Two-branch coset CSS code[1]

Description

Qubit CSS code whose base stabilizer generator matrices place the support of each check on translated cosets of a multiplicative subgroup of a finite field, distributed over two column classes called branches that each carry their own coefficient arrays. The construction also provides a cyclic circulant-permutation-matrix lifting stage that can randomize the edge connections while preserving the sparse CSS constraints. Expressing the incidences through coset arithmetic reduces both the CSS commutation condition and the absence of same-type four-cycles to sufficient equalities and disjointness conditions on multiplicative cosets, turning the search for a regular base pair into a finite check on field coefficients.

Let \(M\) be a multiplicative subgroup of the multiplicative group of a finite field. Columns are indexed by a branch label, a field translation, and a subgroup element, and the \(X\)- and \(Z\)-type rows are incident through coefficient arrays, giving column weight \(J\) and row weight \(L=2|M|\). CSS commutation follows from an equality of cosets of differences of the two coefficient arrays across the branches, and freedom from same-type four-cycles from disjointness of the corresponding difference cosets. A base pair can then be given a \(G\)-lift over a cyclic group, realized by circulant permutation matrices whose shifts are chosen to raise the girth and to remove targeted low-weight logical operators. The worked finite-length construction applies this stage to one \((3,10)\)-regular base [1].