Classical locally recoverable codes, which permit highly efficient recovery from localized errors as well as global recovery from larger errors, provide some of the most useful codes for distributed data storage in practice. In this paper, we initiate the study of quantum locally recoverable codes (qLRCs). In the long term, like their classical counterparts, such qLRCs may be used for large-scale quantum data storage. Our results also have concrete implications for quantum LDPC codes, which are applicable to near-term quantum error-correction. After defining quantum local recoverability, we provide an explicit construction of qLRCs based on the classical LRCs of Tamo and Barg (2014), which we show have (1) a close-to-optimal rate-distance tradeoff (i.e. near the Singleton bound), (2) an efficient decoder, and (3) permit good spatial locality in a physical implementation. Although the analysis is significantly more involved than in the classical case, we obtain close-to-optimal parameters by introducing a "folded" version of our quantum Tamo-Barg (qTB) codes, which we then analyze using a combination of algebraic techniques. We furthermore present and analyze two additional constructions using more basic techniques, namely random qLRCs, and qLRCs from AEL distance amplification. Each of these constructions has some advantages, but neither achieves all 3 properties of our folded qTB codes described above. We complement these constructions with Singleton-like bounds that show our qLRC constructions achieve close-to-optimal parameters. We also apply these results to obtain Singleton-like bounds for qLDPC codes, which to the best of our knowledge are novel. We then show that even the weakest form of a stronger locality property called local correctability, which permits more robust local recovery and is achieved by certain classical codes, is impossible quantumly.

翻译：经典局域可恢复码不仅能够高效地从局部错误中恢复数据，还可在大范围错误时实现全局恢复，是目前分布式数据存储实践中最重要的编码类型之一。本文首次系统研究量子局域可恢复码（qLRCs）。长期来看，与经典对应物类似，此类qLRCs有望用于大规模量子数据存储。我们的研究结果对近期量子纠错可应用的量子LDPC码也具有具体意义。在定义量子局域可恢复性后，我们基于Tamo与Barg（2014）的经典LRCs提出了一种显式qLRCs构造方法，该构造具有：（1）接近最优的速率-距离权衡（即接近Singleton界）；（2）高效解码器；（3）在物理实现中具有良好的空间局域性。尽管分析过程比经典情形复杂得多，我们通过引入量子Tamo-Barg（qTB）码的"折叠"版本，并结合代数技巧进行分析，获得了接近最优的参数。此外，我们利用更基础的方法提出了另外两种构造方案并进行了分析，即随机qLRCs和基于AEL距离放大的qLRCs。每种构造各有优势，但均无法同时实现上述折叠qTB码的全部三个特性。我们通过Singleton类界限补充了这些构造，证明所提出的qLRCs构造达到了接近最优的参数。这些结果还被应用于获得qLDPC码的Singleton类界限——据我们所知，这是全新的结论。最后我们证明，即使是最弱形式的更强局域性性质——局域可纠正性（经典码可实现更强健的局域恢复），在量子情形下也是不可能的。