Modular-qudit GKP code[1; Sec. II] 


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 opeartors 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 \(X^{r_2 n}\).


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 \(|b| < r_2/2\).



  • 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].


D. Gottesman, A. Kitaev, and J. Preskill, “Encoding a qubit in an oscillator”, Physical Review A 64, (2001) arXiv:quant-ph/0008040 DOI
Y. Xu, Y. Wang, and V. V. Albert, “Clifford operations and homological codes for rotors and oscillators”, (2024) arXiv:2311.07679
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Zoo Code ID: qudit_gkp

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“Modular-qudit GKP code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.
@incollection{eczoo_qudit_gkp, title={Modular-qudit GKP code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={} }
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“Modular-qudit GKP code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.