Three-block CSS code[1,2]

Description

Galois-qudit CSS code constructed from three pairwise commuting square matrices \(A\), \(B\), and \(C\); the case \(t=3\) of multi-block CSS codes. The four-term chain complex underlying the code provides a complete set of metachecks in one basis, given by explicit closed-form check matrices.

The complex \(\text{mbc}(A,B,C)\) has boundary matrices [1] \begin{align} Q_1=\left[C \,|\, B \;\, A\right],\quad Q_2=\left[ \begin{array}{cc|c} B& A&0\\ \hline -C&0 &A\\ 0& -C& -B \end{array} \right],\quad Q_3=\left[ \begin{array}{c} A\\-B\\ \hline C \end{array} \right]~, \tag*{(1)}\end{align} which satisfy \(Q_1 Q_2 = 0\) and \(Q_2 Q_3 = 0\) whenever the three blocks commute. The level-one code \(\text{CSS}(Q_1,Q_2^T)\) has \(H_X = Q_1\) and \(H_Z = Q_2^T\), with qudits arranged in three blocks. The matrix \(M_Z = Q_3^T\) satisfies \(M_Z H_Z = 0\) and provides a complete set of \(Z\)-type metachecks; the mirror level-two code \(\text{CSS}(Q_2,Q_3^T)\) instead admits \(X\)-type metachecks provided by \(Q_1\).

For qubit codes, minus signs can be dropped, and the level-one code can be written as [2] \begin{align} H_X = \left[A^T\, B^T\, C^T\right],\quad H_Z = \left[ \begin{array}{ccc} C&0&A\\ 0&C&B\\ B&A&0 \end{array} \right],\quad H_{\text{meta}} = \left[B\, A\, C\right]~, \tag*{(2)}\end{align} with metachecks satisfying \(H_{\text{meta}} H_Z = 0\). There is one block of \(X\)-type checks and three blocks of \(Z\)-type checks, so only the \(Z\)-type checks are redundant; the codes can admit single-shot decoding in the \(Z\) basis but not, in general, in the \(X\) basis.