4.0.…”

Quantum double aspects of surface code models

“…For completeness we prove this in the Appendix A, mostly following [13], and in the process presenting an orthogonal basis for H vac . This implies, in particular: Corollary 3.3.…”

“…The Kitaev model may be used to perform fault-tolerant quantum computation -indeed, the D(G) model corresponds to a class of quantum error-correcting codes in the sense of [19], according to [13]. If we consider the vacuum to be the logical space of a quantum computer and by following the proof of Theorem 3.2, we observe that the only non-trivial operators in End(H vac ) are non-contractible closed loops on the lattice.…”