Pair-partition CPM code[1,2]

Description

Qubit CSS code whose \(X\)- and \(Z\)-type stabilizer generator matrices are \(J\)-by-\(L\) arrays of \(P\)-by-\(P\) circulant permutation matrices (CPMs), with the CSS condition enforced by pairing up block columns [1,2]. Each block of \(H_XH_Z^T\) is a sum of \(L\) CPMs, one per block column. A pair partition, one per block, splits the block columns into \(L/2\) pairs, and the CPM exponents are constrained so that the two CPMs in each pair coincide and cancel over \(\mathbb{F}_2\). CSS orthogonality thereby becomes a homogeneous linear condition on the exponents [1].

Fix a column weight \(J\), an even row weight \(L>J\), and a lift size \(P\). Let \(C(s)\) be the \(P\)-by-\(P\) CPM with shift \(s\in\mathbb{Z}_P\), and let \((e_{i\ell})\) and \((d_{j\ell})\) be \(J\)-by-\(L\) exponent arrays over \(\mathbb{Z}_P\). The generator matrices \(H_X=(C(e_{i\ell}))\) and \(H_Z=(C(d_{j\ell}))\) are \(G\)-lifts of the complete \(J\)-by-\(L\) protograph over the cyclic group of order \(P\), with length \(n=LP\), row weight \(L\), and column weight \(J\). A \(J\)-by-\(J\) array \((M_{ij})\) of pair partitions, one for each \(X\)-type block row \(i\) and \(Z\)-type block row \(j\), is fixed before the exponents. For each pair \(\{u,v\}\in M_{ij}\), the mixed-difference equation \(e_{iu}-d_{ju}=e_{iv}-d_{jv}\) is imposed. It makes the contributions of block columns \(u\) and \(v\) to block \((i,j)\) of \(H_XH_Z^T\) identical. The exponents are solutions of the resulting homogeneous system over \(\mathbb{Z}_P\) [1]. The rate is at least the design rate \(1-2J/L\) [2].

Suppose the exponent arrays satisfy \(d_{j\ell}=\eta\,e_{\rho(j)\sigma(\ell)}+\alpha_j+\beta_\ell\) for a block-row permutation \(\rho\), a fixed-point-free involution \(\sigma\) of the block columns, \(\eta^2\equiv 1 \pmod P\), and offsets with \(\beta_{\sigma(\ell)}=-\eta\beta_\ell\). Then the qubit involution \((\ell,t)\mapsto(\sigma(\ell),\eta(t+\beta_\ell))\) exchanges the two check spaces, and the code can be symplectically halved into a non-CSS stabilizer code on \(LP/2\) qubits [2]. Halved codes reach a given distance at shorter length than pair-partition CPM codes with the same \((J,L)\) [2; Sec. IV.B]. Examples include the \([[90,21,11]]\), \([[140,31,15]]\), and \([[200,43,20]]\) codes with check weight ten [2; Table II].

For odd \(P\), suppose both check matrices have rank \(J\) over each nontrivial field factor of \(R=\mathbb{F}_2[\mathbb{Z}_P]\). Then each logical space is the \(R\)-module \(R^{L-2J}\oplus\mathbf{1}^{2J-2}\), i.e., \(L-2J\) regular orbits of the cyclic shift plus \(2J-2\) shift-invariant logical classes [2].

Reported instances include the \((3,10)\)-regular \([[2230,896,24]]\) code with \(P=223\) and Tanner girth eight [1; Table I]. See Refs. [1; Table I][2; Table VIII] for lists of instances.