Modular-qudit shift-resistant code[1]

Description

Monolithic code encoding a qubit into a single modular qudit and protecting against either \(Z\)-type or \(X\)-type modular-qudit Pauli shifts.

The simplest example requires a 6-dimensional qudit. The bit-flip version admits codewords \(|0\rangle\) and \(|3\rangle\) and corrects a single \(X\)-type shift. The phase-flip version admits codewords \begin{align} \begin{split} |\overline{0}\rangle&=\frac{1}{\sqrt{6}}\sum_{j=0}^{5}|j\rangle\\|\overline{1}\rangle&=\frac{1}{\sqrt{6}}\sum_{j=0}^{5}(-1)^{j}|j\rangle\,. \end{split} \tag*{(1)}\end{align} Both codes are modular-qudit CSS codes with stabilizer generators \(Z^2\) and \(X^2\), respectively.