Perturbative stability and error correction thresholds of quantum codes

Understand

Topologically-ordered phases are stable to local perturbations, and topological quantum error-correcting codes enjoy thresholds to local errors.

• We connect the two notions of stability by constructing classical statistical mechanics models for decoding general CSS codes and classical linear codes.
• Our construction encodes correction success probabilities under uncorrelated bit-flip and phase-flip errors, and simultaneously describes a generalized Z2 lattice gauge theory with quenched disorder.
• We observe that the clean limit of the latter is precisely the discretized imaginary time path integral of the corresponding quantum code Hamiltonian when the errors are turned into a perturbative X or Z magnetic field.