Designer Codes from GALA: Compact, Self-Dual, and Rate-1/2 QEC on Reconfigurable Atom Arrays

Abstract

High rate quantum low-density parity-check codes on reconfigurable neutral-atom arrays can reduce the overhead of quantum error correction, but near-term devices support only hundreds of qubits with limited reconfigurability from a few crossed acousto-optic deflectors (AOD). A practical code must be compact in addition to low-overhead, with checks and logical gates mapping onto hardware-compatible physical instructions. We introduce the GALA codes, or Group-Action Lifts with Active orthogonality, that lifts over a product group $G = H_k \times C_m$ (or $H_k \ltimes C_m^k$). The small non-abelian factor $H_k$ supplies active orthogonality, reaching $1/2$ rate with above-weight distance, while the large abelian factor $C_m$ supplies symmetries that give code automorphisms and explicit, simple AOD move schedules. Hardware compatibility and logical capability thereby become customizable inputs to a code search rather than properties verified post-hoc, making GALA designer codes by construction. The GALA family contains several previously discovered rate-1/2 Kasai codes of Ref. [arXiv:2601.08824, arXiv:2604.16209] while exposing simpler parameter bounds, logical operations, and ZX-dual variants with AOD-compatible fold-transversal Clifford gates. Our search yields a compact self-dual $[[132, 30, 12]]$ with a small number of $4$-cycles (almost girth-6) and below $10^{-8}$ logical error rate (LER) for memory at $10^{-3}$ physical error rate, 3.1ms syndrome-extraction cycle and transversal Clifford gates; a girth-6, rate-$1/2$ $[[672, 336, 12]]$ with a 6.76ms cycle with below $10^{-10}$ LER (extrapolated) and rate-$1/2$ barrier-breaking $[[1752, 880, 14]]$ and $[[2232, 1120, 16]]$ with exactly certified distances greater than check weights and all smaller than previously known hardware compatible rate-1/2 codes.