Bosonic quantum Fourier code[1]

Description

Two-mode non-uniform QSC encoding two logical qubits whose projection is onto a copy of an irreducible representation of the single-qubit Pauli group. This code is an extension of the single-logical-qubit code in [2; Eq. (10)], storing an extra logical qubit in the multiplicity space of the Pauli group.

The code admits the following basis of codewords for complex \(\alpha > 0\), up to normalization: \begin{align} \begin{split} |\overline{0,0}\rangle&\propto\left(\left|\alpha\right\rangle -\left|-\alpha\right\rangle \right)\left(\left|i\alpha\right\rangle +\left|-i\alpha\right\rangle \right)\\ |\overline{0,1}\rangle&\propto\left(\left|i\alpha\right\rangle -\left|-i\alpha\right\rangle \right)\left(\left|\alpha\right\rangle +\left|-\alpha\right\rangle \right)\\ |\overline{1,0}\rangle&\propto\left(\left|i\alpha\right\rangle +\left|-i\alpha\right\rangle \right)\left(\left|\alpha\right\rangle -\left|-\alpha\right\rangle \right)\\ |\overline{1,1}\rangle&\propto\left(\left|\alpha\right\rangle +\left|-\alpha\right\rangle \right)\left(\left|i\alpha\right\rangle -\left|-i\alpha\right\rangle \right) \end{split} \tag*{(1)}\end{align}