Concatenated GKP code[1] 

Description

A concatenated code whose outer code is a GKP code. In other words, a bosonic code that can be thought of as a concatenation of an arbitrary inner code and another bosonic outer code. Most examples encode physical qubits of an inner stabilizer code into the square-lattice GKP code.

Protection

The analog syndrome information of the outer GKP code can improve protection of the inner code. As an example, concatenating a three-qubit quantum repetition code with GKP codes can correct some two-bit-flip errors [1].

Rate

Recursively concatenating the \([[6,2,2]]\) and \([[4,2,2]]\) codes with GKP codes achieves the hashing bound of the displacement channel [2]. Concatenating Abelian LP codes with GKP codes can surpass the CSS Hamming bound [3].

Gates

Linear-optical computation [4].

Code Capacity Threshold

\(0.599\) threshold displacement standard deviation for GKP-repetition code [5].\(0.59\) threshold displacement standard deviation for GKP-color code [6].

Notes

Bosonic Pauli+ model is a numerical simulation tool for concatenated GKP codes [7].

Parent

Children

Cousins

References

[1]
K. Fukui, A. Tomita, and A. Okamoto, “Analog Quantum Error Correction with Encoding a Qubit into an Oscillator”, Physical Review Letters 119, (2017) arXiv:1706.03011 DOI
[2]
K. Fukui et al., “High-Threshold Fault-Tolerant Quantum Computation with Analog Quantum Error Correction”, Physical Review X 8, (2018) arXiv:1712.00294 DOI
[3]
N. Raveendran et al., “Finite Rate QLDPC-GKP Coding Scheme that Surpasses the CSS Hamming Bound”, Quantum 6, 767 (2022) arXiv:2111.07029 DOI
[4]
B. W. Walshe et al., “Linear-optical quantum computation with arbitrary error-correcting codes”, (2024) arXiv:2408.04126
[5]
M. P. Stafford and N. C. Menicucci, “Biased Gottesman-Kitaev-Preskill repetition code”, Physical Review A 108, (2023) arXiv:2212.11397 DOI
[6]
J. Zhang et al., “Quantum error correction with the color-Gottesman-Kitaev-Preskill code”, Physical Review A 104, (2021) arXiv:2112.14447 DOI
[7]
F. Hopfmueller et al., “Bosonic Pauli+: Efficient Simulation of Concatenated Gottesman-Kitaev-Preskill Codes”, (2024) arXiv:2402.09333
[8]
B. Royer, S. Singh, and S. M. Girvin, “Encoding Qubits in Multimode Grid States”, PRX Quantum 3, (2022) arXiv:2201.12337 DOI
[9]
M. P. C. Fossorier, “Quasi-Cyclic Low-Density Parity-Check Codes From Circulant Permutation Matrices”, IEEE Transactions on Information Theory 50, 1788 (2004) DOI
[10]
K. Fukui, T. Matsuura, and N. C. Menicucci, “Efficient Concatenated Bosonic Code for Additive Gaussian Noise”, Physical Review Letters 131, (2023) arXiv:2102.01374 DOI
[11]
M. Lin and K. Noh, “Exploring the quantum capacity of a Gaussian random displacement channel using Gottesman-Kitaev-Preskill codes and maximum likelihood decoding”, (2024) arXiv:2411.04277
[12]
Y. Xu et al., “Qubit-Oscillator Concatenated Codes: Decoding Formalism and Code Comparison”, PRX Quantum 4, (2023) arXiv:2209.04573 DOI
Page edit log

Your contribution is welcome!

on github.com (edit & pull request)— see instructions

edit on this site

Zoo Code ID: gkp_concatenated

Cite as:
“Concatenated GKP code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/gkp_concatenated
BibTeX:
@incollection{eczoo_gkp_concatenated, title={Concatenated GKP code}, booktitle={The Error Correction Zoo}, year={2024}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/gkp_concatenated} }
Share via:
Twitter | Mastodon |  | E-mail
Permanent link:
https://errorcorrectionzoo.org/c/gkp_concatenated

Cite as:

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

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