The Algebra for Stabilizer Codes

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 tableaus are exactly the bases for affine Lagrangian subspaces. This correspondence extends to an isomorphism of props where the composition of stabilizer circuits becomes the relational composition of affine subspaces and the tensor product becomes the direct sum. In this paper, we extend this correspondence between stabilizer circuits and tableaus to the mixed setting; by 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.