# Number-phase code[1]

## Description

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 \(k \geq 0\).

## Protection

## Decoding

## Fault Tolerance

## Parent

- Bosonic rotation code — Number-phase codes are bosonic rotation codes with the primitive state is a Pegg-Barnett phase state [2].

## Cousins

- Rotor GKP code — Number phase codes are a manifestation of planar-rotor GKP codes in an oscillator. Both codes protect against small shifts in angular degrees of freedom.
- Binomial code — In the limit as \(N,S \to \infty\), phase measurement in the binomial code has vanishing variance, just like in a number-phase code [1].
- Cat code — In the limit as \(N,S \to \infty\), phase measurement in the cat code has vanishing variance, just like in a number-phase code [1].

## Zoo code information

## References

- [1]
- A. L. Grimsmo, J. Combes, and B. Q. Baragiola, “Quantum Computing with Rotation-Symmetric Bosonic Codes”, Physical Review X 10, (2020). DOI; 1901.08071
- [2]
- S. M. Barnett and D. T. Pegg, “Phase in quantum optics”, Journal of Physics A: Mathematical and General 19, 3849 (1986). DOI

## Cite as:

“Number-phase code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/number_phase