Quantum error-correcting codes are crucial for quantum computing and communication. Currently, these codes are mainly categorized into additive, non-additive, and surface codes. Additive and non-additive codes utilize one or more invariant subspaces of the stabilizer G to construct quantum codes. Therefore, the selection of these invariant subspaces is a key issue. In this paper, we propose a solution to this problem by introducing quotient space codes and a construction method for quotient space quantum codes. This new framework unifies additive and non-additive quantum codes. We demonstrate the codeword stabilizer codes as a special case within this framework and supplement its error-correction distance. Furthermore, we provide a simple proof of the Singleton bound for this quantum code by establishing the code bound of quotient space codes and discuss the code bounds for pure and impure codes. The quotient space approach offers a concise and clear mathematical form for the study of quantum codes.

翻译：量子纠错码对于量子计算和量子通信至关重要。目前，这些编码主要分为加法型、非加法型和表面码三类。加法型和非加法型编码利用稳定子群G的一个或多个不变子空间来构造量子码，因此这些不变子空间的选择成为关键问题。本文通过引入商空间码及一种商空间量子码的构造方法，提出了对该问题的解决方案。这一新框架统一了加法型与非加法型量子码。我们展示了码字稳定子码作为该框架的一个特例，并补充了其纠错距离。此外，我们通过建立商空间码的码界，为这类量子码提供了Singleton界的简单证明，并讨论了纯码与不纯码的码界。商空间方法为量子码研究提供了简洁清晰的数学形式。