## Description

Quantum lattice code for a non-degenerate lattice, thereby admitting a finite-dimensional logical subspace. Codes on \(n\) modes can be constructed from lattices with \(2n\)-dimensional full-rank Gram matrices \(A\).

The centralizer for the stabilizer group within the displacement operators case can be identified with the symplectic dual lattice \({\mathcal{L}}^{\perp}\) (i.e. all points in \(\mathbb{R}^{2n}\) that have integer symplectic inner product with all points in \({\mathcal{L}}\) ), such that logical operations are identified with the dual quotients \({\mathcal{L}}^{\perp}/{\mathcal{L}}\). The size of this dual quotient is the determinant of the Gram matrix, yielding the logical dimension \(d=\sqrt{\| \det{A}\|}\) [1].

## Protection

## Rate

## Encoding

## Gates

## Decoding

## Fault Tolerance

## Notes

## Parent

- Quantum lattice code — GKP codes are \(n\)-mode quantum lattice codes with \(2n\) stabilizers, i.e., constructed using a non-degenerate lattice.

## Children

- \(D_4\) hyper-diamond GKP code
- GKP cluster-state code — The GKP cluster-state code is a concatenation of a cluster-state stabilizer code with a single-mode GKP code. A GKP-based cluster state is a multimode GKP codeword, although other codewords are not utilized in CV MBQC.
- Square-lattice GKP code
- Hexagonal GKP code
- NTRU-GKP code

## Cousins

- Approximate quantum error-correcting code (AQECC) — Approximate error-correction offered by GKP codes yields achievable rates that are a constant away from the capacity of Guassian loss channels [2,4–6].
- Quadrature-amplitude modulation (QAM) code — Finite-energy GKP codes are quantum analogues of lattice-based QAM codes in that both use a subset of points on a lattice.
- Numerically optimized bosonic code — Numerically optimizing GKP code lattices yields codes for three, seven, and nine modes with larger distances and fidelities than known GKP codes [18].

## References

- [1]
- D. Gottesman, A. Kitaev, and J. Preskill, “Encoding a qubit in an oscillator”, Physical Review A 64, (2001) arXiv:quant-ph/0008040 DOI
- [2]
- J. Harrington and J. Preskill, “Achievable rates for the Gaussian quantum channel”, Physical Review A 64, (2001) arXiv:quant-ph/0105058 DOI
- [3]
- J. Conrad, J. Eisert, and F. Arzani, “Gottesman-Kitaev-Preskill codes: A lattice perspective”, Quantum 6, 648 (2022) arXiv:2109.14645 DOI
- [4]
- K. Sharma et al., “Bounding the energy-constrained quantum and private capacities of phase-insensitive bosonic Gaussian channels”, New Journal of Physics 20, 063025 (2018) arXiv:1708.07257 DOI
- [5]
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- [6]
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- [7]
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- [17]
- N. Raveendran et al., “Finite Rate QLDPC-GKP Coding Scheme that Surpasses the CSS Hamming Bound”, Quantum 6, 767 (2022) arXiv:2111.07029 DOI
- [18]
- 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
- [19]
- L. Babai, “On Lovász’ lattice reduction and the nearest lattice point problem”, Combinatorica 6, 1 (1986) DOI
- [20]
- B. M. Terhal, J. Conrad, and C. Vuillot, “Towards scalable bosonic quantum error correction”, Quantum Science and Technology 5, 043001 (2020) arXiv:2002.11008 DOI
- [21]
- A. L. Grimsmo and S. Puri, “Quantum Error Correction with the Gottesman-Kitaev-Preskill Code”, PRX Quantum 2, (2021) arXiv:2106.12989 DOI
- [22]
- A. J. Brady et al., “Advances in bosonic quantum error correction with Gottesman–Kitaev–Preskill Codes: Theory, engineering and applications”, Progress in Quantum Electronics 100496 (2024) arXiv:2308.02913 DOI

## Page edit log

- Jonathan Conrad (2022-07-05) — most recent
- Victor V. Albert (2022-07-05)
- Victor V. Albert (2022-03-24)

## Cite as:

“Gottesman-Kitaev-Preskill (GKP) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/multimodegkp