Covariant codes are quantum codes such that a symmetry transformation on the logical system could be realized by a symmetry transformation on the physical system, usually with limited capability of performing quantum error correction (an important case being the Eastin–Knill theorem). The need for understanding the limits of covariant quantum error correction arises in various realms of physics including fault-tolerant quantum computation, condensed matter physics and quantum gravity. Here, we explore covariant quantum error correction with respect to continuous symmetries from the perspectives of quantum metrology and quantum resource theory, establishing solid connections between these formerly disparate fields. We prove new and powerful lower bounds on the infidelity of covariant quantum error correction, which not only extend the scope of previous no-go results but also provide a substantial improvement over existing bounds. Explicit lower bounds are derived for both erasure and depolarizing noises. We also present a type of covariant codes which nearly saturates these lower bounds.

中文翻译：

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协变量子纠错的新观点

协变码是量子码，逻辑系统上的对称变换可以通过物理系统上的对称变换来实现，通常具有有限的量子纠错能力（一个重要的例子是Eastin-Knill定理）。理解协变量子纠错极限的需求出现在物理学的各个领域，包括容错量子计算、凝聚态物理学和量子引力。在这里，我们从量子计量学和量子资源理论的角度探索关于连续对称性的协变量子误差校正，在这些以前不同的领域之间建立牢固的联系。我们证明了协变量子纠错不忠的新的和强大的下界，这不仅扩展了先前不通过结果的范围，而且还提供了对现有界限的实质性改进。为擦除和去极化噪声导出了明确的下限。我们还提出了一种几乎使这些下界饱和的协变代码。