## Description

Also can be called a quantum lattice code. Bosonic stabilizer code on \(n\) bosonic modes whose stabilizer group is an infinite countable group of oscillator displacement operators which implement lattice translations in phase space.

Displacement operators on \(n\) modes can be written as \begin{align} D(\xi) = \exp \left\{-i \sqrt{2\pi} {\xi}^\mathrm{T} J \hat{q} \right\} , \quad \xi \in \mathbb{R}^{2n}~, \tag*{(1)}\end{align} where \(\hat{q}\) is a \(2n\)-dimensional vector position and momentum operators of the modes, the symplectic form \begin{align} J = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} \otimes I_n = \begin{pmatrix} 0 & I_n \\ -I_n & 0 \end{pmatrix}~, \tag*{(2)}\end{align} and \(I_n\) is the identity matrix. A group generated by a set of independent displacement operators is given by a lattice \({\mathcal{L}}\) \begin{align} \langle D(\xi_1) ,\dots, D(\xi_{m}) \rangle = \{ e^{ i \phi_M (\xi) } D(\xi) ~\vert~ \xi \in {\mathcal{L}} \} \tag*{(3)}\end{align} and becomes a valid stabilizer group when every symplectic inner product between lattice vectors yields an integer. In other words, the corresponding lattice is symplectically integral, corresponding to an integer-valued symplectic Gram matrix \(A\), \begin{align} A_{ij}={\xi}^T_i J \xi_j \in \mathbb{Z}~. \tag*{(4)}\end{align} The \(m=2n\) case yields multimode GKP codes encoding a finite-dimensional logical subspace, while removing some displacements yields GKP-stabilizer codes encoding an infinite-dimensional logical subspace. Codes defined on a hyper-rectangular lattice are CSS GKP codes, and more general lattices, obtained by Gaussian transformations, yield non-CSS codes.

The centralizer for the stabilizer group within the displacement operators for the \(m=2n\) 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

## Realizations

## Parents

- Bosonic stabilizer code
- Coherent-state constellation code — GKP codewords can be written as superpositions of coherent states lying on a lattice in phase space [1][16].

## 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.
- GKP-stabilizer code — GKP-stabilizer codes are \(n\)-mode GKP codes with less than \(2n\) stabilizers. Equivalently, they correspond to multimode GKP codes constructed using a degenerate lattice (see Appx. A of Ref. [3]).
- Square-lattice GKP code
- Hexagonal 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][5][6].
- Lattice-based code — GKP codes can be thought of as quantum lattice codes because they store information in quantum superpositions of points on a lattice in quantum phase space.
- 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 code — Numerically optimizing GKP code lattices yields codes for three and nine modes with larger distances and fidelities than known GKP codes [14].

## 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]
- M. Rosati, A. Mari, and V. Giovannetti, “Narrow bounds for the quantum capacity of thermal attenuators”, Nature Communications 9, (2018) arXiv:1801.04731 DOI
- [6]
- K. Noh, V. V. Albert, and L. Jiang, “Quantum Capacity Bounds of Gaussian Thermal Loss Channels and Achievable Rates With Gottesman-Kitaev-Preskill Codes”, IEEE Transactions on Information Theory 65, 2563 (2019) arXiv:1801.07271 DOI
- [7]
- N. C. Menicucci, “Fault-Tolerant Measurement-Based Quantum Computing with Continuous-Variable Cluster States”, Physical Review Letters 112, (2014) arXiv:1310.7596 DOI
- [8]
- B. Royer, S. Singh, and S. M. Girvin, “Stabilization of Finite-Energy Gottesman-Kitaev-Preskill States”, Physical Review Letters 125, (2020) arXiv:2009.07941 DOI
- [9]
- B. Royer, S. Singh, and S. M. Girvin, “Encoding Qubits in Multimode Grid States”, PRX Quantum 3, (2022) arXiv:2201.12337 DOI
- [10]
- B. Q. Baragiola et al., “All-Gaussian Universality and Fault Tolerance with the Gottesman-Kitaev-Preskill Code”, Physical Review Letters 123, (2019) arXiv:1903.00012 DOI
- [11]
- C. Vuillot et al., “Quantum error correction with the toric Gottesman-Kitaev-Preskill code”, Physical Review A 99, (2019) arXiv:1810.00047 DOI
- [12]
- 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
- [13]
- N. Raveendran et al., “Finite Rate QLDPC-GKP Coding Scheme that Surpasses the CSS Hamming Bound”, Quantum 6, 767 (2022) arXiv:2111.07029 DOI
- [14]
- M. Lin, C. Chamberland, and K. Noh, “Closest lattice point decoding for multimode Gottesman-Kitaev-Preskill codes”, (2023) arXiv:2303.04702
- [15]
- O. Regev, “On lattices, learning with errors, random linear codes, and cryptography”, Journal of the ACM 56, 1 (2009) DOI
- [16]
- V. V. Albert et al., “Performance and structure of single-mode bosonic codes”, Physical Review A 97, (2018) arXiv:1708.05010 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