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}

An alternative basis for general codewords is \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 states \(|2J,0\rangle\) and \(|0,2J\rangle\) [3].

Parent

Cousins

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
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Zoo Code ID: two-mode_binomial

Cite as:
“Two-mode binomial code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/two-mode_binomial
BibTeX:
@incollection{eczoo_two-mode_binomial, title={Two-mode binomial code}, booktitle={The Error Correction Zoo}, year={2023}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/two-mode_binomial} }
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Cite as:

“Two-mode binomial code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/two-mode_binomial

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/oscillators/fock_state/constant_excitation/two-mode_binomial.yml.