There is a bijection between odd prime dimensional qudit pure stabilizer states modulo invertible scalars and affine Lagrangian subspaces of finite dimensional symplectic $\mathbb{F}_p$-vector spaces. In the language of the stabilizer formalism, full rank stabilizer tableaux are exactly the bases for affine Lagrangian subspaces. This correspondence extends to an isomorphism of props: the composition of stabilizer circuits corresponds to the relational composition of affine subspaces spanned by the tableaux, the tensor product corresponds to the direct sum. In this paper, we extend this correspondence between stabilizer circuits and tableaux to the mixed setting; regarding stabilizer codes as affine coisotropic subspaces (again only in odd prime qudit dimension/for qubit CSS codes). We show that by splitting the projector for a stabilizer code we recover the error detection protocol and the error correction protocol with affine classical processing power.

翻译：奇素数维度qudit纯稳定器状态（模可逆标量）与有限维辛$\mathbb{F}_p$向量空间的仿射拉格朗日子空间之间存在双射。在稳定器形式化语言中，满秩稳定器表格恰好是仿射拉格朗日子空间的基。这种对应关系推广到pro同构：稳定器电路的复合对应于由表格张成的仿射子空间的关系复合，张量积则对应于直和。在本文中，我们将这种稳定器电路与表格之间的对应关系扩展到混合情形——将稳定器码视为仿射余迷向子空间（同样仅适用于奇素数qudit维度/量子比特CSS码）。我们证明，通过分裂稳定器码的投影算子，可恢复具有仿射经典处理能力的错误检测协议与纠错协议。