# 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}. \tag*{(1)}\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.
- Bosonic stabilizer code — Number-phase codewords span the joint right eigenspace of the \(N\)th power of the Susskind–Glogower phase operator and the bosonic rotation operator [1]. These operators no longer form a group since the phase operator is not unitary.
- Square-lattice GKP code — Square-lattice GKP codes utilize translational symmetry in phase space, while number-phase codes utilize rotational symmetry. The two are related via a mapping [9].
- 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].
- Homological number-phase code — Homological number-phase codes and number-phase codes are both manifestations of certain rotor codes, namely, the homological rotor codes and rotor GKP codes, respectively.
- 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].

## References

- [1]
- A. L. Grimsmo, J. Combes, and B. Q. Baragiola, “Quantum Computing with Rotation-Symmetric Bosonic Codes”, Physical Review X 10, (2020) arXiv:1901.08071 DOI
- [2]
- S. M. Barnett and D. T. Pegg, “Phase in quantum optics”, Journal of Physics A: Mathematical and General 19, 3849 (1986) DOI
- [3]
- C. M. Dawson, H. L. Haselgrove, and M. A. Nielsen, “Noise thresholds for optical cluster-state quantum computation”, Physical Review A 73, (2006) arXiv:quant-ph/0601066 DOI
- [4]
- E. Knill, “Quantum computing with realistically noisy devices”, Nature 434, 39 (2005) arXiv:quant-ph/0410199 DOI
- [5]
- E. Knill, “Scalable Quantum Computation in the Presence of Large Detected-Error Rates”, (2004) arXiv:quant-ph/0312190
- [6]
- T. Hillmann et al., “Performance of Teleportation-Based Error-Correction Circuits for Bosonic Codes with Noisy Measurements”, PRX Quantum 3, (2022) arXiv:2108.01009 DOI
- [7]
- Carl W. Helstrom. Quantum Detection and Estimation Theory. Elsevier, 1976.
- [8]
- A. Holevo, Probabilistic and Statistical Aspects of Quantum Theory (Edizioni della Normale, 2011) DOI
- [9]
- A. D. C. Tosta, T. O. Maciel, and L. Aolita, “Grand Unification of continuous-variable codes”, (2022) arXiv:2206.01751

## Page edit log

- Victor V. Albert (2022-07-08) — most recent
- Victor V. Albert (2021-12-30)
- Joseph T. Iosue (2021-12-19)

## Cite as:

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