Overlapped-repetition Shor codes achieving fourfold asymptotic rate

TL;DR

High-rate quantum error-correcting codes with manageable stabilizer weights are needed for practical quantum hardware. The paper introduces overlapped-repetition Shor constructions that concatenate overlapped outer and inner repetition codes to achieve a fourfold rate improvement while preserving an average stabilizer weight of $4$, and extends the framework with LDPC outer codes, constant-excitation protection against collective coherent errors, and bosonic oscillator encodings. It demonstrates three integration strategies, analyzes distance preservation, and shows asymptotic rates of $1/[d-le]$ or $1/(d-le)^2$ depending on the construction, with explicit comparisons to surface and BB codes. The results provide a versatile family of high-rate, low-weight QECCs that maintain straightforward syndrome-to-recovery mappings and offer practical pathways toward memory-rich and oscillator-based quantum information processing. Overall, the work offers a concrete route to balance rate, stabilizer weight, and decoding practicality in scalable quantum architectures.

Abstract

Introducing controlled overlap among a few repetition blocks yields a fourfold asymptotic rate improvement while preserving an average stabilizer weight of \(4\). Substituting the overlapped outer layer with an LDPC code further produces a family of constructions with asymptotic rate \(2/d\). We also describe a constant-excitation variant that suppresses collective coherent errors without additional overhead, as well as a bosonic generalization that extends the framework to oscillator encodings. The resulting family achieves a code rate intermediate between that of the rotated surface code (average stabilizer weight \(4\)) and the BB code (average stabilizer weight \(6\)), while remaining free of the performance degradation typical of iterative decoders.