Actively orthogonal CSS code[1,2]

Description

Qubit CSS code whose stabilizer generator matrices are the first \(J\) block rows, the active rows, of a pair of block-circulant parent matrices \(\hat{H}_X\) and \(\hat{H}_Z\), with the CSS orthogonality condition imposed only among those rows [1]. The discarded latent rows need not commute with the active checks of the opposite type, so neither they nor their low-weight combinations automatically become logical operators. The distance can therefore exceed the stabilizer generator weight at rate at or above one half.

If the parents were orthogonal, every latent row would commute with all active checks, and a latent row outside the stabilizer row space would be a logical operator of weight equal to its row weight. This caps the distance of any row-deleted fully orthogonal construction at the stabilizer generator weight [1]. The parents have \(L/2\) block rows and \(L\) block columns and are assembled by a \(G\)-lift, in which every nonzero entry of a circulant proto-matrix is replaced by a permutation matrix drawn from a group \(G\). Each block row of \(\hat{H}_X\) is a cyclic shift of the lifts \(\mathcal{F}=\{F_i\}\) in its left half and of the lifts \(\mathcal{G}=\{G_i\}\) in its right half. The matrix \(\hat{H}_Z\) holds the transposed lifts with the two halves exchanged and the shift reversed [1]. Because both parents are block circulant, the \((i,j)\) block of \(\hat{H}_X\hat{H}_Z^T\) depends only on the offset \(j-i\) and collects the commutators of all lift pairs whose indices sum to that offset. Requiring \([F_i,G_j]=0\) for every index pair \((i,j)\) in a chosen active set \(\Gamma\) that covers the offsets between active rows makes these blocks vanish for all \(0 \leq i,j < J\), so the retained rows define a CSS code [1]. Commutation is deliberately broken for at least one pair outside \(\Gamma\), which keeps some block between an active and a latent row nonzero.

Kasai codes draw the lifts from the affine permutation group, and GALA codes from a product of a small non-Abelian top factor, which decides the commutation pattern, and an Abelian bottom factor, which supplies code automorphisms [1,2]. When the top factor is Abelian or trivial, no pair fails to commute and the parent matrices are fully orthogonal [2]. Polynomial variants replace a single permutation in each lift entry by a sum of group elements [2]. Loose active orthogonality then requires only the aggregate commutator sum at each active offset to vanish, rather than every contributing pair commuting individually [2]. These variants remain actively orthogonal, but the pairwise condition above and the sharp monomial bounds below no longer apply.