## Description

Generalization of the GKP code to \(n\) bosonic modes whose stabilizer group is an infinite countable group of oscillator displacement operators.

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}~, \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}~, \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_{2n}) \rangle = \{ e^{ i \phi_M (\xi) } D(\xi) ~\vert~ \xi \in {\mathcal{L}} \} \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}~. \end{align}

The centralizer for this stabilizer group within the displacement operators 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

## Parents

- Qudit-into-oscillator code
- Bosonic stabilizer code — GKP codes defined on rectangular lattice are bosonic CSS codes, which more general lattices yield bosonic stabilizer codes.

## Children

- GKP-stabilizer code — GKP-stabilizer codes correspond to multimode GKP codes constructed using a degenerate lattice; see Appx. A of Ref. [3].
- Gottesman-Kitaev-Preskill (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 — Multimode GKP codes are quantum analogues of lattice-based codes.

## Zoo code information

## References

- [1]
- D. Gottesman, A. Kitaev, and J. Preskill, “Encoding a qubit in an oscillator”, Physical Review A 64, (2001). DOI; quant-ph/0008040
- [2]
- J. Harrington and J. Preskill, “Achievable rates for the Gaussian quantum channel”, Physical Review A 64, (2001). DOI; quant-ph/0105058
- [3]
- J. Conrad, J. Eisert, and F. Arzani, “Gottesman-Kitaev-Preskill codes: A lattice perspective”, Quantum 6, 648 (2022). DOI; 2109.14645
- [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). DOI; 1708.07257
- [5]
- M. Rosati, A. Mari, and V. Giovannetti, “Narrow bounds for the quantum capacity of thermal attenuators”, Nature Communications 9, (2018). DOI; 1801.04731
- [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). DOI; 1801.07271
- [7]
- B. Royer, S. Singh, and S. M. Girvin, “Stabilization of Finite-Energy Gottesman-Kitaev-Preskill States”, Physical Review Letters 125, (2020). DOI; 2009.07941
- [8]
- B. Royer, S. Singh, and S. M. Girvin, “Encoding Qubits in Multimode Grid States”, PRX Quantum 3, (2022). DOI; 2201.12337
- [9]
- B. Q. Baragiola et al., “All-Gaussian Universality and Fault Tolerance with the Gottesman-Kitaev-Preskill Code”, Physical Review Letters 123, (2019). DOI; 1903.00012
- [10]
- C. Vuillot et al., “Quantum error correction with the toric Gottesman-Kitaev-Preskill code”, Physical Review A 99, (2019). DOI; 1810.00047
- [11]
- K. Noh and C. Chamberland, “Fault-tolerant bosonic quantum error correction with the surface–Gottesman-Kitaev-Preskill code”, Physical Review A 101, (2020). DOI; 1908.03579
- [12]
- Nithin Raveendran et al., “Finite Rate QLDPC-GKP Coding Scheme that Surpasses the CSS Hamming Bound”. 2111.07029

## Cite as:

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