Extension beyond translation-invariant codes

Establish whether every finite-range quantum low-density parity-check code in D-dimensional space described by an irreducible length-D chain complex is equivalent to copies of a D-dimensional toric code, without assuming translation invariance.

Background

The main theorem assumes D-dimensional translation invariance and a length-D chain complex with suitable exactness properties for the complex and its ZX-dual. The authors explicitly ask whether an analogous toric-code classification can hold for finite-range qLDPC codes in D dimensions when translation invariance is dropped but irreducibility and the length-D condition are retained.