Storage efficiency of layer codes

Determine whether any layer-code construction can use fewer physical qubits per logical qubit than rotated surface-code patches at equal distance and logical error rate.

Background

The paper derives an optimistic necessary condition for a layer code to outperform rotated surface-code patches in physical-qubit overhead. The condition depends on the input-code rate and on amplification factors describing how efficiently patch dimensions translate into output distance.

The screened input-code families do not satisfy this condition: the most favorable candidate remains above the storage-overhead threshold, and the authors find no distance-at-least-five comparison at equal logical error rate. These negative results are explicitly limited to the families examined, so the general storage-efficiency question remains unresolved.