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


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].


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][4][5][6]. Particular 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].


By applying GKP error correction to Gaussian input states, universality can be achieved without non-Gaussian elements [10].


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) or by using other effective iterative schemes when e.g. the lattice represents a concatenated GKP code [11][12][3][13].Closest lattice point decoding [14].

Fault Tolerance

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


GKP states are similar to the lattice Gaussian states featured in the proof of hardness of LWE [15; pg. 12].





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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.