Modular-qudit GKP code[1; Sec. II]
Description
Modular-qudit analogue of the GKP code. Encodes a qudit into a larger qudit and protects against Pauli shifts up to some maximum value.
The simplest example requires a 15-dimensional qudit and admits stabilizer generators \(Z^6\) and \(X^6\). The logical codewords are \begin{align} \begin{split} |\overline{0}\rangle&=\frac{1}{\sqrt{3}}\left(|0\rangle+|6\rangle+|12\rangle\right)\\|\overline{1}\rangle&=\frac{1}{\sqrt{3}}\left(|3\rangle+|9\rangle+|15\rangle\right)~, \end{split} \tag*{(1)}\end{align} and logical operators are \(Z^3\) and \(X^3\).
More generally, for qudit dimension \(q = r_1 r_2 K\) for some positive integers \(r_1\), \(r_2\), and logical dimension \(K\), the stabilizer generators are \(Z^{r_1 n}\) and \(X^{r_2 n}\).
Protection
The above simple code corrects any Pauli string \(X^{a}Z^{b}\) with \(|a|,|b|\leq 1\). A general code protects against any strings for which \(|a| < r_1/2\) and \(|b| < r_2/2\).Cousins
- Square-lattice GKP code— The square-lattice GKP code can be obtained from the modular-qudit code by taking the physical qudit dimension to be infinite [1; Sec. II].
- Perfect quantum code— The modular-qudit GKP code is not a block code, but it is perfect in the sense that each correctable error maps the logical space into a distinct error space.
- Rotor GKP code— The rotor GKP code can be thought of as a concatenation of a homological rotor code and a modular-qudit GKP code [2; Fig. 3].
- Modular-qudit shift-resistant code— The modular-qudit shift-resistant code requires a smaller physical qudit dimension but protects against only one type of error [3].
Primary Hierarchy
References
- [1]
- D. Gottesman, A. Kitaev, and J. Preskill, “Encoding a qubit in an oscillator”, Physical Review A 64, (2001) arXiv:quant-ph/0008040 DOI
- [2]
- Y. Xu, Y. Wang, and V. V. Albert, “Multimode rotation-symmetric bosonic codes from homological rotor codes”, Physical Review A 110, (2024) arXiv:2311.07679 DOI
- [3]
- S. Pirandola, S. Mancini, S. L. Braunstein, and D. Vitali, “Minimal qudit code for a qubit in the phase-damping channel”, Physical Review A 77, (2008) arXiv:0705.1099 DOI
Page edit log
- Victor V. Albert (2022-11-03) — most recent
Cite as:
“Modular-qudit GKP code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/qudit_gkp