Actively orthogonal CSS code[1,2]

Description

Qubit CSS code whose stabilizer generators are an active subset of the rows of a larger pair of block-circulant parent matrices, so that the CSS orthogonality condition is imposed only on the retained rows. The construction is designed so that low-weight combinations of the discarded latent rows fail to commute with the active checks, and therefore do not automatically become logical operators as they would were orthogonality enforced on every row. Evading this mechanism circumvents a barrier that otherwise caps the distance of a row-deleted low-density construction at the stabilizer generator weight, allowing sparse, regular, high-girth checks to be combined with encoding rate at or above one half.

In the strict monomial construction, two block-circulant parent matrices \(\hat{H}_X\) and \(\hat{H}_Z\), with \(L/2\) block rows and \(L\) block columns, are assembled by a \(G\)-lift in which every nonzero proto-matrix entry is replaced by a permutation matrix drawn from a group \(G\). Writing \(\mathcal{F}=\{F_i\}\) and \(\mathcal{G}=\{G_i\}\) for the two lists of lifts and \(\Gamma\) for a chosen active set of index pairs, the lifts are required to commute, \([F_i,G_j]=0\), for every \((i,j)\in\Gamma\), while commutation is deliberately broken for at least one pair outside \(\Gamma\). The lift group is generally a product of two factors with distinct roles. The top factor is the one whose non-commutativity creates the broken commutators that active orthogonality relies on, while the remaining bottom factor is Abelian and supplies code automorphisms. In the sectors whose top factor is Abelian or trivial no pair need fail to commute at all, and the parent matrices are then fully orthogonal [2]. The stabilizer generator matrices \(H_X\) and \(H_Z\) are then the first \(J\) block rows of the parents. Because the block-circulant structure makes the \((i,j)\) block of \(\hat{H}_X\hat{H}_Z^T\) depend only on the offset \(j-i\), collecting the commutators of all lift pairs whose indices sum to that offset, a suitable choice of \(\Gamma\) makes these blocks vanish for all \(0 \leq i,j < J\), so the retained rows define a CSS code [1].

Polynomial variants replace a single permutation in each lift entry by a sum of group elements, while loose active orthogonality requires only the aggregate commutator sum at each active offset to vanish, rather than requiring every contributing pair to commute individually [2]. These variants remain actively orthogonal, but the pairwise condition above and the sharp monomial bounds below no longer apply.