Perturbative Stability and Error-Correction Thresholds of Quantum Codes

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 Calderbank-Shor-Steane codes and classical linear codes. Our construction encodes correction success probabilities under uncorrelated bit-flip and phase-flip errors, and simultaneously describes a generalized <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><a:msub><a:mrow><a:mi mathvariant="double-struck">Z</a:mi></a:mrow><a:mn>2</a:mn></a:msub></a:math> lattice-gauge theory with quenched disorder. We observe that the clean limit of the latter is precisely the imaginary-time path integral of the corresponding quantum code Hamiltonian when the errors are turned into a perturbative <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><e:mi>X</e:mi></e:math> or <h:math xmlns:h="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><h:mi>Z</h:mi></h:math> magnetic field. Motivated by error-correction considerations, we define general order parameters for all such generalized <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><k:msub><k:mrow><k:mi mathvariant="double-struck">Z</k:mi></k:mrow><k:mn>2</k:mn></k:msub></k:math> lattice-gauge theories, and show that they are generally lower bounded by success probabilities of error correction. For CSS codes satisfying the low-density parity-check condition and with a sufficiently large code distance, we prove the existence of a low-temperature ordered phase of the corresponding lattice-gauge theories, particularly for those lacking Euclidean spatial locality and/or when there is a nonzero code rate. We further argue that these results provide evidence for stable phases in the corresponding perturbed quantum Hamiltonians, obtained in the limit of continuous imaginary time. To do so, we distinguish space- and timelike defects in the lattice-gauge theory. A high free-energy cost of spacelike defects corresponds to a successful “memory experiment” and suppresses the energy splitting among the ground states, while a high free-energy cost of timelike defects corresponds to a successful “stability experiment” and points to a nonzero gap to local excitations.