Single-block CSS codes realizing the full logical Clifford group

Construct single-block CSS codes that transversally realize the full logical Clifford group with shorter physical blocklengths than the inefficient construction based on combining \(\ell\) independent code blocks.

Background

The paper notes that its construction of CSS codes realizing the full logical Clifford group combines multiple independent code blocks, which is physically inefficient. It identifies the construction of a single-block alternative with shorter physical blocklength as an important unresolved direction.

The problem concerns reducing the blocklength overhead while retaining transversal implementations of the full logical Clifford group; it is separate from the further question of optimizing the asymptotic rate and minimum distance.