Nested HA-MN CSS code [1]

Description

CSS code whose \(X\)-type stabilizer group is \(B\,\mathrm{Ker}A_X\) and whose \(Z\)-type stabilizer group is dual to the HA-LDPC code \(B\,\mathrm{Ker}A_Z\), where \(A_Z\), \(A_\Delta\), and \(B\) are random regular sparse matrices and \(A_X=[A_Z;A_\Delta]\) [1; Thm. 2.2]. Stacking gives \(\mathrm{Ker}A_X\subseteq\mathrm{Ker}A_Z\), so the two groups commute, and the extra block \(A_\Delta\) yields design quantum rate \(m_\Delta/n\), with \(m_\Delta\) the number of rows of \(A_\Delta\) [1; Prop. 2.7].

The extended parity-check matrices with hidden variables, \([A_Z\ 0;\ B\ I_n]\) and \([A_X^{T}\ B^{T}]\), are sparse, but the stabilizer generator matrices on the visible qubits are generally dense [1; Sec. 2.4]. For \((j_Z,k)\)-regular \(A_Z\), \((j_\Delta,k)\)-regular \(A_\Delta\), and \((k,k)\)-regular \(B\), the degree triple is \((j_Z,j_X,k)\) with \(j_X=j_Z+j_\Delta\). The triple is called balanced when \(j_X+j_Z=k\), in which case both classical design rates equal \(j_X/k\) [1; Sec. 2.3].