Quantum error correction and symmetry arise in many areas of physics, including many-body systems, metrology in the presence of noise, fault-tolerant computation, and holographic quantum gravity. Here we study the compatibility of these two important principles. If a logical quantum system is encoded into $n$ physical subsystems, we say that the code is covariant with respect to a symmetry group $G$ if a $G$ transformation on the logical system can be realized by performing transformations on the individual subsystems. For a $G$-covariant code with $G$ a continuous group, we derive a lower bound on the error correction infidelity following erasure of a subsystem. This bound approaches zero when the number of subsystems $n$ or the dimension $d$ of each subsystem is large. We exhibit codes achieving approximately the same scaling of infidelity with $n$ or $d$ as the lower bound. Leveraging tools from representation theory, we prove an approximate version of the Eastin-Knill theorem for quantum computation: If a code admits a universal set of transversal gates and corrects erasure with fixed accuracy, then, for each logical qubit, we need a number of physical qubits per subsystem that is inversely proportional to the error parameter. We construct codes covariant with respect to the full logical unitary group, achieving good accuracy for large $d$ (using random codes) or $n$ (using codes based on $W$-states). We systematically construct codes covariant with respect to general groups, obtaining natural generalizations of qubit codes to, for instance, oscillators and rotors. In the context of the AdS/CFT correspondence, our approach provides insight into how time evolution in the bulk corresponds to time evolution on the boundary without violating the Eastin-Knill theorem, and our five-rotor code can be stacked to form a covariant holographic code.

中文翻译：

---

连续对称性和近似量子误差校正

量子误差校正和对称性出现在物理学的许多领域，包括多体系统，存在噪声的计量学，容错计算和全息量子引力。在这里，我们研究这两个重要原理的兼容性。如果将逻辑量子系统编码为$ñ$ 物理子系统，我们说代码相对于对称组是协变的 $G$ 如果一个 $G$逻辑系统上的转换可以通过对各个子系统执行转换来实现。为一个$G$-协变代码 $G$一个连续的组，我们在擦除子系统后得出纠错不忠的下限。当子系统的数量时，此界限接近零$ñ$ 或尺寸 $d$每个子系统的数量很大。我们展示的代码可以实现与$ñ$ 要么 $d$作为下限。利用表示理论的工具，我们证明了用于量子计算的Eastin-Knill定理的近似形式：如果代码允许通用的横向门集合并以固定的精度校正擦除，那么对于每个逻辑量子位，我们需要多个每个子系统的物理量子位与错误参数成反比。我们构造相对于完整逻辑unit组协变的代码，对于大型$d$ （使用随机代码）或 $ñ$ （使用基于 $\mathrm{w\; ^}$-状态）。我们系统地构造了关于一般群协变的代码，从而获得了对例如振荡器和转子的量子位代码的自然概括。在AdS / CFT对应关系的背景下，我们的方法可以洞悉主体中的时间演化如何与边界上的时间演化相对应，而又不违反Eastin-Knill定理，并且我们的五转子代码可以堆叠以形成协变全息图码。