Quantum error correction is rapidly seeing first experimental implementations, but there is a significant gap between asymptotically optimal error-correcting codes and codes that are experimentally feasible. Quantum LDPC codes range from the surface code, which has a vanishing encoding rate, to very promising codes with constant encoding rate and linear distance. In this work, motivated by current small-scale experimental quantum processing units, we devise small quantum codes that are inspired by a subset of quantum LDPC codes, known as generalized bicycle (GB) codes. We introduce a code construction based on algebraic manipulation of the parity-check matrix of GB codes, rather than manipulation of Tanner graphs. Our construction leads to families of quantum LDPC codes of small size, and we demonstrate numerically that their performance scales comparably to the performance of surface codes for similar sizes under a phenomenological noise model. The advantage of our code family is that they encode many logical qubits in one code, at the expense of non-local connectivity. We then explore three variants of the code construction focusing on reducing the long-range connectivity by bringing it closer to the current experimental capabilities of short-range connectivity devices.

翻译：量子纠错正迅速进入首次实验实现阶段，但渐近最优纠错码与实验可行编码之间仍存在显著差距。量子LDPC码涵盖了从编码率为零的表面码到具有恒定编码率与线性距离的极具前景的编码方案。本研究受当前小规模实验量子处理单元启发，基于量子LDPC码子类——广义自行车码（GB码）——设计新型小型量子码。我们提出一种基于GB码奇偶校验矩阵代数操作（而非Tanner图操作）的编码构造方法。该构造方法可生成小尺寸量子LDPC码族，数值模拟表明，在唯象噪声模型下，其性能尺度与同等尺寸表面码相当。该编码族的优势在于单个编码可编码多逻辑量子比特，但需以非局域连通性为代价。在此基础上，我们进一步探索三种变体构造方案，重点通过降低长程连通性使其更接近当前短程连通器件的实验能力。