Gottesman-Kitaev-Preskill (GKP) code[1,2] 


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].


The level of protection against displacement errors is quantified by the Euclidean code distance \(\Delta=\min_{x\in {\mathcal{L}}^{\perp}\setminus {\mathcal{L}}} \|x\|_2\) [3].


Transmission schemes with multimode GKP codes achieve, up to a constant-factor offset, the capacity of displacement-noise and thermal-noise Gaussian loss channels [2,46]. Particular random lattice families of multimode GKP codes achieve the hashing bound of the displacement noise channel [2].


GKP codes with fixed \(n\) and prime-dimensional logical Hilbert space are symplectically related to a disjoint product of single-mode GKP codes on \(n\) modes, such that encoding via Gaussian unitaries is possible.Dissipative stabilization of finite-energy GKP states using stabilizers conjugated by cooling ([7], Appx. B) or damping operator, i.e., a damped exponential of the total occupation number [8,9].


Gaussian operations and homodyne measurements on GKP states are classically simulable, and there is a sufficient condition for an additional element to achieve universal quantum computation [10].There is a relation between magic (i.e., how far away a state is from being a stabilizer state) and non-Gaussianity for GKP codewords [11,12]. In particular, implementing a non-Clifford logical gate requires a higher degree of non-Gaussianity than that expressed by ideal non-normalizable GKP states [12].By applying GKP error correction to Gaussian input states, computational universality can be achieved without additional non-Gaussian elements [13].


The MLD decoder for Gaussian displacement errors is realized by evaluating a lattice theta function, and in general the decision can be approximated by either solving (approximating) the closest vector problem (CVP) [14] (a.k.a. closest lattice point problem) or by using other effective iterative schemes when, e.g., the lattice represents a concatenated GKP code [3,1517]. While the decoder time scales exponentially with number of modes \(n\) generically, the time can be polynomial in \(n\) for certain codes [18].Babai's nearest plane algorithm [19] can be used for bounded-distance decoding [18].

Fault Tolerance

Logical Clifford operations are given by Gaussian unitaries, which map bounded-size errors to bounded-size errors [1].


Reviews on GKP codes presented in Refs. [2022].


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




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O. Hahn et al., “Quantifying Qubit Magic Resource with Gottesman-Kitaev-Preskill Encoding”, Physical Review Letters 128, (2022) arXiv:2109.13018 DOI
O. Hahn, G. Ferrini, and R. Takagi, “Bridging magic and non-Gaussian resources via Gottesman-Kitaev-Preskill encoding”, (2024) arXiv:2406.06418
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
E. Agrell et al., “Closest point search in lattices”, IEEE Transactions on Information Theory 48, 2201 (2002) DOI
C. Vuillot et al., “Quantum error correction with the toric Gottesman-Kitaev-Preskill code”, Physical Review A 99, (2019) arXiv:1810.00047 DOI
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
N. Raveendran et al., “Finite Rate QLDPC-GKP Coding Scheme that Surpasses the CSS Hamming Bound”, Quantum 6, 767 (2022) arXiv:2111.07029 DOI
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
L. Babai, “On Lovász’ lattice reduction and the nearest lattice point problem”, Combinatorica 6, 1 (1986) DOI
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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
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“Gottesman-Kitaev-Preskill (GKP) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.
@incollection{eczoo_multimodegkp, title={Gottesman-Kitaev-Preskill (GKP) code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={} }
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“Gottesman-Kitaev-Preskill (GKP) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.