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

Description

Actively orthogonal CSS code whose lifts are drawn from a direct product \(H_k \times C_m\), or a semidirect product \(H_k \ltimes C_m^k\), of a small non-Abelian group \(H_k\) and a large Abelian group \(C_m\) [1]. For a direct product, two lifts commute exactly when their \(H_k\) parts commute, so the non-Abelian top factor alone decides the active orthogonality pattern. The Abelian bottom factor is unconstrained by that pattern, and its shifts are code automorphisms. They also make the qubit permutation between syndrome-extraction rounds one rigid row move and one rigid column move, the parallel moves native to crossed acousto-optic deflectors [1].

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 the non-Abelian permutation group \(H_k\) and \(f_i,g_i\) in the Abelian permutation group \(C_m\), such that \([F_i,G_j]=0\) for every \((i,j)\in\Gamma\). The stabilizer generator matrices are the first \(J\) block rows of the parents \(\hat{H}_X=[F|G]\) and \(\hat{H}_Z=[G^T|F^T]\). The Abelian entries are unconstrained by the commutation pattern and can be chosen freely to improve girth and distance [1]. Semidirect product GALA codes replace the direct product with \(H_k \ltimes C_m^k\), in which \(H_k\) permutes the \(k\) blocks and \(C_m\) acts within each block [1]. Polynomial GALA codes replace each single-element lift by a sum of group elements, which decouples the stabilizer generator weight from \(L\), and admit loose active orthogonality [1]. 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].

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

The \([[2232,1120,16]]\) code has stabilizer generator weight \(12\) and an exactly certified distance above that weight at rate above one half [1]. See Ref. [1; Tables S3, S4, and S5] for all certified instances.