Two-mode binomial code[1]
Description
Two-mode constant-energy CLY code whose coefficients are square-roots of binomial coefficients.
The simplest two-mode \(S=1\) code is an analogue of the "0-2-4" single-mode binomial 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}
The general codewords are \begin{align} |\overline{\mu}\rangle=\frac{1}{2^{J}}\sum_{m=0}^{2J}\left(-1\right)^{\mu m}\sqrt{{2J \choose m}}\left|2J-(S+1)m,(S+1)m\right\rangle~, \tag*{(2)}\end{align} with spacing \(S\) and dephasing error parameter \(N\) such that \(J = \frac{1}{2}(N+1)(S+1)\) [2]. The \(S=0\) version can be obtained by applying a \(50:50\) beamsplitter to the highest-weight Fock state \(|2J,0\rangle\) [3].
Parent
Cousins
- Binomial code
- \(\chi^{(2)}\) code — Two-mode binomial codes [4; Eqs. (90-91)] are closely related to three-mode \(\chi^2\) binomial codes [4; Eqs. (61-62)].
- 2T-qutrit code — The \(2T\)-qutrit code reduces to the two-mode "0-2-4" binomial code as \(\alpha\to 0\).
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
- [3]
- M. Bergmann and P. van Loock, “Quantum error correction against photon loss using NOON states”, Physical Review A 94, (2016) arXiv:1512.07605 DOI
- [4]
- 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 (2023-11-17) — most recent
Cite as:
“Two-mode binomial code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/two-mode_binomial