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 \(L_{X}L_{Z}^{T}=I\).


If the CSS codes have parameters \([[n_i,k_i,d_{i},d_{i}]]\) and sparsity \(\{r_i,c_i\}\), for \(i=A,B\) respectively, the SP code have parameters \([[n_an_b,k_ak_b,d\leq d_ad_b]]\) and sparsity \(\{\max (r_a,r_b), c_a+c_b\}\). Note the distance relation holds for both \(X\) and \(Z\), hence we omit the \(X/Z\) subscript.





W. Zeng and L. P. Pryadko, “Minimal distances for certain quantum product codes and tensor products of chain complexes”, Physical Review A 102, (2020) arXiv:2007.12152 DOI
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Zoo Code ID: subsystem_product

Cite as:
“Subsystem homological product code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024.
@incollection{eczoo_subsystem_product, title={Subsystem homological product code}, booktitle={The Error Correction Zoo}, year={2024}, editor={Albert, Victor V. and Faist, Philippe}, url={} }
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“Subsystem homological product code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024.