  We study the spectral gap of frustrated spin (qubit) Hamiltonians constructed
from quantum subsystem (gauge) codes. Such a Hamiltonian can be block
diagonalized, with blocks labelled by eigenvalues of extensively many integrals
of motion (stabilizers of the code.) Of particular interest is the 3D gauge
color code, whose Hamiltonian has been conjectured to act as a quantum memory
at finite temperature. Using Perron-Frobenius theory we constrain the location
of first and second eigenvalues among the blocks of the Hamiltonian. This
allows us to numerically find the gap of some large instances of the 3D gauge
color code, which is compared to other frustrated spin Hamiltonians. Finally,
we suggest a relation between bounded stabilizers and gapped spectra.
