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Four-rotor code [1; Sec. VIII]

Description

\([[4,2,2]]_{\mathbb Z}\) CSS rotor code that is an extension of the four-qubit code to the integer alphabet, i.e., the angular momentum states of a rotor.

The code is \(U(1)\)-covariant and its ideal logical-rotor codewords, \begin{align} |\overline{a,b}\rangle = \sum_{j,k,l\in\mathbb{Z}} \delta_{a,j+k}\delta_{b,l} \left| j,k,j+l,k+l \right\rangle~,\tag*{$(1)$} \end{align}where \(a,b\in\mathbb{Z}\), are not normalizable.

Cousin

  • Homological number-phase code— After suitable rotor-parity flips and projection onto the non-negative angular-momentum orthant, the four-rotor current-mirror code yields a homological number-phase code [2; Ex. 4].

Member of code lists

Primary Hierarchy

Quantum Domain
Group quantum Kingdom
Group-based quantum codeQECC Quantum
Rotor code
Rotor stabilizer codeStabilizer Hamiltonian-based QECC Quantum
Homological rotor codeCSS Stabilizer Hamiltonian-based QECC Quantum
Parents
Homological rotor code
\([[4,2,2]]_{G}\) four group-qudit codeTopological Hamiltonian-based Small-distance block quantum QECC Quantum
The four group-qudit code reduces to the four-rotor code for \(G= \mathbb{Z}\).
Four-rotor code

References

  1. [1]

    P. Faist, S. Nezami, V. V. Albert, G. Salton, F. Pastawski, P. Hayden, and J. Preskill, “Continuous Symmetries and Approximate Quantum Error Correction”, Physical Review X 10, (2020) arXiv:1902.07714 DOI

  2. [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

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Zoo Code ID: rotor_4_2_2

Cite as:
“Four-rotor code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/rotor_4_2_2, arXiv:2606.11484
BibTeX:
@incollection{eczoo_rotor_4_2_2,
title={Four-rotor code},
booktitle={The Error Correction Zoo},
year={2026},
editor={Albert, Victor V. and Faist, Philippe},
eprint={2606.11484},
doi={10.48550/arXiv.2606.11484},
url={https://errorcorrectionzoo.org/c/rotor_4_2_2}
}
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Permanent link:
https://errorcorrectionzoo.org/c/rotor_4_2_2

Cite as:

“Four-rotor code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/rotor_4_2_2, arXiv:2606.11484

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/groups/rotors/stabilizer/css/rotor_4_2_2.yml.