Quantum error correction (QEC) is a key concept in quantum computation as well as many areas of physics. There are fundamental tensions between continuous symmetries and QEC. One vital situation is unfolded by the Eastin–Knill theorem, which forbids the existence of QEC codes that admit transversal continuous symmetry actions (transformations). Here, we systematically study the competition between continuous symmetries and QEC in a quantitative manner. We first define a series of meaningful measures of approximate symmetries motivated from different perspectives, and then establish a series of trade-off bounds between them and QEC accuracy utilizing multiple different methods. Remarkably, the results allow us to derive general quantitative limitations of transversally implementable logical gates, an important topic in fault-tolerant quantum computation. As concrete examples, we showcase two explicit types of quantum codes, obtained from quantum Reed–Muller codes and thermodynamic codes, respectively, that nearly saturate our bounds. Finally, we discuss several potential applications of our results in physics.

量子纠错（QEC）是量子计算以及许多物理学领域的关键概念。连续对称性和 QEC 之间存在着根本性的紧张关系。Eastin-Knill 定理揭示了一种重要情况，该定理禁止存在允许横向连续对称作用（变换）的 QEC 代码。在这里，我们以定量的方式系统地研究连续对称性和 QEC 之间的竞争。我们首先从不同角度定义了一系列有意义的近似对称性度量，然后利用多种不同的方法在它们和 QEC 精度之间建立了一系列权衡界限。值得注意的是，这些结果使我们能够得出横向可实现逻辑门的一般定量限制，这是容错量子计算中的一个重要主题。作为具体的例子，我们展示了两种显式的量子代码，分别从量子里德-穆勒代码和热力学代码获得，它们几乎饱和了我们的界限。最后，我们讨论了我们的结果在物理学中的几个潜在应用。