# Topological quantum error correction in the Kitaev honeycomb model

## TL;DR

This paper analyzes the topological quantum error correction capabilities of the Kitaev honeycomb model, providing rigorous bounds and numerical insights, and discovering a potential lifetime scaling advantage over the toric code in certain regimes.

## Contribution

It offers the first explicit exponential bounds on degeneracy, indistinguishability, and correctability for the honeycomb model, and explores its error correction performance near the toric code fixed point.

## Key findings

- Exponential bounds on code properties are tighter than general topological phase bounds.
- Thermal noise has minimal impact on error correction in studied regimes.
- Low-temperature, finite-size effects may improve lifetime scaling compared to the toric code.

## Abstract

The Kitaev honeycomb model is an approximate topological quantum error correcting code in the same phase as the toric code, but requiring only a 2-body Hamiltonian. As a frustrated spin model, it is well outside the commuting models of topological quantum codes that are typically studied, but its exact solubility makes it more amenable to analysis of effects arising in this noncommutative setting than a generic topologically ordered Hamiltonian. Here we study quantum error correction in the honeycomb model using both analytic and numerical techniques. We first prove explicit exponential bounds on the approximate degeneracy, local indistinguishability, and correctability of the code space. These bounds are tighter than can be achieved using known general properties of topological phases. Our proofs are specialized to the honeycomb model, but some of the methods may nonetheless be of broader interest. Following this, we numerically study noise caused by thermalization processes in the perturbative regime close to the toric code renormalization group fixed point. The appearance of non-topological excitations in this setting has no significant effect on the error correction properties of the honeycomb model in the regimes we study. Although the behavior of this model is found to be qualitatively similar to that of the standard toric code in most regimes, we find numerical evidence of an interesting effect in the low-temperature, finite-size regime where a preferred lattice direction emerges and anyon diffusion is geometrically constrained. We expect this effect to yield an improvement in the scaling of the lifetime with system size as compared to the standard toric code.