 Solution

For sufficiently weak local fields the bulk gap remains open, so quasi-adiabatic continuation supplies a quasi-local unitary $U$ mapping the unperturbed ground space to the span of the $2^k$ lowest eigenstates:If $S_j$, $j=1,\dots ,n-k$, are independent original stabilizer generators, defineThen ${\tilde{S}}_j|{\phi }_{\alpha }⟩=|{\phi }_{\alpha }⟩$, and independence is preserved by conjugation. The operators ${\tilde{S}}_j$ are quasilocal, with exponentially decaying tails, but a generic perturbation makes them non-Pauli; this is the dressed stabilizer under a weak local perturbation.

Exact error correction is transported with the code: the exactly correctable dressed errors are $U{E}_a{U}^{†}$. A bare Pauli error is generally not one of these dressed operators. Its expansion in the dressed algebra has exponentially small long-range components that can act within the logical space, soThus a set of bare Pauli errors satisfies the KL conditions only approximately, with deviations suppressed exponentially by the code distance relative to the dressing length, except at specially tuned points.