Gapped Quantum Matter and Error-Correction with Adiabatic Quantum Channels

The codespace of a quantum error-correcting code can often be identified with the ground-space of a gapped quantum many-body Hamiltonian in which the energy gap, and thus the ground-space, can be stable to a class of perturbations. In this case, the ground-space is representative of a stable phase of quantum matter. We argue that the stability of this quantum phase is directly related to a set of coherent error processes against which this quantum error-correcting code is robust. Specifically, we argue that such a quantum code can recover from adiabatic quantum channels, corresponding to adiabatic drift of code states through the phase, with asymptotically perfect fidelity in the thermodynamic limit. In the case of the one-dimensional repetition code, we construct an explicit decoder against this error which permits the preservation of a logical qubit.