\(k\)-orthogonal code[1–3]

Description

Qubit stabilizer code for which the binary space \(S_X\) of \(X\) components of stabilizers is \(k\)-orthogonal in the symplectic representation. In other words, the overlap of any \(j\) vectors in \(S_X\) is even for every \(1\leq j\leq k\) [3; Def. 4]. This definition applies to general qubit stabilizer codes and does not require a CSS presentation. This entry is formulated for qubits, but an extension exists for modular qudits [1].

Equivalently, a generator matrix for \(S_X\) is \(k\)-orthogonal if \begin{align} |x^1|&\equiv 0 \mod 2 \tag*{(1)}\\ |x^1\cdot x^2|&\equiv 0 \mod 2 \tag*{(2)}\\ |x^1\cdot x^2\cdot x^3|&\equiv 0 \mod 2 \tag*{(3)}\\ &\vdots \tag*{(4)}\\ |x^1\cdot x^2\cdot x^3\cdot\ldots\cdot x^k|&\equiv 0 \mod 2 \tag*{(5)}\end{align} for all vectors \(x^j\) in its row space, where the generalized dot-product notation means a sum of products of the respective coordinates of all vectors.