# Chuang-Leung-Yamamoto (CLY) code[1]

## Description

Bosonic Fock-state code that encodes \(k\) qubits into \(n\) oscillators, with each oscillator restricted to having at most \(N\) excitations. Codewords are superpositions of oscillator Fock states which have exactly \(N\) total excitations, and are either uniform (i.e., balanced) superpositions or unbalanced superpositions. Codes can be denoted as \([[N,n,2^k,d]]\), which conflicts with stabilizer code notation.

A simple example of a CLY code is a two-mode "0-2-4" qubit code with codewords \begin{align} \begin{split} |\overline{0}\rangle&=\frac{1}{\sqrt{2}}\left(|40\rangle+|04\rangle\right)\\ |\overline{1}\rangle&=|22\rangle~. \end{split} \tag*{(1)}\end{align}

## Protection

## Rate

## Encoding

## Decoding

## Parents

- Fock-state bosonic code — Chuang-Leung-Yamamoto code are multi-mode Fock-state codes.
- Constant-excitation (CE) code — Chuang-Leung-Yamamoto codewords are constructed out of Fock states with the same total excitation number.

## Child

## Cousins

- 2T-qutrit code — The \(2T\)-qutrit code reduces to the two-mode "0-2-4" CLY code as \(\alpha\to 0\).
- Binomial code — Two-mode version of binomial codes correspond to two-mode "0-2-4" CLY codes (see Sec. IV.A of Ref. [2]).

## References

- [1]
- I. L. Chuang, D. W. Leung, and Y. Yamamoto, “Bosonic quantum codes for amplitude damping”, Physical Review A 56, 1114 (1997) DOI
- [2]
- M. H. Michael et al., “New Class of Quantum Error-Correcting Codes for a Bosonic Mode”, Physical Review X 6, (2016) arXiv:1602.00008 DOI

## Page edit log

- Victor V. Albert (2022-03-02) — most recent
- Dhruv Devulapalli (2021-12-17)
- Jonathan Kunjummen (2021-12-07)

## Cite as:

“Chuang-Leung-Yamamoto (CLY) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/chuang-leung-yamamoto