Analog surface code

Analog surface code[1]

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

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

Shift-based decoder [2].

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

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].

[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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“Analog surface code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/analog_surface