Perturbed bivariate bicycle (PBB) code[1]

Description

Qubit stabilizer code defined over the ring \(R=\mathbb{F}_2[x,y]/(x^{\ell}-1,y^{m}-1)\) by four polynomials \(A,B,C,D\in R\) [1]. The polynomials \(C\) and \(D\) add \(Z\)-type support to the \(X\)-type checks of the BB code defined by \(A\) and \(B\). Such codes are not CSS in general.

With each polynomial represented by an \(\ell m\times\ell m\) circulant matrix, the stabilizer matrix in symplectic \((X|Z)\) form is \begin{align} H=\begin{pmatrix} A & B & C & D \\ 0 & 0 & B^{\top} & A^{\top} \end{pmatrix}~, \tag*{(1)}\end{align} where \(A^{\top}\) is the image of \(A\) under \(x\mapsto x^{-1}\) and \(y\mapsto y^{-1}\). All generators commute iff \(AC^{\top}+BD^{\top}\) is symmetric over \(\mathbb{F}_2\) [1]. Some PBB codes are equivalent to CSS codes under single-qubit Hadamard or \(S\) gates [1; Appx. F].