Subsystem homological product code[1] 

A CSS subsystem code constructed from a product of two (subspace) CSS codes. The case for qubits is formulated below, but these codes have also been extended to Galois qudits [1].

Denote the two CSS codes' parity-check matrix blocks as \(H_X^i, H_Z^i\) for \(i \in \{A, B\}\). SP codes can be constructed by the following gauge generating matrices \begin{align} \begin{split} \label{sub:gauge} G_X=\left(\begin{array}{c}H_X^A \otimes I \\ I \otimes H_X^B \end{array}\right) G_Z=\left(\begin{array}{c}H_Z^A \otimes I \\ I \otimes H_Z^B \end{array}\right)~, \end{split} \tag*{(1)}\end{align} where \(I\) is the identity matrix with size chosen to match the dimensions.

A stabilizer generator matrix can be written in terms of the codes' generating matrices, \(L_X^i, L_Z^i\) for \(i \in \{A, B\}\): \begin{align} \begin{split} \label{sub:stabilizer} H_X=\left(\begin{array}{c}H_X^A \otimes H_X^B \\ H_X^A \otimes L_X^B \\ L_X^A \otimes H_X^B \end{array}\right), H_Z=\left(\begin{array}{c}H_Z^A \otimes H_Z^B \\ H_Z^A \otimes L_Z^B \\ L_Z^A \otimes H_Z^B \end{array}\right)~. \end{split} \tag*{(2)}\end{align} The null space of \(G\) excluding \(H\) gives logical generating matrices in canonical pairs \begin{align} \begin{split} L_{X}&=\left(L_{X}^{A}\otimes L_{X}^{B}\right)\\ L_{Z}&=\left(L_{Z}^{A}\otimes L_{Z}^{B}\right)~, \end{split} \tag*{(3)}\end{align} which satisfy \(L_{X}L_{Z}^{T}=I\).