Analog surface code[1] 

Also known as \(\mathbb{R}\) gauge theory code, Continuous-variable (CV) surface code.

Description

An analog CSS version of the Kitaev surface code realizing a phase of 2D \(\mathbb{R}\) gauge theory.

Decoding

Shift-based decoder [2].

Notes

See [3; Sec. III.C2] for an exposition.

Parents

Cousins

  • Modular-qudit surface code — The tiger surface code can be thought of as a realization of the \(q\to\infty\) \(\mathbb{R}\) oscillator limit [4] of the qudit surface code as a bosonic stabilizer code.
  • Abelian topological code — The analog surface code realizes a straightforward extension of the modular-qudit surface code to infinite local dimension, \(q\to\infty\) [4]. The code realizes a phase of 2D \(\mathbb{R}\) gauge theory. There are two types of anyons, \(e\) and \(m\), with each type being valued in a continuous domain as opposed to \(\mathbb{Z}_q\) for the qudit surface code.
  • Compactified \(\mathbb{R}\) gauge theory code — The compactified \(\mathbb{R}\) gauge theory code can be obtained from the analog surface code by condensing certain anyons [5]. This results in a pinning of each mode to the space of periodic functions, which make up a physical rotor, and can be thought of as compactification of the 2D \(\mathbb{R}\) gauge theory phase realized by the analog surface code.
  • \(U(1)_{2n} \times U(1)_{-2m}\) Chern-Simons GKP code — The \(U(1)_{2n} \times U(1)_{-2m}\) Chern-Simons GKP code can be obtained from the analog surface code by condensing certain anyons [5].
  • GKP-surface code — The GKP-surface code can be obtained from the analog surface code by condensing certain anyons [5].

References

[1]
J. Zhang, C. Xie, K. Peng, and P. van Loock, “Anyon statistics with continuous variables”, Physical Review A 78, (2008) arXiv:0711.0820 DOI
[2]
C. Vuillot, H. Asasi, Y. Wang, L. P. Pryadko, and B. M. Terhal, “Quantum error correction with the toric Gottesman-Kitaev-Preskill code”, Physical Review A 99, (2019) arXiv:1810.00047 DOI
[3]
B. M. Terhal, “Quantum error correction for quantum memories”, Reviews of Modern Physics 87, 307 (2015) arXiv:1302.3428 DOI
[4]
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
[5]
J. C. M. de la Fuente, T. D. Ellison, M. Cheng, and D. J. Williamson, “Topological stabilizer models on continuous variables”, (2024) arXiv:2411.04993
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Zoo Code ID: analog_surface

Cite as:
“Analog surface code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/analog_surface
BibTeX:
@incollection{eczoo_analog_surface, title={Analog surface code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/analog_surface} }
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“Analog surface code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/analog_surface

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/oscillators/stabilizer/hyperplane/analog_surface.yml.