Fig. 1: Schematics of three types of generalized NP codes in phase space defined by canonical number and phase variables.
From: Generalized number-phase lattice encoding of a bosonic mode for quantum error correction
a–c show the generalized NP codes with rectangle, oblique, and diamond lattice coding. Red and blue circles represent probability peaks of NP Wigner function of dual logical states \({\left\vert+\right\rangle }_{L}\) and \({\left\vert -\right\rangle }_{L}\), respectively. \(\hat{{{{\mathcal{D}}}}}({{{{\bf{n}}}}}_{x})\) and \(\hat{{{{\mathcal{D}}}}}({{{{\bf{n}}}}}_{z})\) are the logical \(\bar{X}\) and \(\bar{Z}\) operations, which are represented as deep green and orange arrows, respectively. \(\hat{{{{\mathcal{D}}}}}({{{{\bf{n}}}}}_{e})\) is the NP-shift error, represented by light green and light blue arrows. (d, e) are schematic diagrams of the generalized NP codes in NP space, which depict the codes rolling along the phase direction to form a cylindrical surface.
