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valbert4 committed Oct 16, 2023
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CSS code formulated using the homological product of two chain complexes (see \ref{topic:CSS-to-homology-correspondence}).
Given a classical code with parity-check matrix \(H\), the homological product yields two classical codes \(C_{X,Z}\) with parity-check matrices
\begin{align}
H_X=\begin{pmatrix}I\ot H\\
H\ot I
\end{pmatrix}\quad\quad\text{and}\quad\quad H_Z=\begin{pmatrix}H\ot I\\
I\ot H
H_X=\begin{pmatrix}I\otimes H\\
H\otimes I
\end{pmatrix}\quad\quad\text{and}\quad\quad H_Z=\begin{pmatrix}H\otimes I\\
I\otimes H
\end{pmatrix}~.
\end{align}
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