Concatenated GKP code[1]
Description
A concatenated code whose outer code is a GKP code. In other words, a bosonic code that can be thought of as a concatenation of an arbitrary inner code and another bosonic outer code. Most examples encode physical qubits of an inner stabilizer code into the square-lattice GKP code.
Protection
The analog syndrome information of the outer GKP code can improve protection of the inner code. As an example, concatenating a three-qubit quantum repetition code with GKP codes can correct some two-bit-flip errors [1].
Rate
Recursively concatenating the \([[6,2,2]]\) and \([[4,2,2]]\) codes with GKP codes achieves the hashing bound of the displacement channel [2]. Concatenating Abelian LP codes with GKP codes can surpass the CSS Hamming bound [3].
Code Capacity Threshold
\(0.599\) threshold displacement standard deviation for GKP-repetition code [4].\(0.59\) threshold displacement standard deviation for GKP-color code [5].
Notes
Bosonic Pauli+ model is a numerical simulation tool for concatenated GKP codes [6].
Parent
Children
- \(D_4\) hyper-diamond GKP code — The \(D_4\) hyper-diamond GKP code can be seen as a concatenation of a rotated square-lattice GKP code with a repetition code [7]. This is related to the fact that the four-bit repetition code yields the \(D_4\) hyper-diamond lattice code via Construction A.
- GKP-surface code
Cousins
- Cluster-state code — GKP codes have been concatenated with cluster-state codes [2].
- Quantum repetition code — Concatenating a three-qubit quantum repetition code with GKP codes can correct some two-bit-flip errors [1] (see also [4]).
- \([[4,2,2]]\) Four-qubit code — Recursively concatenating the \([[6,2,2]]\) and \([[4,2,2]]\) codes with GKP codes achieves the hashing bound of the displacement channel [2].
- \([[6,2,2]]\) \(C_6\) code — Recursively concatenating the \([[6,2,2]]\) and \([[4,2,2]]\) codes with GKP codes achieves the hashing bound of the displacement channel [2].
- Abelian LP code — GKP codes have been concatenated with Abelian LP codes [3] that are in turn based on QC-LDPC codes [8].
- Quantum parity code (QPC) — GKP codes have been concatenated with QPCs [9].
- Square-octagon (4.8.8) color code — GKP codes have been concatenated with 4.8.8 color codes [5].
- GKP CV-cluster-state code — GKP CV-cluster-state codes reduce to cluster-state codes concatenated with single-mode GKP codes [2] when all physical modes are initialized in GKP states.
- Oscillator-into-oscillator GKP code — Oscillator-into-oscillator GKP codes concantenated with qubit-into-oscillator GKP codes can outperform more conventional concatenations of qubit-into-oscillator GKP codes with qubit stabilizer codes [10].
References
- [1]
- K. Fukui, A. Tomita, and A. Okamoto, “Analog Quantum Error Correction with Encoding a Qubit into an Oscillator”, Physical Review Letters 119, (2017) arXiv:1706.03011 DOI
- [2]
- K. Fukui et al., “High-Threshold Fault-Tolerant Quantum Computation with Analog Quantum Error Correction”, Physical Review X 8, (2018) arXiv:1712.00294 DOI
- [3]
- N. Raveendran et al., “Finite Rate QLDPC-GKP Coding Scheme that Surpasses the CSS Hamming Bound”, Quantum 6, 767 (2022) arXiv:2111.07029 DOI
- [4]
- M. P. Stafford and N. C. Menicucci, “Biased Gottesman-Kitaev-Preskill repetition code”, Physical Review A 108, (2023) arXiv:2212.11397 DOI
- [5]
- J. Zhang et al., “Quantum error correction with the color-Gottesman-Kitaev-Preskill code”, Physical Review A 104, (2021) arXiv:2112.14447 DOI
- [6]
- F. Hopfmueller et al., “Bosonic Pauli+: Efficient Simulation of Concatenated Gottesman-Kitaev-Preskill Codes”, (2024) arXiv:2402.09333
- [7]
- B. Royer, S. Singh, and S. M. Girvin, “Encoding Qubits in Multimode Grid States”, PRX Quantum 3, (2022) arXiv:2201.12337 DOI
- [8]
- M. P. C. Fossorier, “Quasi-Cyclic Low-Density Parity-Check Codes From Circulant Permutation Matrices”, IEEE Transactions on Information Theory 50, 1788 (2004) DOI
- [9]
- K. Fukui, T. Matsuura, and N. C. Menicucci, “Efficient Concatenated Bosonic Code for Additive Gaussian Noise”, Physical Review Letters 131, (2023) arXiv:2102.01374 DOI
- [10]
- Y. Xu et al., “Qubit-Oscillator Concatenated Codes: Decoding Formalism and Code Comparison”, PRX Quantum 4, (2023) arXiv:2209.04573 DOI
Page edit log
- Victor V. Albert (2024-07-17) — most recent
Cite as:
“Concatenated GKP code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/gkp_concatenated