Two-mode binomial code

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]; in this case, codewords are two-mode binomial coherent states [4–6].

Parent

Chuang-Leung-Yamamoto (CLY) code

Cousins

Binomial code
\(\chi^{(2)}\) code — Two-mode binomial codes [7; Eqs. (90-91)] are closely related to three-mode \(\chi^2\) binomial codes [7; Eqs. (61-62)].
Concatenated bosonic code — Two-mode binomial codes can be concatenated with repetition codes to yield bosonic analogues of QPCs [3].
Quantum parity code (QPC) — Two-mode binomial codes can be concatenated with repetition codes to yield bosonic analogues of QPCs [3].
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]: J. M. Radcliffe, “Some properties of coherent spin states”, Journal of Physics A: General Physics 4, 313 (1971) DOI
[5]: F. T. Arecchi et al., “Atomic Coherent States in Quantum Optics”, Physical Review A 6, 2211 (1972) DOI
[6]: M. Calixto, A. Mayorgas, and J. Guerrero, “Entanglement and U(D)-spin squeezing in symmetric multi-quDit systems and applications to quantum phase transitions in Lipkin–Meshkov–Glick D-level atom models”, Quantum Information Processing 20, (2021) arXiv:2104.10581 DOI
[7]: 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

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