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We present a family of non-CSS quantum error-correcting code consisting ofgeometrically local stabilizer generators on a 3D lattice. We study theHamiltonian constructed from ferromagnetic interaction of overcomplete set oflocal stabilizer generators. The degenerate ground state of the system ischaracterized by a quantum error-correcting code whose number of encoded qubitsare equal to the second Betti number of the manifold. These models (i) havesolely local interactions; (ii) admit a strong-weak duality relation with anIsing model on a dual lattice; (iii) have topological order in the groundstate, some of which survive at finite temperature; and (iv) behave asclassical memory at finite temperature.

Local non-CSS quantum error correcting code on a three-dimensional lattice