Number-phase code[1]

Bosonic rotation code consisting of superpositions of Pegg-Barnett phase states [2], \begin{align} |\phi\rangle \equiv \frac{1}{\sqrt{2\pi}}\sum_{n=0}^{\infty} \mathrm{e}^{\mathrm{i} n \phi} \ket{n}. \end{align} Since phase states and thus the ideal codewords are not normalizable, approximate versions need to be constructed. The codes' key feature is that, in the ideal case, phase measurement has zero uncertainty, making it a good canditate for a syndrome measurement.

Logical states of an order-\(N\) number-phase qubit encoding are \(|\overline{0}\rangle= \sum_{m=0}^{2N-1} |\phi = m\pi/N\rangle\) and \(|\overline{1}\rangle = \sum_{m=0}^{2N-1} (-1)^m |\phi=m\pi/N\rangle\). By performing the summation over \(m\), one finds that \(|\overline{0}\rangle\) is supported on Fock states \(|2kN\rangle\), while \(|\overline{1}\rangle\) is supported on states \(|(2k+1)N\rangle\), for \(k \geq 0\).