The Pauli stabilizer formalism is perhaps the most thoroughly studied means of procuring quantum error-correcting codes, whereby the code is obtained through commutative Pauli operators and ``stabilized'' by them. In this work we will show that every quantum error-correcting code, including Pauli stabilizer codes and subsystem codes, has a similar structure, in that the code can be stabilized by commutative ``Paulian'' operators which share many features with Pauli operators and which form a \textbf{Paulian stabilizer group}. By facilitating a controlled gate we can measure these Paulian operators to acquire the error syndrome. Examples concerning codeword stabilized codes and bosonic codes will be presented; specifically, one of the examples has been demonstrated experimentally and the observable for detecting the error turns out to be Paulian, thereby showing the potential utility of this approach. This work provides a possible approach to implement error-correcting codes and to find new codes.

翻译：泡利稳定子形式化方法或许是研究最透彻的量子纠错码构造手段，其通过交换性泡利算子获得码字并使其“稳定”。本文将证明每个量子纠错码（包含泡利稳定子码和子系统码）均具有类似结构：码字可由交换性的“类泡利”算子稳定，这些算子与泡利算子共享众多特性，并构成\textbf{类泡利稳定子群}。通过构建受控门，可测量这些类泡利算子以获取错误症状。文中将给出关于码字稳定子码和玻色子码的实例；特别地，其中一个实验已验证的示例中，用于检测错误的可观测量即为类泡利算子，从而展示了该方法的潜在实用性。本研究为纠错码的实现与新码的发现提供了可行途径。