# \(\chi^{(2)}\) code[1]

## Description

A \(3n\)-mode bosonic Fock-state code that requires only linear optics and the \(\chi^{(2)}\) optical nonlinear interaction for encoding, decoding, and logical gates. Codewords lie in Fock-state subspaces that are invariant under Hermitian combinations of the \(\chi^{(2)}\) nonlinearities \(abc^\dagger\) and \(i abc^\dagger\), where \(a\), \(b\), and \(c\) are lowering operators acting on one of the \(n\) triples of modes on which the codes are defined. Codewords are also \(+1\) eigenstates of stabilizer-like symmetry operators, and photon parities are error syndromes.

## Protection

Codes protect against loss, gain, and dephasing errors conditional on the knowledge of the total number of photons lost.

## Encoding

Linear optics and \(\chi^{(2)}\) interactions.

## Gates

Linear optics and \(\chi^{(2)}\) interactions yield a universal set of gates.

## Decoding

Linear optics and \(\chi^{(2)}\) interactions.

## Parent

## Cousin

- Two-mode binomial code — Two-mode binomial codes [1; Eqs. (90-91)] are closely related to three-mode \(\chi^2\) binomial codes [1; Eqs. (61-62)].

## References

- [1]
- M. Y. Niu, I. L. Chuang, and J. H. Shapiro, “Hardware-efficient bosonic quantum error-correcting codes based on symmetry operators”, Physical Review A 97, (2018) arXiv:1709.05302 DOI

## Page edit log

- Victor V. Albert (2023-01-10) — most recent

## Cite as:

“\(\chi^{(2)}\) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/chi2