## 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]. Stabilizer generator matrices equivalent under symplectic transformations are classified by distinct Hermite normal forms [3].

The space of all single-mode GKP codes is the moduli space of elliptic curves, i.e., the three sphere with a trefoil knot removed [4].

## 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

## 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 [2,5–7].
- Amplitude-damping (AD) code — Transmission schemes with multimode GKP codes achieve, up to a constant-factor offset, the capacity of AD, displacement-noise, and thermal-noise Gaussian loss channels [2,5–7]. Combining AD noise with amplification yields displacement noise, the noise that GKP codes are designed to correct [7].
- 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 [19].

## References

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- [2]
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- [3]
- J. Conrad, J. Eisert, and F. Arzani, “Gottesman-Kitaev-Preskill codes: A lattice perspective”, Quantum 6, 648 (2022) arXiv:2109.14645 DOI
- [4]
- J. Conrad, A. G. Burchards, and S. T. Flammia, “Lattices, Gates, and Curves: GKP codes as a Rosetta stone”, (2024) arXiv:2407.03270
- [5]
- 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
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- [17]
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- [18]
- N. Raveendran et al., “Finite Rate QLDPC-GKP Coding Scheme that Surpasses the CSS Hamming Bound”, Quantum 6, 767 (2022) arXiv:2111.07029 DOI
- [19]
- 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
- [20]
- L. Babai, “On Lovász’ lattice reduction and the nearest lattice point problem”, Combinatorica 6, 1 (1986) DOI
- [21]
- 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
- [22]
- D. Gottesman and L. L. Zhang, “Fibre bundle framework for unitary quantum fault tolerance”, (2017) arXiv:1309.7062
- [23]
- 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
- [24]
- A. L. Grimsmo and S. Puri, “Quantum Error Correction with the Gottesman-Kitaev-Preskill Code”, PRX Quantum 2, (2021) arXiv:2106.12989 DOI
- [25]
- 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