A new study examines how to correct quantum errors in topological codes deformed away from their stabilizer limit, particularly as they approach a quantum phase transition into a trivial phase. The work focuses on quantum information stored in the ground-state manifold of the transverse-field toric code (TFTC), subjected to independent bit-flip and phase-flip decoherence. Although the transverse field induces strong quantum fluctuations of m anyons in the ground states, the authors show that projective measurements of the bare toric code stabilizers can be used to build an efficient decoding algorithm with a finite threshold for both types of decoherence, even when the transversal field is tuned infinitesimally close to the quantum critical point.

The construction of the optimal postmeasurement decoders, that is, the optimal decoders assuming the stabilizers are measured first, crucially relies on the stoquastic nature of the TFTC Hamiltonian, which allows its ground states to be described in terms of a classical lattice gauge theory. Using stoquasticity, the authors first prove that stabilizer measurements act unitarily, to excellent approximation, on the undecohered TFTC ground states, and that the optimal postmeasurement decoder can be efficiently implemented via Monte Carlo simulation of a constrained lattice gauge theory.

From there, they construct the optimal postmeasurement decoder for TFTC states under bit-flip and phase-flip decoherence, numerically demonstrating a finite threshold close to the quantum critical point. The authors provide theoretical evidence, though without a rigorous proof, that stabilizer measurements constitute an optimal decoding strategy in this setting. For phase-flip decoherence, they further prove that the optimal postmeasurement decoding threshold is independent of the transverse field strength throughout the topological phase.