2T-qutrit code[1]
Description
Two-mode qutrit code constructed out of superpositions of coherent states whose amplitudes make up the binary tetrahedral group \(2T\), a.k.a. the 24-cell.
The codespace is a particular three-dimensional subspace of the 24-dimensional two-mode coherent-state subspace, \begin{align} \mathrm{Span}( \{|\sqrt{2} e^{i (2k+1) \pi/4} \alpha\rangle |0\rangle, |0\rangle |\sqrt{2} e^{i (2k+1) \pi/4} \alpha\rangle, |e^{i k\pi/2} \alpha\rangle |e^{i \ell \pi/2} \alpha\rangle \: : \: 0\leq k, \ell \leq 3\}) \tag*{(1)}\end{align} for any \(\alpha \geq 0\). A basis can be constructed whose elements are equal superpositions of coherent states whose amplitudes make up cosets of the quaternion subgroup \(Q\) in \(2T\).
Gates
Logical phase-flip can be implemented using an excitation-preserving Gaussian transformation. Degree-four polynomial in the lowering operators of the two modes serves as a non-unitary logical bit-flip. Rotations of either mode by \(\pi/4\) are logical gates that swap two logical codewords.Cousins
- Two-mode binomial code— The \(2T\)-qutrit code reduces to the two-mode "0-2-4" binomial code as \(\alpha\to 0\).
- 24-cell code— The \(2T\)-qutrit code is constructed out of superpositions of coherent states whose amplitudes make up the binary tetrahedral group \(2T\), a.k.a. the 24-cell.
Member of code lists
Primary Hierarchy
References
- [1]
- A. Denys and A. Leverrier, “The 2T-qutrit, a two-mode bosonic qutrit”, Quantum 7, 1032 (2023) arXiv:2210.16188 DOI
Page edit log
- Shubham P. Jain (2023-03-01) — most recent
- Victor V. Albert (2022-10-31)
Cite as:
“2T-qutrit code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/2t_qutrit