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 \(C_6\) and \([[4,2,2]]\) codes with GKP codes achieves the hashing bound of the displacement channel [1]. Concatenating Abelian LP codes with GKP codes can surpass the CSS Hamming bound [2]. Particular families of GKP codes achieve the capacity of AD and amplification channels for some loss rates [3]. Concatenations of square-lattice GKP codes with Hermitian Galois-qudit codes achieve the capacity for all loss rates [4]. Concatenation of GKP codes with quantum polar codes achieves a rate against the displacement channel [4].Gates
Linear-optical computation [5].Decoding
Circuit-level soft information decoder [6].Code Capacity Threshold
\(0.599\) threshold displacement standard deviation for GKP-repetition code [7].\(0.59\) threshold displacement standard deviation for GKP-color code [8].A concatenated threshold with GKP codes on the lowest level exists for general Markovian noise [9].There is an upper bound on the threshold under local update recovery that is derived via quantum optimal transport [10].Notes
Bosonic Pauli+ model is a numerical simulation tool for concatenated GKP codes [11].Cousins
- Cluster-state code— GKP codes have been concatenated with cluster-state codes [12].
- Quantum repetition code— Concatenating a three-qubit quantum repetition code with GKP codes can correct some two-bit-flip errors [1] (see also [7]).
- \([[4,2,2]]\) Four-qubit code— Recursively concatenating the \(C_6\) and \([[4,2,2]]\) codes with GKP codes achieves the hashing bound of the displacement channel [1].
- \([[6,2,2]]\) \(C_6\) code— Recursively concatenating the \(C_6\) and \([[4,2,2]]\) codes with GKP codes achieves the hashing bound of the displacement channel [1].
- Abelian LP code— GKP codes have been concatenated with Abelian LP codes [2] that are in turn based on QC-LDPC codes [13]. Concatenating Abelian LP codes with GKP codes can surpass the CSS Hamming bound [2].
- Quantum parity code (QPC)— GKP codes have been concatenated with QPCs [14].
- Square-octagon (4.8.8) color code— GKP codes have been concatenated with 4.8.8 color codes [8].
- Honeycomb (6.6.6) color code— GKP codes have been concatenated with the 6.6.6 color code [15].
- \([[5,1,3]]\) Five-qubit perfect code— GKP codes have been concatenated with the five-qubit code [15].
- Quantum polar code— Concatenation of GKP codes with quantum polar codes achieves a rate against the displacement channel [4].
- Hermitian Galois-qudit code— Concatenations of square-lattice GKP codes with Hermitian Galois-qudit codes achieve the capacity for all loss rates [4].
- Hyperbolic tessellation code— The qubit-Pauli tessellation GKP code [16] is the Euclidean \(\{2,4,4\}\) member of the curvature-dependent tessellation-code framework. It is a two-mode code in which each Cartesian direction is a single-mode qubit GKP code, making the full code a 2-to-1 concatenated qubit encoding. The logical single-qubit Pauli group is implemented geometrically by one \(\pi\) rotation and two \(\pi/2\) rotations on the Euclidean tessellation [16].
- GKP CV-cluster-state code— GKP CV-cluster-state codes reduce to cluster-state codes concatenated with single-mode GKP codes [12] when all physical modes are initialized in GKP states. Analog QEC with GKP codes concatenated with surface-code and 3D-cluster-state fault-tolerant schemes can reach a threshold displacement standard deviation of about \(0.607\) for ideal syndrome measurements and reduce the squeezing required for topologically protected MBQC to \(9.8\) dB [12].
- Oscillator-into-oscillator GKP code— Oscillator-into-oscillator GKP codes concatenated with qubit-into-oscillator GKP codes can outperform more conventional concatenations of qubit-into-oscillator GKP codes with qubit stabilizer codes [17].
Primary Hierarchy
Parents
Concatenated GKP code
Children
The \(D_4\) hyper-diamond GKP code can be seen as a concatenation of a rotated square-lattice GKP code with a repetition code [18]. This is related to the fact that the four-bit repetition code yields the \(D_4\) hyper-diamond lattice via Construction A.
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]
- N. Raveendran, N. Rengaswamy, F. Rozpędek, A. Raina, L. Jiang, and B. Vasić, “Finite Rate QLDPC-GKP Coding Scheme that Surpasses the CSS Hamming Bound”, Quantum 6, 767 (2022) arXiv:2111.07029 DOI
- [3]
- G. Zheng, W. He, G. Lee, K. Noh, and L. Jiang, “Performance and achievable rates of the Gottesman-Kitaev-Preskill code for pure-loss and amplification channels”, (2024) arXiv:2412.06715
- [4]
- M. Subramanian, G. Zheng, and L. Jiang, “Achievable Rates for Concatenated Square Gottesman-Kitaev-Preskill Codes”, PRX Quantum 6, (2025) arXiv:2505.10499 DOI
- [5]
- B. W. Walshe et al., “Linear-Optical Quantum Computation with Arbitrary Error-Correcting Codes”, Physical Review Letters 134, (2025) arXiv:2408.04126 DOI
- [6]
- S. K. Borah, A. K. Pradhan, N. Raveendran, M. Pacenti, and B. Vasic, “Fault Tolerant Decoding of QLDPC-GKP Codes with Circuit Level Soft Information”, (2025) arXiv:2505.06385
- [7]
- M. P. Stafford and N. C. Menicucci, “Biased Gottesman-Kitaev-Preskill repetition code”, Physical Review A 108, (2023) arXiv:2212.11397 DOI
- [8]
- J. Zhang, J. Zhao, Y.-C. Wu, and G.-P. Guo, “Quantum error correction with the color-Gottesman-Kitaev-Preskill code”, Physical Review A 104, (2021) arXiv:2112.14447 DOI
- [9]
- T. Matsuura, N. C. Menicucci, and H. Yamasaki, “Continuous-Variable Fault-Tolerant Quantum Computation under General Noise”, (2024) arXiv:2410.12365
- [10]
- R. König and C. Rouzé, “Limitations of local update recovery in stabilizer-GKP codes: a quantum optimal transport approach”, (2023) arXiv:2309.16241
- [11]
- F. Hopfmueller, M. Tremblay, P. St-Jean, B. Royer, and M.-A. Lemonde, “Bosonic Pauli+: Efficient Simulation of Concatenated Gottesman-Kitaev-Preskill Codes”, Quantum 8, 1539 (2024) arXiv:2402.09333 DOI
- [12]
- K. Fukui, A. Tomita, A. Okamoto, and K. Fujii, “High-Threshold Fault-Tolerant Quantum Computation with Analog Quantum Error Correction”, Physical Review X 8, (2018) arXiv:1712.00294 DOI
- [13]
- M. P. C. Fossorier, “Quasi-Cyclic Low-Density Parity-Check Codes From Circulant Permutation Matrices”, IEEE Transactions on Information Theory 50, 1788 (2004) DOI
- [14]
- 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
- [15]
- M. Lin and K. Noh, “Exploring the quantum capacity of a Gaussian random-displacement channel using Gottesman-Kitaev-Preskill codes and maximum-likelihood decoding”, Physical Review A 111, (2025) arXiv:2411.04277 DOI
- [16]
- Y. Wang, Y. Xu, and Z.-W. Liu, “Tessellation Codes: Encoded Quantum Gates by Geometric Rotation”, Physical Review Letters 135, (2025) arXiv:2410.18713 DOI
- [17]
- Y. Xu, Y. Wang, E.-J. Kuo, and V. V. Albert, “Qubit-Oscillator Concatenated Codes: Decoding Formalism and Code Comparison”, PRX Quantum 4, (2023) arXiv:2209.04573 DOI
- [18]
- B. Royer, S. Singh, and S. M. Girvin, “Encoding Qubits in Multimode Grid States”, PRX Quantum 3, (2022) arXiv:2201.12337 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