Rotor code 

Description

Encodes a logical Hilbert space, finite- or infinite-dimensional, into a physical Hilbert space of \(\ell^2\)-normalizable functions on either the integers \(\mathbb Z\) or the circle group \(U(1)\). Ideal codewords may not be normalizable because the space is infinite-dimensional, so approximate versions have to be constructed in practice.

Protection

Rotor generalized Pauli error basis

A rotor analogue of the Pauli string basis for qubit codes consists of rotor generalized Pauli operators.

Rotor generalized Pauli strings: For a single rotor, its elements are products of exponentials of the rotor's angular position (\(\hat\phi\)) and angular momentum (\(\hat L\)) operators, acting on the rotor's angular position states \(|\phi\rangle\) for \(\phi\in U(1)\) as \begin{align} e^{-i\varphi\hat{L}}\left|\phi\right\rangle =\left|\phi+\varphi\right\rangle \,\,\text{ and }\,\,e^{i\ell\hat{\phi}}\left|\phi\right\rangle =e^{i\ell\phi}\left|\phi\right\rangle ~, \tag*{(1)}\end{align} where \(\varphi\in U(1)\) and \(\ell\in\mathbb{Z}\). For multiple rotors, error set elements are tensor products of elements of the single-rotor error set, characterized by vectors of angle and integer coefficients multiplying vectors of angular momentum \(\hat{\boldsymbol{L}}\) and angular position \(\hat{\boldsymbol{\phi}}\) operators.

Notes

See Refs. [1][2; Sec. IV] for introductions to rotor Hilbert spaces.

Parent

  • Group-based quantum code — Group quantum codes whose physical spaces are constructed using either the group of the integers \(\mathbb{Z}\) or the circle group \(U(1)\) are rotor codes.

Child

Cousin

  • Square-lattice GKP code — Because square-lattice GKP error states are parameterized by two modular (i.e., periodic) variables of position and momentum, measuring one of the GKP stabilizers constrains the oscillator Hilbert space into that of a rotor.

References

[1]
V. V. Albert, S. Pascazio, and M. H. Devoret, “General phase spaces: from discrete variables to rotor and continuum limits”, Journal of Physics A: Mathematical and Theoretical 50, 504002 (2017) arXiv:1709.04460 DOI
[2]
V. V. Albert, J. P. Covey, and J. Preskill, “Robust Encoding of a Qubit in a Molecule”, Physical Review X 10, (2020) arXiv:1911.00099 DOI
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Zoo Code ID: rotor

Cite as:
“Rotor code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/rotor
BibTeX:
@incollection{eczoo_rotor, title={Rotor code}, booktitle={The Error Correction Zoo}, year={2024}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/rotor} }
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Permanent link:
https://errorcorrectionzoo.org/c/rotor

Cite as:

“Rotor code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/rotor

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