Efficient Quantum Error Correction from Three Dimensional Qubit Control

Abstract: High-rate quantum low-density parity-check (qLDPC) codes can substantially reduce qubit overhead relative to surface codes, but their advantage depends on efficiently realizing nonlocal syndrome extraction. We study the ([[144,12,12]]) bivariate bicycle code on a neutral-atom architecture with native three-dimensional (3D) geometry, comparing planar and 3D embeddings while holding the code fixed. We characterize spatial efficiency using the logical-qubit density, defined as the number of encoded logical qubits per unit spatial footprint. Because the optical controller's field of view limits the transverse extent of an array, this metric estimates the number of logical qubits that can be accommodated within a fixed optical field of view. The 3D embedding achieves approximately (4\times) greater areal logical-qubit density than the planar layout and (42\times) greater than a surface-code baseline. It also reduces the bivariate bicycle syndrome-extraction time by roughly (2\times) compared to a planar baseline, with fewer movement operations and substantially shorter atom-transport distance. Native 3D geometry can improve both the packing density and executable realization of nonlocal qLDPC codes, making practical performance depend jointly on code structure, optical geometry, transport scheduling, and hardware-level noise. This motivates further development of control techniques in 3D.