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].
Cousins
- Tiger code— The two-mode binomial code for \(S=0\) is a tiger code with \(G = (2,-2)\) and \(H = (1,1)\) [7]. It can be generalized [7] to an \(n\)-mode code encoding a qu\(n\)it in generalized \(SU(n)\) coherent states [8].
- Binomial code
- \(\chi^{(2)}\) code— Two-mode binomial codes [9; Eqs. (90-91)] are closely related to three-mode \(\chi^2\) binomial codes [9; 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 \(\alpha\to 0\).
Primary Hierarchy
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, M. Silveri, R. T. Brierley, V. V. Albert, J. Salmilehto, L. Jiang, and S. M. Girvin, “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, E. Courtens, R. Gilmore, and H. Thomas, “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]
- 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
- [8]
- D. M. Gitman and A. L. Shelepin, “Coherent states of SU(N) groups”, Journal of Physics A: Mathematical and General 26, 313 (1993) DOI
- [9]
- 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