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.

References

This work raises a number of questions we leave for future work. Our results show that all DDTI CSS codes derived from length-$D$ resolutions with ZX-dual resolutions fall into gapped quantum liquid phases of matter. Does our result provide a full classification of all TI gapped quantum liquid codes?

— A Classification of Translation-Invariant Quantum Codes in Any Dimension  (2608.20981 - Li et al., 21 Aug 2026) in Discussion

Can the approach used here be extended to the classification of fracton codes?

— A Classification of Translation-Invariant Quantum Codes in Any Dimension  (2608.20981 - Li et al., 21 Aug 2026) in Discussion

In, we constructed this pairing using an algebraic \v{C}ech complex, but left its nondegeneracy open.

— Poincaré Duality and Quadratic Refinements over Laurent Rings  (2609.25767 - Ruba et al., 22 Sep 2026) in Section 1, subsection “Motivation from physics”