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

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].

Protection

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].

Rate

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].

Encoding

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].

Gates

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].

Decoding

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].

Notes

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

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

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]
C. Calcluth et al., “Sufficient condition for universal quantum computation using bosonic circuits”, (2024) arXiv:2309.07820
[11]
O. Hahn et al., “Quantifying Qubit Magic Resource with Gottesman-Kitaev-Preskill Encoding”, Physical Review Letters 128, (2022) arXiv:2109.13018 DOI
[12]
O. Hahn, G. Ferrini, and R. Takagi, “Bridging magic and non-Gaussian resources via Gottesman-Kitaev-Preskill encoding”, (2024) arXiv:2406.06418
[13]
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
[14]
E. Agrell et al., “Closest point search in lattices”, IEEE Transactions on Information Theory 48, 2201 (2002) DOI
[15]
C. Vuillot et al., “Quantum error correction with the toric Gottesman-Kitaev-Preskill code”, Physical Review A 99, (2019) arXiv:1810.00047 DOI
[16]
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
[17]
N. Raveendran et al., “Finite Rate QLDPC-GKP Coding Scheme that Surpasses the CSS Hamming Bound”, Quantum 6, 767 (2022) arXiv:2111.07029 DOI
[18]
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
[19]
L. Babai, “On Lovász’ lattice reduction and the nearest lattice point problem”, Combinatorica 6, 1 (1986) DOI
[20]
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
[21]
A. L. Grimsmo and S. Puri, “Quantum Error Correction with the Gottesman-Kitaev-Preskill Code”, PRX Quantum 2, (2021) arXiv:2106.12989 DOI
[22]
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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Zoo Code ID: multimodegkp

Cite as:
“Gottesman-Kitaev-Preskill (GKP) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/multimodegkp
BibTeX:
@incollection{eczoo_multimodegkp, title={Gottesman-Kitaev-Preskill (GKP) code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/multimodegkp} }
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“Gottesman-Kitaev-Preskill (GKP) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/multimodegkp

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/oscillators/stabilizer/lattice/multimodegkp.yml.