Locality Bounds on Hamiltonians for Stabilizer Codes

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