Group-action lift with active orthogonality (GALA) code[1]

Description

Qubit actively orthogonal CSS code whose block-circulant parent matrices are obtained by a \(G\)-lift. In the direct-product monomial construction, a small, generally non-Abelian factor supplies the controlled non-commutativity that makes active orthogonality possible, while a larger Abelian factor supplies code automorphisms and parallelizable syndrome-extraction schedules. Polynomial and special sectors can instead have an Abelian or trivial top factor. Separating the two roles makes the rate, distance, and girth bounds, the automorphisms, and the available dualities designable inputs to a code search rather than properties verified after the fact.

Given \(L\), \(J \leq L/2\), and an active set \(\Gamma\), a direct product GALA code \(\mathrm{GALA}_{L,J}(H_k \times C_m)\) is defined on \(n=Lkm\) qubits by choosing lifts \begin{align} \mathcal{F}=\{(F_i,f_i)\}_{i\in[L/2]}~,\qquad \mathcal{G}=\{(G_i,g_i)\}_{i\in[L/2]}~, \tag*{(1)}\end{align} with \(F_i,G_i\) in a non-Abelian permutation group \(H_k\) and \(f_i,g_i\) in an Abelian permutation group \(C_m\), such that \([F_i,G_j]=0\) for every \((i,j)\in\Gamma\). When the top factor is Abelian or trivial the group ring is commutative and no pair fails to commute, so the parents are fully orthogonal [1]. Commutativity of a pair of lifts depends only on their non-Abelian parts, so the entire orthogonality pattern is decided inside \(H_k\) and the Abelian entries are unconstrained by it and may be chosen freely to improve girth and distance. The stabilizer generator matrices are the first \(J\) block rows of the parents \(\hat{H}_X=[F|G]\) and \(\hat{H}_Z=[G^T|F^T]\).

Several generalizations are given in Ref. [1]. Semidirect product GALA codes replace the direct product with \(H_k \ltimes C_m^k\), polynomial GALA codes replace each single-element lift by a sum of group elements, and loose active orthogonality requires only the aggregate commutator sums to cancel rather than each term individually.

When \(4 \mid L\), requiring \(F_i G_{r(i)} = t\) for a group element \(t\) and one of the four block-index involutions \(i\mapsto i\) and \(i\mapsto i+L/4\) (translations) or \(i\mapsto -i\) and \(i\mapsto L/4-i\) (reflections) yields a ZX duality, a qubit permutation \(\tau\) with \(H_Z = H_X \tau\) [1]. For monomial GALA codes, one reflection-type involution forces girth four whenever \(J\geq 2\), while the same conclusion is proved for the other reflection in the balanced case \(J=L/4\). In that balanced case the latter reflection also forces full parent orthogonality and, subject to the latent-row proviso below, \(d\leq L\) [1]. Within the parameter range covered by these results, only translation-type involutions are compatible with girth at least six.