Quotient Space Quantum Codes

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.