The stabilizer formalism for quantum error-correcting codes has been, without doubt, the most successful at producing examples of quantum codes with strong error-correcting properties. In this paper, we discuss strong automorphism groups of stabilizer codes, beginning with the analogous notion from the theory of classical codes. Two weakenings of this concept, the weak automorphism group and Clifford-twisted automorphism group, are also discussed, along with many examples highlighting the possible relationships between the types of "automorphism groups". In particular, we construct an example of a $[[10,0,4]]$ stabilizer code showing how the Clifford-twisted automorphism groups might be connected to the Mathieu groups. Finally, nonexistence results are proved regarding stabilizer codes with highly transitive strong and weak automorphism groups, suggesting a potential inverse relationship between the error-correcting properties of a quantum code and the transitivity of those automorphism groups.
翻译:毫无疑问,量子误差校正代码的稳定性化形式主义是生成具有强烈错误校正特性的量子代码实例最成功的范例。在本文中,我们讨论稳定器代码的强大自动化组,首先从古典代码理论的类似概念开始。还讨论了这一概念的两个弱点,即薄弱的自制组和扭曲的克利福德自制组,以及许多突出“自制组”类型之间可能存在的关系的例子。特别是,我们构建了一个$[10,0,4]美元稳定器代码的例子,以表明克里夫德自制组如何与马蒂厄组相联系。最后,在稳定器代码方面,与高度中转强和弱的自制组证明不存在结果,这表明量子代码错误校正特性与这些自制组的过渡性之间可能存在反关系。