Analyzing the Toric Code using High-Temperature Series Expansions*

The decoding of several topological quantum codes (TQC) can be mapped onto statistical physics models (SPM). In this mapping, a successful decoding of the error syndrome of the TQC corresponds to a certain phase of the corresponding SPM. The error-correction performance of several TQC-s have been analyzed using Monte Carlo (MC) simulations.
We, on the other hand, use high-temperature series expansion to analyze the decoding performance of the toric code. In contrast to zero temperature simulations, which estimate the threshold of the minimum-weight perfect-matching decoder, our method naturally provides an estimate of that of the maximum-likelihood decoder. First, we analyze the phase diagram of the 2D random-bond Ising model to a higher order than previously performed. Our results provide an estimate of the decoding threshold of the toric code in absence of measurement imperfections. We compare our result to those obtained by MC simulations and network model analysis. Then, we perform the analysis of the free-energy and the Wilson loop order parameter in the 3D Ising gauge theory in the presence of quenched disorder. The latter model describes the decoding of the toric code subject to measurement errors.