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 [2,4–6]. 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
By applying GKP error correction to Gaussian input states, universality can be achieved without non-Gaussian elements [10].
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 [3,11–13].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].
Parent
- Quantum lattice code — GKP codes are \(n\)-mode quantum lattice codes with \(2n\) stabilizers, i.e., constructed using a non-degenerate lattice.
Children
- \(D_4\) hyper-diamond GKP code
- GKP cluster-state code — The GKP cluster-state code is a concatenation of a cluster-state stabilizer code with a single-mode GKP code. A GKP-based cluster state is a multimode GKP codeword, although other codewords are not utilized in CV MBQC.
- Square-lattice GKP code
- Hexagonal GKP code
- NTRU-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 [2,4–6].
- Quadrature-amplitude modulation (QAM) code — Finite-energy GKP codes are quantum analogues of lattice-based QAM codes in that both use a subset of points on a lattice.
- Numerically optimized code — Numerically optimizing GKP code lattices yields codes for three and nine modes with larger distances and fidelities than known GKP codes [14].
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]
- 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
- [11]
- C. Vuillot et al., “Quantum error correction with the toric Gottesman-Kitaev-Preskill code”, Physical Review A 99, (2019) arXiv:1810.00047 DOI
- [12]
- 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
- [13]
- N. Raveendran et al., “Finite Rate QLDPC-GKP Coding Scheme that Surpasses the CSS Hamming Bound”, Quantum 6, 767 (2022) arXiv:2111.07029 DOI
- [14]
- M. Lin, C. Chamberland, and K. Noh, “Closest lattice point decoding for multimode Gottesman-Kitaev-Preskill codes”, (2023) arXiv:2303.04702
Page edit log
- Jonathan Conrad (2022-07-05) — most recent
- Victor V. Albert (2022-07-05)
- Victor V. Albert (2022-03-24)
Cite as:
“Gottesman-Kitaev-Preskill (GKP) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/multimodegkp