Locality bounds on hamiltonians for stabilizer codes

In this paper, we study the complexity of Hamiltonians whose groundstate is a stabilizer code. Weintroduce various notions of k-locality of a stabilizer code, inherited from the associated stabilizergroup. A choice of generators leads to a Hamiltonian with the code in its groundspace. We establishbounds on the locality of any other Hamiltonian whose groundspace contains such a code, whetheror not its Pauli tensor summands commute. Our results provide insight into the cost of creating anenergy gap for passive error correction and for adiabatic quantum computing. The results simplify inthe cases of XZ-split codes such as Calderbank-Shor-Steane stabilizer codes and topologically-orderedstabilizer codes arising from surface cellulations.