Multi-mode GKP code[1][2]

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

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][4][5][6].

Encoding

Multi-mode 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 a damping operator, i.e., a damped exponential of the total occupation number [7][8].

Gates

By applying GKP error correction to Gaussian input states, universality can be achieved without non-Gaussian elements [9].

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) or by using other effective iterative schemes when e.g. the lattice represents a concatenated GKP code [10][11][3][12].

Fault Tolerance

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

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Zoo code information

Internal code ID: multimodegkp

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Zoo Code ID: multimodegkp

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

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

Github: https://github.com/errorcorrectionzoo/eczoo_data/tree/main/codes/quantum/oscillators/multimodegkp.yml.