## Description

Qudit-into-oscillator code whose codewords can succinctly be expressed as superpositions of a countable set of coherent states that is called a constellation. Some useful constellations form a group (see gkp, cat or \(2T\)-qutrit codes) while others make up a Gaussian quadrature rule [1,2].

## Rate

Coherent-state constellation codes consisting of points from a Gaussian quadrature rule can be concatenated with quantum polar codes to achieve the Gaussian coherent information of the thermal noise channel [1,2].

## Parent

## Children

- Renormalization group (RG) cat code
- Quantum spherical code (QSC) — Coherent-state QSCs are coherent-state constellation codes constrained to lie on a sphere.
- Quantum lattice code — Quantum lattice codewords can be written as superpositions of coherent states lying on a lattice in phase space [3,4].

## Cousins

- Concatenated bosonic code — Coherent-state constellation codes consisting of points from a Gaussian quadrature rule can be concatenated with quantum polar codes to achieve the Gaussian coherent information of the thermal noise channel [1,2].
- \(t\)-design — Coherent-state constellation codes consisting of points from a Gaussian quadrature rule can be concatenated with quantum polar codes to achieve the Gaussian coherent information of the thermal noise channel [1,2].
- Modulation scheme — Coherent-state constellation codes are quantum versions of modulation schemes in that their codewords are superpositions of points in a constellation. Additionally, analog codes that achieve AGWN capacity [5] can be used to develop capacity-achieving concatenations of coherent-state constellation codes with quantum polar codes [1,2].
- Coherent-state c-q code — Coherent-state c-q codes encode classical alphabets into constellations of coherent states, while coherent-state constellation codes encode quantum information into superpositions of coherent states.

## References

- [1]
- F. Lacerda, J. M. Renes, and V. B. Scholz, “Coherent-state constellations and polar codes for thermal Gaussian channels”, Physical Review A 95, (2017) arXiv:1603.05970 DOI
- [2]
- F. Lacerda, J. M. Renes, and V. B. Scholz, “Coherent state constellations for Bosonic Gaussian channels”, 2016 IEEE International Symposium on Information Theory (ISIT) (2016) DOI
- [3]
- D. Gottesman, A. Kitaev, and J. Preskill, “Encoding a qubit in an oscillator”, Physical Review A 64, (2001) arXiv:quant-ph/0008040 DOI
- [4]
- V. V. Albert et al., “Performance and structure of single-mode bosonic codes”, Physical Review A 97, (2018) arXiv:1708.05010 DOI
- [5]
- Y. Wu and S. Verdu, “The impact of constellation cardinality on Gaussian channel capacity”, 2010 48th Annual Allerton Conference on Communication, Control, and Computing (Allerton) 620 (2010) DOI

## Page edit log

- Armin Gerami (2023-02-22) — most recent
- Victor V. Albert (2023-02-22)
- Victor V. Albert (2022-10-31)

## Cite as:

“Coherent-state constellation code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/coherent_constellation