True Galois-qudit stabilizer code[1,2]

Cousins

• Linear \(q\)-ary code— A true Galois-qudit stabilizer code is the closest quantum analogue of a linear code over \(\mathbb{F}_q\) because the \(q\)-ary vectors corresponding to the Galois symplectic representation of the stabilizers form a linear subspace.
• \([[2m,2m-2,2]]\) error-detecting code— A naive extension of the iceberg code to Galois qudits keeps only two CSS-type generators, \(M_1(1)=X_{\alpha_1}\otimes\cdots\otimes X_{\alpha_n}\) and \(M_2(1)=Z_{\beta_1}\otimes\cdots\otimes Z_{\beta_n}\), with nonzero \(\alpha_i,\beta_i\in\mathbb{F}_q\) satisfying \(\sum_i \alpha_i\beta_i=0\). For prime-power dimensions with \(q=p^m\) and \(m>1\), this yields a Galois-qudit code of distance one that is generally not a true stabilizer code because the stabilizer is not closed under multiplication by arbitrary \(\gamma\in\mathbb{F}_q\). Adding all \(M_1(\gamma)\) and \(M_2(\gamma)\) to the stabilizer group recovers the corresponding true Galois-qudit CSS code of distance two [2; Sec. 8.2.2].