Dressed stabilizer under a weak local perturbation

Suppose a quasi-adiabatic continuation maps a stabilizer code projector $P_0$ to a perturbed low-energy projector $P=U{P}_0{U}^{†}$. Each original stabilizer generator $S_i$ has a dressed operator ${\tilde{S}}_i=U{S}_i{U}^{†}$. Conjugation preserves mutual commutation and ${\tilde{S}}_{i}P=P$, so the dressed checks characterize the transported subspace.

They generally have spatial tails rather than finite Pauli operator support. They need not commute with the entire perturbed Hamiltonian operator; preserving its selected spectral subspace does not make them global conserved quantities. Likewise, a non-Pauli component alone does not destroy quantum error correction, since correctable error sets have a linear span. For bare local errors, accuracy instead depends on the spread of the corresponding conjugated errors, the original quantum code distance and the Knill--Laflamme condition. Bounds on spatial tails can control approximate local correction under the stated gap and locality assumptions.