Description
A concatenated code whose outer code is a GKP code and whose inner code is a surface code, including toric surface-code variants [1,2], rotated surface codes [3–6], and XZZX surface codes [7].Rate
The error threshold under ML decoding of GKP-rotated-surface codes comes close to \(\sigma\approx 0.6065\), at which the best-known lower bound [8] on the capacity vanishes [9].Decoding
Minimum-energy and random-plaquette-gauge-model decoders for the toric-GKP code [2; Secs. IV-V].MWPM closest point decoder [6].Code Capacity Threshold
\(0.55\) (\(0.54\)) threshold displacement standard deviation for GKP-toric (GKP-surface) codes without using GKP analog information [2; Sec. IV.B] ([1]). Using the continuous GKP syndrome information raises the GKP-toric threshold to \(\sigma_0\approx 0.6\), corresponding to a qubit error rate of about \(14\%\) [2; Sec. IV.B].Analog QEC on GKP-surface codes with ideal syndrome measurements yields a threshold displacement standard deviation of about \(0.607\), close to the hashing bound for the Gaussian quantum channel [1].\(0.67\) threshold displacement standard deviation for GKP-XZZX-surface code [7].\(0.602\) threshold displacement standard deviation for GKP-surface codes with analog side information using MWPM closest point decoder [6].Threshold
The ML decoding problem for the toric-GKP code maps to a 3D compact QED model in the presence of a quenched random gauge field [2; Secs. V.B-V.C]. A decoder based on this mapping yields a threshold displacement standard deviation of \(\sigma_0\approx 0.243\) when toric-code measurements, data errors, and GKP ancilla errors are all noisy [2; Sec. V.D.3], but this noise model did not properly take into account error propagation [3].\(11.2\)dB of squeezing under displacement noise using MWPM decoding for GKP-rotated-surface codes [3,5]. The error threshold under ML decoding of GKP-rotated-surface codes comes close to \(\sigma\approx 0.6065\), at which the best-known lower bound [8] on the capacity vanishes [9].Cousins
- Toric code— GKP codes have been concatenated with toric codes [2].
- Rotated surface code— GKP codes have been concatenated with rotated surface codes [3–6].
- XZZX surface code— GKP codes have been concatenated with XZZX surface codes [7].
- Analog surface code— Condensing pure fluxes and charges in the analog surface code yields toric-GKP codes [10].
Member of code lists
- 2D stabilizer codes
- Bosonic stabilizer codes
- Concatenated quantum codes and friends
- Hamiltonian-based codes and friends
- Quantum codes
- Quantum codes with a rate
- Quantum codes with code capacity thresholds
- Quantum codes with notable decoders
- Quantum codes with other thresholds
- Quantum LDPC codes
- Stabilizer codes
Primary Hierarchy
Parents
GKP-surface code
References
- [1]
- K. Fukui, A. Tomita, A. Okamoto, and K. Fujii, “High-Threshold Fault-Tolerant Quantum Computation with Analog Quantum Error Correction”, Physical Review X 8, (2018) arXiv:1712.00294 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]
- K. Noh and C. Chamberland, “Fault-tolerant bosonic quantum error correction with the surface–Gottesman-Kitaev-Preskill code”, Physical Review A 101, (2020) arXiv:1908.03579 DOI
- [4]
- M. V. Larsen, C. Chamberland, K. Noh, J. S. Neergaard-Nielsen, and U. L. Andersen, “Fault-Tolerant Continuous-Variable Measurement-based Quantum Computation Architecture”, PRX Quantum 2, (2021) arXiv:2101.03014 DOI
- [5]
- K. Noh, C. Chamberland, and F. G. S. L. Brandão, “Low-Overhead Fault-Tolerant Quantum Error Correction with the Surface-GKP Code”, PRX Quantum 3, (2022) arXiv:2103.06994 DOI
- [6]
- M. Lin, C. Chamberland, and K. Noh, “Closest Lattice Point Decoding for Multimode Gottesman-Kitaev-Preskill Codes”, PRX Quantum 4, (2023) arXiv:2303.04702 DOI
- [7]
- J. Zhang, Y.-C. Wu, and G.-P. Guo, “Concatenation of the Gottesman-Kitaev-Preskill code with the XZZX surface code”, Physical Review A 107, (2023) arXiv:2207.04383 DOI
- [8]
- A. S. Holevo and R. F. Werner, “Evaluating capacities of Bosonic Gaussian channels”, (1999) arXiv:quant-ph/9912067
- [9]
- M. Lin and K. Noh, “Exploring the quantum capacity of a Gaussian random-displacement channel using Gottesman-Kitaev-Preskill codes and maximum-likelihood decoding”, Physical Review A 111, (2025) arXiv:2411.04277 DOI
- [10]
- J. C. Magdalena de la Fuente, T. D. Ellison, M. Cheng, and D. J. Williamson, “Topological Stabilizer Models on Continuous Variables”, Physical Review X 16, (2026) arXiv:2411.04993 DOI
Page edit log
- Victor V. Albert (2024-07-17) — most recent
Cite as:
“GKP-surface code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/gkp_surface_concatenated