\(\chi^{(2)}\) 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.Cousins
- Tiger code— Certain chi-squared codes are supported on the same Fock states as particular tiger codes [2].
- 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)].
Primary Hierarchy
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
- [2]
- Y. Xu, Y. Wang, C. Vuillot, and V. V. Albert, “Letting the tiger out of its cage: bosonic coding without concatenation”, (2024) arXiv:2411.09668
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