Classification of all translation-invariant gapped quantum liquid codes

Determine whether the structure theorem for D-dimensional translation-invariant CSS codes derived from length-D resolutions with ZX-dual resolutions provides a full classification of all translation-invariant gapped quantum liquid codes.

Background

The paper proves that a class of D-dimensional translation-invariant CSS codes arising from length-D free resolutions whose ZX-dual complexes are also resolutions are Hamiltonian-equivalent to copies of D-dimensional toric codes. The authors identify as unresolved whether this result extends beyond that algebraically defined class to every translation-invariant code in a gapped quantum liquid phase.